Search: a046118 -id:a046118
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A046119
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Middle member of a sexy prime triple: value of p+6 such that p, p+6 and p+12 are all prime, but p+18 is not (although p-6 might be).
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+10
6
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13, 23, 37, 53, 73, 103, 107, 157, 173, 233, 263, 277, 353, 373, 563, 593, 613, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1283, 1297, 1367, 1433, 1453, 1493, 1613, 1663, 1753, 1783, 1873, 1907, 1993, 2137, 2287, 2383, 2417, 2683, 2693, 2713
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OFFSET
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1,1
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COMMENTS
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p-6 will be prime if the prime triple contains the last 3 primes of a sexy prime quadruple.
If a sexy prime triple happens to include the last 3 members of a sexy prime quadruple, this sequence will contain the sexy prime triple's middle member; e.g., a(4)=53 is the middle member of the sexy prime triple (47, 53, 59), but is also the third member of the sexy prime quadruple (41, 47, 53, 59). - Daniel Forgues, Aug 05 2009
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LINKS
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Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- N. J. A. Sloane, Mar 07 2021].
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FORMULA
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MATHEMATICA
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Select[Prime[Range[400]], And@@PrimeQ[{#-6, #+6}]&&!PrimeQ[#+12]&] (* Harvey P. Dale, Nov 01 2011 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A046120
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Largest member of a sexy prime triple; value of p+12 where p, p+6 and p+12 are all prime, but p+18 is not.
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+10
6
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19, 29, 43, 59, 79, 109, 113, 163, 179, 239, 269, 283, 359, 379, 569, 599, 619, 659, 739, 953, 983, 1109, 1129, 1193, 1229, 1289, 1303, 1373, 1439, 1459, 1499, 1619, 1669, 1759, 1789, 1879, 1913, 1999, 2143, 2293, 2389, 2423, 2689, 2699, 2719
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OFFSET
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1,1
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COMMENTS
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If a sexy prime triple happens to include the last 3 members of a sexy prime quadruple, this sequence will contain the sexy prime triple's largest member; e.g., a(4)=59 is the largest member of the sexy prime triple (47, 53, 59), but is the fourth member of the sexy prime quadruple (41, 47, 53, 59). - Daniel Forgues, Aug 05 2009
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LINKS
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Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- N. J. A. Sloane, Mar 07 2021].
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FORMULA
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MATHEMATICA
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#+12&/@Select[Prime[Range[400]], PrimeQ[#+{6, 12, 18}]=={True, True, False}&] (* Harvey P. Dale, Dec 08 2012 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A275681
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Table read by rows: list of sexy prime triples (p, p+6, p+12) such that p+18 is composite.
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+10
2
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7, 13, 19, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 73, 79, 97, 103, 109, 101, 107, 113, 151, 157, 163, 167, 173, 179, 227, 233, 239, 257, 263, 269, 271, 277, 283, 347, 353, 359, 367, 373, 379, 557, 563, 569, 587, 593, 599, 607, 613, 619, 647, 653, 659, 727, 733, 739
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OFFSET
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1,1
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LINKS
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FORMULA
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EXAMPLE
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The table starts:
7, 13, 19;
17, 23, 29;
31, 37, 43;
...
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MAPLE
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N:= 10^4: # to get all entries <= N
Primes:= select(isprime, {seq(i, i=1..N+18, 2)}):
S:= select(`<=`, Primes, N) intersect map(t -> t-6, Primes) intersect map(t -> t-12, Primes) minus map(t -> t-18, Primes):
map(t ->(t, t+6, t+12), sort(convert(S, list))); # Robert Israel, Aug 05 2016
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MATHEMATICA
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Most[#]&/@Select[Table[n+{0, 6, 12, 18}, {n, Prime[Range[200]]}], PrimeQ[#] == {True, True, True, False}&]//Flatten (* Harvey P. Dale, Jan 19 2017 *)
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PROG
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(Magma) lst:=[]; for p in PrimesUpTo(727) do b:=p+6; if IsPrime(b) then c:=b+6; if IsPrime(c) and not IsPrime(c+6) then lst:=lst cat [p, b, c]; end if; end if; end for; lst;
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A163858
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Number of sexy prime triples (p, p+6, p+12) where p+18 is not prime (although p-6 might be), with p <= n.
