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Odd numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) > 1, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.
5

%I #17 May 19 2024 14:35:24

%S 135,285,435,455,855,885,1185,1287,1305,1335,1425,1435,1485,1635,2235,

%T 2275,2295,2655,2685,2905,2985,3105,3135,3185,3311,3395,3435,3555,

%U 3585,4005,4035,4185,4425,4785,4865,4905,4995,5385,5685,5805,5835,5845,5925,6135,6237,6335,6345,6585,6675,6735,7125,7155,7175,7185

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

%H Antti Karttunen, <a href="/A349176/b349176.txt">Table of n, a(n) for n = 1..20000</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>

%e For n = 135 = 3^3 * 5, sigma(135) = 240 = 2^4 * 3 * 5, A003961(135) = 5^3 * 7 = 875, and gcd(135,875) = gcd(240,875) = 5, which is larger than 1, therefore 135 is included in the sequence.

%t Select[Range[1, 7200, 2], And[#1/#2 == #1/#3, #2 > 1] & @@ {#3, 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 isA349176(n) = if(!(n%2),0,my(u=A003961(n),t=gcd(u,n)); (t>1)&&(gcd(u,sigma(n))==t));

%Y Intersection of A104210 and A349174, or equally, intersection of A349166 and A349174.

%Y Subsequence of A372567.

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

%K nonn

%O 1,1

%A _Antti Karttunen_, Nov 11 2021