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Euclid-Mullin sequence ( A000945) with initial value a(1)=5 instead of a(1)=2.
+20
39
5, 2, 11, 3, 331, 19, 199, 53, 21888927391, 29833, 101, 71, 23, 311, 7, 72353, 13, 227, 96014559769, 5641, 41, 82107739003, 67, 169637539, 61, 29, 31319, 17, 97, 238591921, 313, 102065429, 157, 37, 595553520313, 244217, 241, 4773229353714971081083834237, 103
COMMENTS
The initial primes 3 and 7 give essentially A000945.
EXAMPLE
5*2*11*3 + 1 = 331, which is prime; the least prime factor of 330*331 + 1 = 109231 = 19*5749 is 19, so a(6) = 19.
MATHEMATICA
a[1]=5; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]
PROG
(PARI) spf(n)=my(f=factor(n)[1, 1]); f;
first(m)=my(v=vector(m)); v[1]=5; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v; \\ Anders Hellström, Aug 15 2015
Euclid-Mullin sequence ( A000945) with initial value a(1)=127 instead of a(1)=2.
+20
33
127, 2, 3, 7, 5, 149, 19, 41, 23899, 139, 43, 761, 281, 17, 53, 2551, 23, 20149, 100720363856036298033578901613089271, 31, 179, 11, 13, 523, 282995646721, 2871347, 83, 10744429, 1031, 427773048135533, 97, 78506876242349, 67
MATHEMATICA
a[1]=127; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]
PROG
(PARI) spf(n)=my(f=factor(n)[1, 1]); f;
first(m)={my(v=vector(m)); v[1]=127; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v; } /* Anders Hellström, Aug 18 2015 */
Euclid-Mullin sequence ( A000945) with initial value a(1)=8191 instead of a(1)=2.
+20
12
8191, 2, 3, 7, 53, 1399, 5, 19, 646843, 26945441, 109, 443, 90670999, 280460690293140589, 907, 16293787, 3655513, 499483, 131, 21067, 143797, 54540542259000816707816058313971443, 392963, 977, 11, 5021, 179, 439, 353, 34417238589462247, 1193114397863177, 13, 59, 31643, 79399, 73, 43, 16639867
MATHEMATICA
a[1]=8191; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]
PROG
(PARI) spf(n)=my(f=factor(n)[1, 1]); f;
first(m)={my(v=vector(m)); v[1]=8191; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v; } /* Anders Hellström, Aug 18 2015 */
4th term in Euclid-Mullin prime sequence started with n-th prime (cf. A000945).
+20
10
43, 43, 3, 43, 3, 79, 3, 5, 3, 3, 11, 223, 3, 7, 3, 3, 827, 367, 13, 3, 439, 5, 3, 3, 11, 5, 619, 3, 5, 3, 7, 3, 3, 5, 5, 907, 23, 11, 3, 3, 3, 1087, 3, 19, 3, 5, 7, 13, 3, 5, 3, 3, 1447, 3, 3, 3, 3767, 1627, 1663, 3, 1699, 3, 19, 5, 1879, 3, 1987, 7, 3, 5, 4943, 3, 2203, 2239, 5, 23
COMMENTS
First term in Euclid-Mullin sequence is p (say), 2nd term (if p odd) is 2, 3rd term is A023592.
EXAMPLE
E.g., (5,2,11,3), (89,2,179,3), (17,2,5,3), (2,3,7,43), (61,2,3,367).
MATHEMATICA
a[n_] := (Clear[f]; f[1] = Prime[n]; f[k_] := f[k] = FactorInteger[Product[f[i], {i, 1, k-1}]+1][[1, 1]]; f[4]); Table[a[n], {n, 1, 76}] (* Jean-François Alcover, Feb 05 2014 *)
Where n-th prime appears in Euclid-Mullin sequence A000945.
