Search: a068435 -id:a068435
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A116086
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Perfect powers n with no primes between n and the next larger perfect power, which is in A116455.
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+10
4
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OFFSET
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1,1
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COMMENTS
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No other n<10^12. There is a conjecture that this sequence is finite.
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LINKS
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EXAMPLE
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The prime-free ranges are (2^3,3^2), (5^2,3^3), (2^5,6^2), (11^2,5^3), (3^7,13^3), (5^5,56^2), (181^2,2^15), (43^3,282^2), (46^3,312^2), (22434^2,55^5).
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MATHEMATICA
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lim=10^12; lst={}; k=2; While[n=Floor[lim^(1/k)]; n>=2, lst=Join[lst, Range[2, n]^k]; k++ ]; lst=Union[lst]; PrimeFree[n1_, n2_] := Module[{n=n1+1}, While[n<n2&&!PrimeQ[n], n++ ]; n ==n2]; lst2={}; Do[If[PrimeFree[lst[[i]], lst[[i+1]]], AppendTo[lst2, lst[[i]]]], {i, Length[lst]-1}]; lst2
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CROSSREFS
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KEYWORD
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hard,nonn
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AUTHOR
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STATUS
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approved
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2, 1, 1, 2, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2
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OFFSET
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1,1
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COMMENTS
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First 0 terms appear at n = 6, 14, 41, 359, 3589, corresponding to consecutive prime powers (8,9), (25,27), (121,125), (2187,2197) and (32761,32768), respectively (cf. A068315 and A068435).
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LINKS
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Michael De Vlieger, 1038 X 1038 raster of a(n), n = 1..1077444, read left to right in rows, then top to bottom, showing a(n) = 0 in white, a(n) = 1 in red, and a(n) = 2 in dark blue.
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EXAMPLE
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a(1) = 2 because in the first prime power pair (2 and 3) there are two primes.
a(14) = 0 because in the 14th prime power pair (25 and 27) there are no primes.
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MATHEMATICA
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With[{upto=500}, Map[Count[#, _?PrimeQ]&, Partition[Select[Range[upto], PrimePowerQ], 2, 1]]] (* Considers prime powers up to 500 *)
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PROG
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(PARI) lista(nn) = my(v=[p| p <- [1..nn], isprimepower(p)]); vector(#v-1, k, isprime(v[k]) + isprime(v[k+1])); \\ Michel Marcus, Oct 26 2023
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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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A116455
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Perfect powers n with no primes between n and the next smaller perfect power, which is in A116086.
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+10
3
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OFFSET
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1,1
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COMMENTS
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LINKS
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CROSSREFS
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KEYWORD
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hard,nonn
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AUTHOR
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STATUS
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approved
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A366833
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Number of times n appears in A362965 (number of primes <= the n-th prime power).
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+10
3
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1, 2, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
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OFFSET
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1,2
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COMMENTS
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a(n) can be only 1, 2, or 3, with the first occurrences of 3 appearing at n = 4, 9, 30, 327 and 3512.
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LINKS
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Paolo Xausa, 1200 X 1200 raster image of a(n), n = 1..1440000, read left to right, top to bottom, showing a(n) = 1 in blue, a(n) = 2 in white and a(n) = 3 in red.
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MATHEMATICA
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With[{upto=1000}, Map[Length, Most[Split[PrimePi[Select[Range[upto], PrimePowerQ]]]]]] (* Considers prime powers up to 1000 *)
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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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A060846
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Smallest prime > a nontrivial power of a prime.
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+10
2
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5, 11, 11, 17, 29, 29, 37, 53, 67, 83, 127, 127, 131, 173, 251, 257, 293, 347, 367, 521, 541, 631, 733, 853, 967, 1031, 1361, 1373, 1693, 1861, 2053, 2203, 2203, 2213, 2411, 2819, 3137, 3491, 3727, 4099, 4493, 4919, 5051, 5333, 6247, 6563, 6863, 6899, 7927
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graph;
refs;
listen;
history;
text;
internal format)
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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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78125=5^7 is followed by 78137.
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PROG
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(PARI) ispp(x) = !isprime(x) && isprimepower(x);
lista(nn) = apply(x->nextprime(x), select(x->ispp(x), [1..nn])); \\ Michel Marcus, Aug 24 2019
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CROSSREFS
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Cf. A025475, A000961, A001597, A001694, A007917, A007918, A013632, A013633, A049711, A060845, A068435, A246547.
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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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OFFSET
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1,1
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COMMENTS
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Equivalently, prime powers (either A000961 or A246655) q such that q and the next prime power are both composite numbers. - Paolo Xausa, Oct 25 2023
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LINKS
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MATHEMATICA
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With[{upto=33000}, Map[First, Select[Partition[Select[Range[upto], PrimePowerQ], 2, 1], NoneTrue[#, PrimeQ]&]]] (* Paolo Xausa, Oct 25 2023 *)
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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