%I #18 Dec 24 2024 22:11:17
%S 8,20,24,27,32,40,44,54,56,60,72,80,88,92,96,100,104,108,116,120,128,
%T 132,135,140,152,160,164,168,171,176,180,184,188,189,196,200,216,224,
%U 232,236,240,248,260,261,264,270,272,276,280,288,296,297,300,308,312,320,325,328,332,342,344,348,351,352,360,368,376
%N Numbers k such that sigma(k) and A003961(k) share a prime power divisor that is not a unitary divisor of A003961(k). Here A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).
%C Numbers k such that A342671(k) [= gcd(sigma(k), A003961(k))] and A349161(k) [= A003961(k)/A342671(k)] share a prime factor.
%C Numbers k for which A349163(k) and A349164(k) are not relatively prime.
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e For n = 8 = 2^3, sigma(8) = 15 = 3*5, while A003961(8) = 3^3 = 27. These share the prime power divisor 3, which however is not a unitary divisor of 27, therefore 8 is included in this sequence.
%e For n = 32 = 2^5, sigma(32) = 63 = 3^2 * 7, while A003961(32) = 3^5 = 243. These share the prime power divisor 3^2, which however is not a unitary divisor of 243, therefore 32 is included.
%e For n = 40 = 2^3 * 5, sigma(40) = 90 = 2 * 3^2 * 5, while A003961(40) = 3^3 * 7 = 189. These share the prime power divisor 3^2, which however is not a unitary divisor of 189, therefore 40 is included.
%o (PARI)
%o A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o isA349168(n) = { my(u=A003961(n), x=gcd(u,sigma(n))); (1!=gcd(x,u/x)); };
%Y Cf. A000203, A003961, A342671, A349161, A349163, A349164.
%Y Subsequence of A349166.
%K nonn
%O 1,1
%A _Antti Karttunen_, Nov 10 2021