A354281 Weird numbers k such that k-1 is the sum of a subset of the aliquot divisors of k.

70, 4030, 5830, 9272, 17272, 243892, 351956, 407132, 1901728, 2189024, 4199030, 11339816, 11547352, 12872512, 13885970, 24450010, 31699430, 32284330, 34041370, 34169630, 34315712, 38546576, 42251930, 50761810, 67727110, 67820390, 68000392, 72986296, 85389368

Offset: 1

Author

Amiram Eldar, May 22 2022

Keywords

• nonn

Comments

There are 17270452 weird numbers below 10^10 and only 68 are in this sequence.

Examples

			70 is a term since it is a weird number, its aliquot divisors are {1, 2, 5, 7, 10, 14, 35} and 69 = 1 + 2 + 7 + 10 + 14 + 35.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..68

Crossrefs

Subsequence of A006037.

A354283 is a subsequence.

Cf. A005835, A354282.

Programs

Mathematica

q[n_] := Module[{d = Most @ Divisors[n], x, s, c}, If[Plus @@ d <= n, False, s = Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}]; c = SeriesCoefficient[s, #] & /@ (n + {-1, 0}); c[[1]] > 0 && c[[2]] == 0]]; Select[Range[10000], q]

PARI

is(n, d=divisors(n)[^-1], s=vecsum(d))={s>n && !is_A005835(n, d, s) && is_A005835(n-1, d, s)}; \\ using is_A005835() by M. F. Hasler at A005835