A354283 Weird numbers k such that k-1 and k+1 are both sums of subsets of the aliquot divisors of k.

70, 4030, 5830, 17272, 243892, 4199030, 11339816, 11547352, 13885970, 24450010, 31699430, 32284330, 34041370, 34169630, 42251930, 50761810, 67727110, 67820390, 85389368, 89283592, 141659096, 146764264, 162079768, 173482552, 259858324, 410832532, 411643576, 486224072

Offset: 1

Author

Amiram Eldar, May 22 2022

Keywords

• nonn

Comments

There are 17270452 weird numbers below 10^10 and only 42 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}, 69 = 1 + 2 + 7 + 10 + 14 + 35, and 71 = 5 + 7 + 10 + 14 + 35.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..42

Crossrefs

Intersection of A354281 and A354282.

Subsequence of A006037.

Cf. A005835.

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 + 1}]; c = SeriesCoefficient[s, #] & /@ (n + {-1, 0, 1}); c[[1]] > 0 && c[[2]] == 0 && c[[3]] > 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) && is_A005835(n+1, d, s)}; \\ using is_A005835() by M. F. Hasler at A005835