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A349165 Numbers k such that sigma(k) and A003961(k) are relatively prime, where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

%I A349165 #16 May 22 2025 11:45:33
%S A349165 1,3,4,5,7,9,11,12,13,15,16,17,19,21,23,25,28,29,31,33,35,36,37,39,41,
%T A349165 43,45,47,48,49,51,52,53,55,59,61,63,64,67,69,71,73,75,76,77,79,81,83,
%U A349165 84,85,89,91,93,95,97,101,103,105,107,109,111,112,113,115,117,119,121,123,124,125,127,129,131,133,137,139
%N A349165 Numbers k such that sigma(k) and A003961(k) are relatively prime, where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).
%C A349165 Includes all odd primes.  A prime power prime(j)^k with k > 1 is a term if and only if k+1 is not divisible by the multiplicative order of prime(j) mod prime(j+1). - _Robert Israel_, May 22 2025
%H A349165 Robert Israel, <a href="/A349165/b349165.txt">Table of n, a(n) for n = 1..10000</a>
%H A349165 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A349165 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%p A349165 filter:= proc(n) local F,a,b,t;
%p A349165    F:= ifactors(n)[2];
%p A349165    b:= convert(map(nextprime,F[..,1]),`*`);
%p A349165    a:= mul((t[1]^(t[2]+1)-1)/(t[1]-1),t=F);
%p A349165    igcd(a,b) = 1
%p A349165 end proc:
%p A349165 select(filter, [$1..1000]); # _Robert Israel_, May 21 2025
%o A349165 (PARI)
%o A349165 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A349165 isA349165(n) = (1==gcd(sigma(n), A003961(n)));
%Y A349165 Cf. A000203, A003961.
%Y A349165 Positions of ones in A342671, and also in A349163.
%Y A349165 Cf. A349166 (complement), A349167 (characteristic function).
%K A349165 nonn
%O A349165 1,2
%A A349165 _Antti Karttunen_, Nov 09 2021