+20
9
1, 2, 7, 3, 12, 5, 13, 36, 25, 33, 50, 18
MATHEMATICA
f[1]=2; f[n_] := f[n] = FactorInteger[Product[f[i], {i, 1, n - 1}] + 1][[1, 1]] ems = Table[f[n], {n, 1, 43}]; Do[Print[Position[ems, Prime[n]][[1, 1]]], {n, 1, 25}]
EXTENSIONS
Gaps that need filling: 1, 2, 7, 3, 12, 5, 13, 36, 25, 33, 50, 18, ?, 4, ?, 6, ?, 42, ?, 22, ?, ?, ?, 35, 26
a(n) is the smallest prime which, if used to start a Euclid-Mullin sequence (like A000945), the resulting sequence contains the n consecutive primes 2, 3, ..., prime(n).
+20
8
2, 2, 19, 199, 2089, 99109, 1960969, 10129129, 87726649, 4549584049, 328034245549, 20584643748679, 666188861477149, 31395465477725359, 894857713367947339, 434392154438254391389, 17934770256689308411399
COMMENTS
Thanks in part to Dirichlet's theorem, a(n) exists for each n. - Don Reble, Oct 07 2006
EXAMPLE
a(1) = a(2) = 2 because they generate {2,3,7,43,13,...};
a(3) = 19 because it generates {19,2,3,5,571,271,...}, see A051312; a(4) = 199 because it generates {199,2,3,5,7,23,881,...};
a(5) = 2089 because it generates {2089,2,3,5,7,11,269,...};
a(6) = 99109 because it generates {99109,2,3,5,7,11,13,2976243271,...};
a(7) = 1960969 because it generates {1960969,2,3,5,7,11,13,17,281,47,419,5539788476533581271,37,19,173,...}
Primes p used as initial values for Euclid-Mullin sequences (variant A000945) instead of 2, such that all provide {p,2,3,5,7,11,13,q,...} initial segments in which the first six primes occur from 2nd to 7th terms.
+20
6
99109, 159169, 189199, 399409, 459469, 609619, 669679, 699709, 819829, 1030039, 1090099, 1150159, 1270279, 1300309, 1390399, 1420429, 1810819, 1870879, 1930939, 1960969, 2021029, 2051059, 2141149, 2201209, 2261269, 2321329
EXAMPLE
Initial segments of Euclid-Mullin sequences provided by
a[33]=3132139, a[34] and a[35] initial values:
{3132139,2,3,5,7,11,13,94058134171}}
{3282289,2,3,5,7,11,13,59}},
{3372379,2,3,5,7,11,13,29}}
MATHEMATICA
b[x_] :=First[Flatten[FactorInteger[Apply[Times, Table[b[j], {j, 1, x - 1}]] +1]]]; b[1] = 1; Do[b[1] = Prime[j], el=8; If[Equal[Table[b[w], {w, 2, 7}], {2, 3, 5, 7, 11, 13}], Print[{j, Table[b[w], {w, 1, el}]}]], {j, 100000, 1000000}]
Euclid-Mullin sequence ( A000945) with initial value a(1)=11 instead of a(1)=2.
+20
5
11, 2, 23, 3, 7, 10627, 433, 17, 13, 10805892983887, 73, 6397, 19, 489407, 2753, 87491, 18618443, 5, 31, 113, 41, 10723, 35101153, 25243, 374399, 966011, 293821591198219762366057, 234947, 4729, 27953, 3256171, 331, 613, 67, 272646324430637, 34281113, 21050393332691947013, 61, 97
MATHEMATICA
a[1]=11; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]
PROG
(PARI) lpf(n)=factor(n)[1, 1]
first(m)=my(v=vector(m)); v[1]=11; for(i=2, m, v[i]=lpf(1+prod(j=1, i-1, v[j]))); v;
a(n) is the position of prime 7 in the Euclid-Mullin (EM) sequence of type A000945, if it were started with prime(n) instead of 2.
