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.

14, 21, 26, 28, 35, 38, 39, 42, 52, 56, 57, 62, 63, 65, 70, 74, 76, 77, 78, 82, 84, 86, 93, 95, 98, 99, 104, 105, 111, 112, 114, 117, 119, 122, 124, 126, 129, 130, 133, 134, 140, 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

Offset

1,1

Comments

Numbers k for which A344695(k^2) > 1.

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.

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.

If k is in the sequence, m*k is also present for any positive integer m coprime to k.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

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:-
sigma(14^2) = sigma(2^2) * sigma(7^2) = 7 * 57 = 7 * (3*19).
psi(14^2) = psi(2^2) * psi(7^2) = 6 * 56 = (2*3) * (2^3*7).
So sigma(14^2) and psi(14^2) share factors 3 and 7, so 14 is in the sequence.
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.

Maple

filter:= proc(k) local n, F, sig, psi, t;
   n:= k^2;
   F:= map(t -> [t[1], 2*t[2]], ifactors(k)[2]);
   sig:= mul((t[1]^(1+t[2])-1)/(t[1]-1), t=F);
   psi:= n*mul(1+1/t[1], t=F);
   igcd(sig, psi) > 1
end proc:
select(filter, [$1..300]); # Robert Israel, Jan 06 2024

Mathematica

filter[k_] := Module[{n, F, sig, psi},
   n = k^2;
   F = {#[[1]], 2 #[[2]]}& /@ FactorInteger[k];
   sig = Product[(t[[1]]^(1 + t[[2]]) - 1)/(t[[1]] - 1), {t, F}];
   psi = n*Product[1 + 1/t[[1]], {t, F}];
   GCD[sig, psi] > 1];
Select[Range[300], filter] // Quiet (* Jean-François Alcover, May 23 2024, after Robert Israel *)

Prog

(PARI)
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
A344695(n) = gcd(sigma(n), A001615(n));
isA344703(n) = (A344695(n^2)>1);

Crossrefs

Cf. A000203, A000290, A001615, A344695.

Subsequences: A344872.

Sequence in context: A255742 A093994 A114985 * A001944 A024803 A004432

Adjacent sequences: A344700 A344701 A344702 * A344704 A344705 A344706

Keyword

nonn

Author

Antti Karttunen and Peter Munn, May 27 2021

Status

approved