A133290 Prime powers of the form (6n+1)^k.

7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 169, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 343, 349, 361, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601

Offset: 1

Author

Mats Granvik, Oct 16 2007, Oct 20 2007

Keywords

• nonn

Comments

1 + sum of the indices of the first two numbers in A003215 that are divisible by n if 1 + the sum of those indices equals n.

From Bernard Schott, Mar 31 2021: (Start)

Positive integers m that can be primitively represented as m = k^2+k*q+q^2 with 1 <= k < q and gcd(k,q)=1 in exactly 1 way. For example: 7 = 1 + 1*2 + 2^2.

Positive integers m such that m^2 can be primitively represented as k^2-k*q+q^2 with 1 <= k < q and gcd(k,q)= 1 in exactly 2 ways. For example: 7^2 = 3^2 - 3*8 + 8^2 = 5^2 - 5*8 + 8^2.

Length of the middle side b of the primitive triangles such that A < B < C with an angle B = 60 degrees and that appears precisely twice consecutively in A335895. (End)

Examples

			A003215(1) = 7 is divisible by 7, A003215(5) = 91 is divisible by 7 and 1+5+1=7, so 7 is a member.
A003215(5) = 91 is divisible by 13, A003215(7) = 169 is divisible by 13 and 5+7+1=13 so 13 is a member.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Hex Number.

Crossrefs

Cf. A003215, A002476, subsequence of A000961.

Programs

Maple

N:= 1000: # for terms <= N
sort(map(p -> seq(p^i,i=1..floor(log[p](N))), select(isprime, [seq(i,i=1..N,6)]))): # Robert Israel, Dec 02 2019

Mathematica

Select[a=6Range@100+1,PrimePowerQ@#&&MemberQ[a,First@@FactorInteger@#]&] (* Giorgos Kalogeropoulos, Mar 31 2021 *)

PARI

a133290(uptolimit)={my(a=vector(uptolimit));
for(n=1,oo,my(j=6*n+1);if(j>#a,break);if(isprime(j),for(k=1,oo,my(m=j^k);if(m>#a,break,a[m]++)))); for(k=1,#a,if(a[k],print1(k,", ")))};
a133290(601) \\ Hugo Pfoertner, Dec 03 2019