%I #38 Jun 30 2021 19:46:36
%S 1,3,4,1,6,12,8,3,1,18,12,4,14,24,24,1,18,3,20,6,32,36,24,12,1,42,4,8,
%T 30,72,32,3,48,54,48,1,38,60,56,18,42,96,44,12,6,72,48,4,1,3,72,14,54,
%U 12,72,24,80,90,60,24,62,96,8,1,84,144,68,18,96,144,72,3,74,114,4,20,96,168,80,6,1,126,84,32,108
%N a(n) = gcd(sigma(n), psi(n)), where sigma is the sum of divisors function, A000203, and psi is the Dedekind psi function, A001615.
%C This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 8, although a(4) = 1 and a(27) = 4. See A344702.
%C A more specific property holds: for prime p that does not divide n, a(p*n) = a(p) * a(n). In particular, on squarefree numbers (A005117) this sequence coincides with sigma and psi, which are multiplicative.
%C If prime p divides the squarefree part of n then p+1 divides a(n). (For example, 20 has square part 4 and squarefree part 5, so 5+1 divides a(20) = 6.) So a(n) = 1 only if n is square. The first square n with a(n) > 1 is a(196) = 21. See A344703.
%C Conjecture: the set of primes that appear in the sequence is A065091 (the odd primes). 5 does not appear as a term until a(366025) = 5, where 366025 = 5^2 * 11^4. At this point, the missing numbers less than 22 are 2, 10 and 17. 17 appears at the latest by a(17^2 * 103^16) = 17.
%H Antti Karttunen, <a href="/A344695/b344695.txt">Table of n, a(n) for n = 1..20000</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F a(n) = gcd(A000203(n), A001615(n)).
%F For prime p, a(p^e) = (p+1)^(e mod 2).
%F For prime p with gcd(p, n) = 1, a(p*n) = a(p) * a(n).
%F a(A007913(n)) | a(n).
%F a(n) = gcd(A000203(n), A244963(n)) = gcd(A001615(n), A244963(n)).
%F a(n) = A000203(n) / A344696(n).
%F a(n) = A001615(n) / A344697(n).
%t Table[GCD[DivisorSigma[1,n],DivisorSum[n,MoebiusMu[n/#]^2*#&]],{n,100}] (* _Giorgos Kalogeropoulos_, Jun 03 2021 *)
%o (PARI)
%o A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
%o A344695(n) = gcd(sigma(n), A001615(n));
%o (Python 3.8+)
%o from math import prod, gcd
%o from sympy import primefactors, divisor_sigma
%o def A001615(n):
%o plist = primefactors(n)
%o return n*prod(p+1 for p in plist)//prod(plist)
%o def A344695(n): return gcd(A001615(n),divisor_sigma(n)) # _Chai Wah Wu_, Jun 03 2021
%Y Cf. A000203, A001615, A005117, A244963, A344696, A344697, A344702, A344703 (numbers k for which a(k^2) > 1).
%Y Cf. also A048250, A291750, A291751, A340070, A323363.
%Y Subsets of range: A008864, A065091 (conjectured).
%K nonn
%O 1,2
%A _Antti Karttunen_ and _Peter Munn_, May 26 2021