login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Prime powers of the form (6n+1)^k.
4

%I #21 Apr 01 2021 14:48:34

%S 7,13,19,31,37,43,49,61,67,73,79,97,103,109,127,139,151,157,163,169,

%T 181,193,199,211,223,229,241,271,277,283,307,313,331,337,343,349,361,

%U 367,373,379,397,409,421,433,439,457,463,487,499,523,541,547,571,577,601

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

%C 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.

%C From _Bernard Schott_, Mar 31 2021: (Start)

%C 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.

%C 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.

%C 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)

%H Robert Israel, <a href="/A133290/b133290.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexNumber.html">Hex Number</a>.

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

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

%p N:= 1000: # for terms <= N

%p 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

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

%o (PARI) a133290(uptolimit)={my(a=vector(uptolimit));

%o 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,", ")))};

%o a133290(601) \\ _Hugo Pfoertner_, Dec 03 2019

%Y Cf. A003215, A002476, subsequence of A000961.

%K nonn

%O 1,1

%A _Mats Granvik_, Oct 16 2007, Oct 20 2007