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A349166 Numbers k such that sigma(k) and A003961(k) share a prime factor, where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

%I A349166 #17 May 22 2025 11:45:28
%S A349166 2,6,8,10,14,18,20,22,24,26,27,30,32,34,38,40,42,44,46,50,54,56,57,58,
%T A349166 60,62,65,66,68,70,72,74,78,80,82,86,87,88,90,92,94,96,98,99,100,102,
%U A349166 104,106,108,110,114,116,118,120,122,126,128,130,132,134,135,136,138,140,142,146,150,152,154,158,160,162,164
%N A349166 Numbers k such that sigma(k) and A003961(k) share a prime factor, where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).
%C A349166 The only prime term is 2. A prime power prime(j)^k with k > 1 is a term if and only if k+1 is divisible by the multiplicative order of prime(j) mod prime(j+1). - _Robert Israel_, May 22 2025
%H A349166 Robert Israel, <a href="/A349166/b349166.txt">Table of n, a(n) for n = 1..10000</a>
%H A349166 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A349166 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e A349166 For n = 2, A000203(2) = A003961(2) = 3, therefore they share a prime factor 3, and 2 is included in this sequence.
%e A349166 For n = 10 = 2*5, sigma(10) = 18 = 2 * 3^2, while A003961(10) = 21 = 3*7, therefore 10 is included, as there is a shared prime factor (3).
%p A349166 filter:= proc(n) local F,a,b,t;
%p A349166    F:= ifactors(n)[2];
%p A349166    b:= convert(map(nextprime,F[..,1]),`*`);
%p A349166    a:= mul((t[1]^(t[2]+1)-1)/(t[1]-1),t=F);
%p A349166    igcd(a,b) <> 1
%p A349166 end proc;
%p A349166 select(filter, [$1..1000]); # _Robert Israel_, May 21 2025
%o A349166 (PARI)
%o A349166 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A349166 isA349166(n) = (1!=gcd(sigma(n), A003961(n)));
%Y A349166 Cf. A000203, A003961.
%Y A349166 Positions of terms larger than ones in A342671, and also in A349163.
%Y A349166 Positions of zeros in A349167.
%Y A349166 Cf. A349165 (complement), A349168 (subsequence).
%K A349166 nonn
%O A349166 1,1
%A A349166 _Antti Karttunen_, Nov 09 2021