A345531 Smallest prime power greater than the n-th prime.

3, 4, 7, 8, 13, 16, 19, 23, 25, 31, 32, 41, 43, 47, 49, 59, 61, 64, 71, 73, 79, 81, 89, 97, 101, 103, 107, 109, 113, 121, 128, 137, 139, 149, 151, 157, 163, 167, 169, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 256, 263, 269, 271, 277

Offset

1,1

Comments

Take the family of correlated prime-indexed conjectures appearing in A343249 - A343253, in which an alternative formula for the p-adic order of positive integers is proposed. There, the general p-indexed conjecture says that v_p(n), the p-adic order of n, is given by the formula: v_p(n) = log_p(n / L_p(k0, n)), where L_p(k0, n) is the lowest common denominator of the elements of the set S_p(k0, n) = {(1/n)*binomial(n, k), with 0 < k <= k0 such that k is not divisible by p}. Evidence suggests that the primality of p is a necessary condition in this general conjecture. So, if a composite number q is used instead of a prime p in the proposed formula for the p-adic (now, q-adic) order of n, the first counterexample (failure) is expected to occur for n = q * a(i), where i is the index of the smallest prime that divides q.

The prime-power a(n) is at most the next prime, so this sequence is strictly increasing. See also A366833. - Gus Wiseman, Nov 06 2024

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.

Formula

a(n) = A000015(1+A000040(n)). - Antti Karttunen, Jul 19 2021

a(n) = A000015(A008864(n)). - Omar E. Pol, Oct 27 2021

Example

a(4) = 8 because the fourth prime number is 7, and the least power of a prime which is greater than 7 is 2^3 = 8.

Maple

f:= proc(n) local p, x;
  p:= ithprime(n);
  for x from p+1 do
    if nops(numtheory:-factorset(x)) = 1 then return x fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Aug 25 2024

Mathematica

a[i_]:= Module[{j, k, N = 0, tab={}}, tab = Sort[Drop[DeleteDuplicates[Flatten[Table[ If[Prime[j]^k > Prime[i], Prime[j]^k], {j, 1, i+1}, {k, 1, Floor[Log[Prime[j], Prime[i+1]]]}]]], 1]]; N = Take[tab, 1][[1]]; N];
tabseq = Table[a[i], {i, 1, 100}];
(* second program *)
Table[NestWhile[#+1&, Prime[n]+1, Not@*PrimePowerQ], {n, 100}] (* Gus Wiseman, Nov 06 2024 *)

Prog

(PARI)
A000015(n) = for(k=n, oo, if((1==k)||isprimepower(k), return(k)));
A345531(n) = A000015(1+prime(n)); \\ Antti Karttunen, Jul 19 2021
(Python)
from itertools import count
from sympy import prime, factorint
def A345531(n): return next(filter(lambda m:len(factorint(m))<=1, count(prime(n)+1))) # Chai Wah Wu, Oct 25 2024

Crossrefs

Cf. A000040, A007814, A007949, A008864, A343249, A343250, A112765, A343251, A214411, A343252, A343253.

Starting with n instead of prime(n): A000015, A031218, A377468, A377780, A377782.

Opposite (greatest prime-power less than): A065514, A377289, A377781.

For squarefree instead of prime-power: A112926, opposite A112925.

The difference from prime(n) is A377281.

The prime terms have indices A377286(n) - 1.

First differences are A377703.

A version for perfect-powers is A378249.

A000961 and A246655 list the prime-powers, differences A057820.

A024619 and A361102 list the non-prime-powers, differences A375735.

Cf. A366833, A053607, A080101, A304521, A366833, A377057, A377283, A377287, A377288, A377432.

Sequence in context: A023054 A374962 A060023 * A357644 A120355 A114210

Adjacent sequences: A345528 A345529 A345530 * A345532 A345533 A345534

Keyword

nonn

Author

Dario T. de Castro, Jun 20 2021

Status

approved