A377929 Quasi-practical numbers: positive integers m such that every k <= sigma(m)-m is a sum of distinct proper divisors of m.

1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 47, 48, 53, 54, 56, 59, 60, 61, 64, 66, 67, 71, 72, 73, 78, 79, 80, 83, 84, 88, 89, 90, 96, 97, 100, 101, 103, 104, 107, 108, 109, 112, 113, 120, 126, 127, 128

Offset

1,2

Comments

Equivalently, positive integers m such that every number k <= d is a sum of distinct proper divisors of m, where d is the largest proper divisor of m (follows from Corollary 2.11 in the Kukla and Miska paper).

Rao and Peng (2013) proved that a number is quasi practical if and only if it is prime or practical (also Theorem 2.9 in Kukla/Miska paper).

Links

Andrzej Kukla, Table of n, a(n) for n = 1..10000

K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.

Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Mathematica

QuasiPracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]] || PrimeQ[n]]; Select[Range[200], QuasiPracticalQ] (* Created based on code by T. D. Noe, Apr 02 2010 *)

Crossrefs

Union of A000040 and A005153.

Cf. A002093, A103288, A033630, A119348, A174533, A174973, A083207.

Sequence in context: A053577 A365127 A093515 * A350242 A249724 A084369

Adjacent sequences: A377926 A377927 A377928 * A377930 A377931 A377932

Keyword

nonn,easy

Author

Andrzej Kukla, Nov 11 2024

Status

approved