A133290 - OEIS

%I #22 Feb 16 2025 08:33:06

%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="https://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,changed

%O 1,1

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