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+10
1
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0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7
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OFFSET
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1,17
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COMMENTS
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p-6 will be prime if the prime triple contains the last 3 primes of a sexy prime quadruple.
There are two sexy prime triples classes, (-1, -1, -1) (mod 6) and (+1, +1, +1) (mod 6). They should asymptotically have the same number of triples, if there is an infinity of such triples, although with a Chebyshev bias expected against the quadratic residue class triples (+1, +1, +1) (mod 6), which doesn't affect the asymptotic result. This sequence counts both classes.
Also the sexy prime triples of class (-1, -1, -1) (mod 6) fall within (11, 17, 23, 29) (mod 30) while the sexy prime triples of class (+1, +1, +1) (mod 6) fall within (1, 7, 13, 19) (mod 30).
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LINKS
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CROSSREFS
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A046118 Smallest member of a sexy prime triple: value of p where (p, p+6, p+12) are all prime but p+18 is not (although p-6 might be.)
A046119 Middle member of a sexy prime triple: value of p+6 where (p, p+6, p+12) are all prime but p+18 is not (although p-6 might be.)
A046120 Largest member of a sexy prime triple, value of p+12 where (p, p+6, p+12) are all prime but p+18 is not (although p-6 might be.)
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A372042
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Monogamously Faithful Primes (primes that are sexy primes with only one other prime in their pair).
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+10
1
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83, 89, 131, 137, 191, 193, 197, 199, 223, 229, 307, 311, 313, 317, 331, 337, 383, 389, 433, 439, 443, 449, 457, 461, 463, 467, 503, 509, 541, 547, 571, 577, 677, 683, 751, 757, 821, 823, 827, 829, 853, 857, 859, 863, 877, 881, 883, 887, 991, 997, 1013, 1019, 1033, 1039, 1063, 1069, 1087
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OFFSET
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0,1
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COMMENTS
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These are all the numbers found in A136207 but not found in A046118, A046119, A046120, A023271, A046122, A046123, or A046124, i.e., members of a sexy prime pair but not members of sexy prime triplets, quadruplets, ...
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LINKS
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EXAMPLE
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83 and 89 are "sexy" with each other, because they differ by 6. They are monogamously faithful, because neither is sexy with any other number.
71 is not "sexy" because it is not in A136207.
67 is "sexy" with both 61 and 73. Therefore, it is not monogamously faithful, since it has multiple numbers that it is sexy with.
43 is "sexy" only with 37. But it is not monogamously faithful, even though it isn't sexy with another number, because 37 is also "sexy" with 31, therefore "cheating" on 43 with 31.
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MAPLE
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isA372042 := proc(n)
if isprime(n) then
if isprime(n+6) then
if not isprime(n-6) and not isprime(n+12) then
true;
else
false;
end if;
elif isprime(n-6) then
if not isprime(n+6) and not isprime(n-12) then
true;
else
false;
end if;
else
false ;
end if;
else
false ;
end if;
end proc:
option remember;
local a;
if n = 1 then
83 ;
else
a := nextprime(procname(n-1)) ;
while true do
if isA372042(a) then
return a;
else
a := nextprime(a) ;
end if;
end do:
end if;
end proc:
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A275686
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Difference between the smallest 10^n-digit member of a sexy prime triple and 10^(10^n - 1).
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+10
0
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OFFSET
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0,1
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LINKS
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CROSSREFS
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KEYWORD
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nonn,base,hard,more
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AUTHOR
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STATUS
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approved
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A286217
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Product of the n-th sexy prime triple.
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+10
0
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1729, 11339, 49321, 146969, 386389, 1089019, 1221191, 3864241, 5171489, 12640949, 18181979, 21243961, 43974269, 51881689, 178433279, 208506509, 230324329, 278421569, 393806449, 849244031, 932539661, 1341880019, 1416207439, 1672403471, 1829232539, 2111885999
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graph;
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listen;
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OFFSET
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1,1
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LINKS
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FORMULA
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a(n) = s(n)*(s(n)+6)*(s(n)+12), where s = A046118.
