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A344703 Numbers k for which sigma(k^2) and psi(k^2) share a factor, where sigma is the sum of divisors, A000203, and psi is the Dedekind psi function, A001615.

%I A344703 #23 May 23 2024 09:12:50
%S A344703 14,21,26,28,35,38,39,42,52,56,57,62,63,65,70,74,76,77,78,82,84,86,93,
%T A344703 95,98,99,104,105,111,112,114,117,119,122,124,126,129,130,133,134,140,
%U A344703 143,146,148,152,154,155,156,158,161,166,168,171,172,175,182,183,185,186,189,190,194,195,198,201,203,206,208,209
%N A344703 Numbers k for which sigma(k^2) and psi(k^2) share a factor, where sigma is the sum of divisors, A000203, and psi is the Dedekind psi function, A001615.
%C A344703 Numbers k for which A344695(k^2) > 1.
%C A344703 It can be shown that sigma(m) and psi(m) share a factor if m is nonsquare. (See A344695 for more detail.) So here we consider only square numbers, m = k^2.
%C A344703 For prime p, sigma(p^2) and psi(p^2) are coprime, since sigma(p^2) = p^2 + p + 1 = psi(p^2) + 1. So all terms are composite. We can say more, since for prime p and positive integer e, psi(p^(2*e)) = p^(2*e-1) * (p+1), whereas sigma(p^(2*e)) can be shown to be congruent to 1 modulo p and to 1 modulo p+1, so shares no factors with p^(2*e-1) * (p+1). So all terms are divisible by more than one prime.
%C A344703 If k is in the sequence, m*k is also present for any positive integer m coprime to k.
%H A344703 Robert Israel, <a href="/A344703/b344703.txt">Table of n, a(n) for n = 1..10000</a>
%e A344703 Sigma (A000203) and the Dedekind psi function (A001615) are both multiplicative, so we gain insight by showing the values of these functions using their multiplicative properties:-
%e A344703 sigma(14^2) = sigma(2^2) * sigma(7^2) = 7 * 57 = 7 * (3*19).
%e A344703 psi(14^2) = psi(2^2) * psi(7^2) = 6 * 56 = (2*3) * (2^3*7).
%e A344703 So sigma(14^2) and psi(14^2) share factors 3 and 7, so 14 is in the sequence.
%e A344703 Looking in particular at the shared factor 3, we see it is present in sigma(7^2) and psi(2^2). For prime p, sigma(p^2) and psi(p^2) equate to polynomials in p, so we deduce 3 divides sigma(p^2) for p congruent to 7 modulo 3, and divides psi(p^2) for p congruent to 2 modulo 3. From this we see all products of a prime of the form 3m+1 and a prime of the form 3m+2 are in the sequence; for instance (3*4+1) * (3*1+2) = 13 * 5 = 65.
%p A344703 filter:= proc(k) local n,F, sig, psi, t;
%p A344703    n:= k^2;
%p A344703    F:= map(t -> [t[1],2*t[2]], ifactors(k)[2]);
%p A344703    sig:= mul((t[1]^(1+t[2])-1)/(t[1]-1),t=F);
%p A344703    psi:= n*mul(1+1/t[1],t=F);
%p A344703    igcd(sig,psi) > 1
%p A344703 end proc:
%p A344703 select(filter, [$1..300]); # _Robert Israel_, Jan 06 2024
%t A344703 filter[k_] := Module[{n, F, sig, psi},
%t A344703    n = k^2;
%t A344703    F = {#[[1]], 2 #[[2]]}& /@ FactorInteger[k];
%t A344703    sig = Product[(t[[1]]^(1 + t[[2]]) - 1)/(t[[1]] - 1), {t, F}];
%t A344703    psi = n*Product[1 + 1/t[[1]], {t, F}];
%t A344703    GCD[sig, psi] > 1];
%t A344703 Select[Range[300], filter] // Quiet (* _Jean-François Alcover_, May 23 2024, after _Robert Israel_ *)
%o A344703 (PARI)
%o A344703 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 A344703 A344695(n) = gcd(sigma(n), A001615(n));
%o A344703 isA344703(n) = (A344695(n^2)>1);
%Y A344703 Cf. A000203, A000290, A001615, A344695.
%Y A344703 Subsequences: A344872.
%K A344703 nonn
%O A344703 1,1
%A A344703 _Antti Karttunen_ and _Peter Munn_, May 27 2021