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

Author

Andrzej Kukla, Nov 11 2024

Keywords

• nonn
• easy

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.

Crossrefs

Union of A000040 and A005153.

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

Programs

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 *)