A389197 Numbers whose sum of unitary divisors (usigma, A034448) is congruent to 2 modulo 4.

5, 9, 10, 13, 17, 18, 20, 25, 26, 29, 34, 36, 37, 40, 41, 49, 50, 52, 53, 58, 61, 68, 72, 73, 74, 80, 81, 82, 89, 97, 98, 100, 101, 104, 106, 109, 113, 116, 121, 122, 125, 136, 137, 144, 146, 148, 149, 157, 160, 162, 164, 169, 173, 178, 181, 193, 194, 196, 197, 200, 202, 208, 212, 218, 226, 229, 232, 233, 241, 242

Offset

1,1

Comments

Numbers k such that the odd part of k [= A000265(k)] is a prime power, p^e, e >= 1, and that prime power is congruent to 1 modulo 4.

Terms that are not in the subsequence A389199 are found in A389205. This is because both A034448 and A048250 obtain odd values only on A000079, thus when the former is of the form 4u+2, the latter must be even, and if it is of the form 4u, their arithmetic mean A325973 is odd.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

{k | A034448(k) == 2 (mod 4)}.

{k | A001221(A000265(k)) == 1 and A000265(k) == 1 (mod 4)}.

Mathematica

A389197Q[k_] := k > 1 && Mod[Times @@ (Power @@@ FactorInteger[k] + 1), 4] == 2;
Select[Range[500], A389197Q] (* Paolo Xausa, Oct 13 2025 *)

Prog

(PARI)
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
is_A389197(k) = (2==(A034448(k)%4));
(PARI)
A000265(n) = (n>>valuation(n, 2));
is_A389197(k) = (1==omega(A000265(k)) && 1==(A000265(k))%4);

Crossrefs

Cf. A000265, A001221, A034448, A191217, A325973, A389199 (subsequence), A389205.

Subsequence of A336101.

Sequence in context: A070873 A234037 A225836 * A140509 A314580 A314581

Adjacent sequences: A389194 A389195 A389196 * A389198 A389199 A389200

Keyword

nonn

Author

Antti Karttunen, Oct 06 2025

Status

approved