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

1,1

Comments

There are 17270452 weird numbers below 10^10 and only 42 are in this sequence.

Links

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

Example

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.

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]

Prog

(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

Crossrefs

Intersection of A354281 and A354282.

Subsequence of A006037.

Cf. A005835.

Sequence in context: A064114 A354281 A348631 * A319735 A258374 A363297

Adjacent sequences: A354280 A354281 A354282 * A354284 A354285 A354286

Keyword

nonn

Author

Amiram Eldar, May 22 2022

Status

approved