+20
5
3, 3, 15, 1, 5, 6, 5, 24, 10, 6, 7, 6, 5, 4, 7, 5, 3, 5, 6, 16, 5, 6, 5, 28, 6, 3, 5, 36, 7, 15, 4, 15, 7, 7, 8, 7, 7, 5, 7, 14, 5, 6, 19, 16, 17, 5, 4, 12, 5, 8, 10, 17, 5, 5, 8, 10, 3, 5, 7, 30, 5, 5, 20, 3, 5, 6, 6, 4, 9, 9, 3, 9, 5, 6, 8, 8
EXAMPLE
n=8: p(8)=19, the corresponding EM sequence is A051312 in which p=7 arises at the 24th position as follows: {19, 2, 3, 5, 571, 271, 457, 397, 1123, 23, 103, 42572757267735264511, 313, 17, 16013177, 7951, 1259, 41, 1531, 11, 83, 53, 67, 7, 21397}, thus a(8)=24.
A variant of Euclid-Mullin ( A000945): a(1)=2, a(n+1) is the least prime dividing [Product_{i in I} a(i) + Product_{i not in I} a(i)], minimized over all subsets I of {1..n}.
+20
4
2, 3, 5, 11, 37, 13, 7, 29, 17, 19, 43, 23, 47, 41, 53, 31, 61, 59, 67, 79, 83, 73, 97, 71, 101, 89, 103, 127, 107, 113, 137, 131, 139, 109, 149, 151, 163, 157, 167, 173, 193, 211, 179, 191, 181, 223, 199, 197, 233, 227, 229, 239, 241, 251, 257, 307, 281, 269, 271, 293
COMMENTS
By Euclid's argument, the a(i) are distinct.
One can ask whether all primes occur in this sequence.
FORMULA
For any n, we have Legendre symbol (-a(1)*a(2)*...*a(n-1) / a(n)) = 1. If p is the smallest prime such that (-a(1)*a(2)*...*a(n-1) / p) = 1, then a(n) >= p. Conjecture: For all n, a(n) = p. Note that if b is such that b^2 == -a(1)*a(2)*...*a(n-1) (mod p) and for some I, b == prod_{i in I} a(i) (mod p), then a(n) = p. Heuristically, I must exist for large enough n, since the number of possible subsets I is much larger than p. - Max Alekseyev, Nov 11 2009, May 20 2015
EXAMPLE
a(4)=11 which is the smallest prime dividing the 4 partitions 2+3*5=17, 3+2*5=13, 5+2*3=11, 1+2*3*5=31.
MAPLE
with(numtheory):p:=proc(N) local S, d : S:=NULL:for d in divisors(N) while d^2<=N do S:=S, divisors(d+N/d)[2] od : return(min(S)) end:
a :=n->if n = 1 then 2 else p(mul(a(i), i = 1 .. n-1)) fi :
seq(a(n), n=1..15);
PROG
(PARI) { A167604_list() = my(a, A, p, b, q, z, m); a = []; A=1; while(1, p=2; while( kronecker(-A, p)!=1, p=nextprime(p+1) ); b=lift(sqrt(-A+O(p))); z=znprimroot(p); m=nextprime(random(10^6)); q=lift(prod(i=1, #a, Mod(1+x^znlog(Mod(a[i], p), z, p-1), (1-x^(p-1))*Mod(1, m)) )); if( polcoeff(q, znlog(Mod(b, p), z, p-1), x)==0, error("conjecture failed mod", m)); a=concat(a, [p]); A*=p; print1(p, ", ") ) } /* Max Alekseyev, May 20 2015 */
CROSSREFS
A167605 lists such n that the first n terms of a(n) is a permutation of the first n primes.
A000945 is the original Euclid-Mullin sequence (where I is restricted to the empty set).
AUTHOR
Kok Seng Chua (chuakokseng(AT)hotmail.com), Nov 07 2009
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