A388986 Numbers k such that {sum of unitary divisors of k} plus {sum of squarefree divisors of k} is less than 2*k.

8, 9, 16, 25, 27, 32, 36, 40, 45, 48, 49, 50, 54, 56, 63, 64, 72, 75, 80, 81, 88, 96, 98, 99, 100, 104, 108, 112, 117, 121, 125, 128, 135, 136, 144, 147, 152, 153, 160, 162, 169, 171, 175, 176, 184, 189, 192, 196, 200, 207, 208, 216, 224, 225, 232, 242, 243, 245, 248, 250, 252, 256, 261, 272, 275, 279, 288, 289

Offset

1,1

Comments

Numbers k for which A325973(k) < k, or equally, A325977(k) < 0.

Question: Is A228058 a subsequence? See the graph of A389217.

If A228058 indeed is a subsequence of this sequence, then A386427 is a subsequence of A389219. See also A048108 and A389079. - Antti Karttunen, Oct 05 2025

Conjecture: Numbers k such that at least 1 number less than k is a multiple of rad(k), but its prime factorization is not a subset of k's. As an example, rad(36)=6 and 30 is a multiple of 6 less than 36, but the prime factorization of 30 is not a subset of the prime factorization of 36, so 36 is in the sequence. - Anthony Browne, May 15 2026

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

{k | A034448(k)+A048250(k) < 2*k}.

Mathematica

Select[Range[300], Apply[Times, 1 + Power @@@ #2] + Apply[Times, 1 + #2[[;; , 1]]] < 2*#1 & @@ {#, FactorInteger[#]} &] (* Michael De Vlieger, Sep 29 2025 *)

Prog

(PARI)
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A048250(n) = factorback(apply(p -> p+1, factor(n)[, 1]));
is_A388986(k) = ((A034448(k)+A048250(k)) < 2*k);

Crossrefs

Cf. A034448, A048108, A048250, A228058 (conjectured subsequence), A325973, A389079, A389215 (complement), A389217, A389219.

Positions of negative terms in A325977.

Setwise difference A388985 \ A325963.

Subsequence of A013929.

Includes the first 42 terms of A360765.

Sequence in context: A169817 A351098 A393869 * A396324 A227649 A227648

Adjacent sequences: A388983 A388984 A388985 * A388987 A388988 A388989

Keyword

nonn

Author

Antti Karttunen, Sep 29 2025

Status

approved