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EXAMPLE
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The first sexy prime triple is (7, 13, 19) so a(1) = 7*13*19 = 1729.
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MATHEMATICA
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Select[Prime@ Range@ 500, PrimeQ[# + {6, 12, 18}] == {True, True, False} &] // # (#+6) (#+12) & (* Giovanni Resta, May 05 2017 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A297847
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Sexiness of p = prime(n): number of iterations of the function f(x) = x + 6 that leave p prime.
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+10
0
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0, 0, 4, 2, 3, 1, 2, 0, 1, 0, 2, 1, 3, 0, 2, 1, 0, 3, 2, 0, 1, 0, 1, 0, 2, 2, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 0, 1, 0, 0, 3, 2, 1, 0, 2, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0
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OFFSET
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1,3
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COMMENTS
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a(n) > 0 iff p is a term of A023201.
a(n) = 0 iff p is a term of A140555.
a(n) = 2 iff p is a term of A046118.
a(n) > 2 iff p is a term of A023271.
a(n) < 4 except for n = 3. Proof: The last digits of the numbers in the progression repeat 1, 7, 3, 9, 5, 1, 7, 3, 9, 5, ..., so a(n) is at most 4, which only happens for p = 5, since A007652(n) = 5 only for n = 3.
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LINKS
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EXAMPLE
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For n = 13: prime(13) = 41 and 41 remains prime through exactly 3 iterations of f(x) = x + 6, since 47, 53 and 59 are prime, but 65 is composite, so a(13) = 3.
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MATHEMATICA
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Array[-2 + Length@ NestWhileList[# + 6 &, Prime@ #, PrimeQ] &, 105] (* Michael De Vlieger, Jan 11 2018 *)
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PROG
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(PARI) a(n) = my(p=prime(n), x=p, i=0); while(1, x=x+6; if(!ispseudoprime(x), return(i), i++))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A358572
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Smallest prime p in a sexy prime triple such that (p-3)/2 is also the smallest prime in a sexy prime triple (A023241).
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+10
0
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17, 97, 1117, 1217, 2897, 130337, 188857, 207997, 221197, 324517, 610817, 900577, 1090877, 1452317, 1719857, 1785097, 2902477, 3069917, 3246317, 4095097, 4536517, 4977097, 5153537, 5517637, 5745557, 6399677, 7168277, 7351957, 7588697, 7661077, 8651537, 8828497, 9153337
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OFFSET
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1,1
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COMMENTS
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Also numbers m such that m-4, m-1, m+5, m+8, m+11 and m+20 cannot be represented as x*y + x + y, with x >= y > 1 (A254636).
Number of terms < 10^k: 0, 2, 2, 5, 5, 12, 34, 150, 655, ...
All terms p and (p-3)/2 have a final decimal digit of 7. This follows from considering possibilities modulo 10 and implies p + 18 and (p-3)/2 + 18 are divisible by 5 and hence composite. Both p and (p-3)/2 are therefore also terms of A046118. - Andrew Howroyd, Dec 31 2022
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LINKS
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EXAMPLE
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97 is the smallest prime in the sexy prime triple (97, 103, 109), and the triple (47 = (97 - 3)/2, 47 + 6, 47 + 12) forms another sexy prime triple. Hence 97 is in the sequence.
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MATHEMATICA
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Select[Prime[Range[700000]], AllTrue[Join[# + {6, 12}, (#-3)/2 + {0, 6, 12}], PrimeQ] &] (* Amiram Eldar, Nov 23 2022 *)
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PROG
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(PARI)
istriple(p)={isprime(p) && isprime(p+6) && isprime(p+12)}
isok(p)={istriple(p) && istriple((p-3)/2)}
{ forprime(p=1, 10^7, if(isok(p), print1(p, ", "))) } \\ Andrew Howroyd, Dec 30 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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