%I #16 Nov 12 2021 22:34:48

%S 1,3,5,7,9,11,13,17,19,21,23,25,29,31,33,37,39,41,43,47,49,51,53,55,

%T 59,61,63,67,69,71,73,79,81,83,85,89,91,93,95,97,101,103,107,109,111,

%U 113,115,117,119,121,123,125,127,129,131,133,135,137,139,141,145,147,149,151,153,155,157,159,161,163,167,169

%N Odd numbers k for which gcd(k, A003961(k)) is equal to gcd(sigma(k), A003961(k)), where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

%C Odd numbers k for which A322361(k) = A342671(k).

%C Odd numbers k for which A348994(k) = A349161(k).

%C Odd numbers k such that A319626(k) = A349164(k).

%C Odd terms of A336702 form a subsequence of this sequence. See also A349169.

%C Ratio of odd numbers residing in this sequence, vs. in A349175 seems to slowly decrease, but still apparently stays > 2 for a long time. E.g., for range 2 .. 2^28, it is 95302074/38915653 = 2.4489...

%H <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>

%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>

%t Select[Range[1, 169, 2], GCD[#1, #3] == GCD[#2, #3] & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* _Michael De Vlieger_, Nov 11 2021 *)

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o isA349174(n) = if(!(n%2),0,my(u=A003961(n)); gcd(u,sigma(n))==gcd(u,n));

%Y Cf. A000203, A003961, A319626, A322361, A336702, A342671, A348994, A349161, A349164, A349169.

%Y Cf. A349175 (complement among the odd numbers).

%Y Union of A349176 and A349177.

%K nonn

%O 1,2

%A _Antti Karttunen_, Nov 10 2021