A307222 Deficient numbers k at least one of whose divisors is the sum of other distinct divisors of k.

45, 63, 99, 105, 110, 117, 130, 135, 154, 165, 170, 182, 189, 195, 225, 231, 238, 255, 266, 273, 285, 286, 297, 315, 322, 345, 351, 357, 374, 385, 399, 405, 418, 429, 441, 455, 459, 475, 483, 494, 495, 506, 513, 525, 561, 567, 585, 595, 598, 609, 621, 627, 646

Offset: 1

Author

Amiram Eldar, Mar 29 2019

Keywords

• nonn

Comments

Erdős used the term "integers with the property P" for numbers k such that all the 2^d(k) sums formed from the d(k) divisors of k are distinct and proved that they are all deficient numbers and have a positive density. This sequence lists deficient numbers not having this property.

Differs from A051773 from n >= 12.

Examples

			45 is in this sequence since its divisors are 1, 3, 5, 9, 15, 45 whose sum is 78 < 90, and thus it is deficient, yet the divisor 15 is the sum of other divisors of 45: 1 + 5 + 9.
		

Links

• S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp., 28 (1974), pp. 617-623. Alternate link; 1975 corrigendum
• Paul Erdős, Some extremal problems in combinatorial number theory, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio (1970), pp. 123-133.

Crossrefs

Cf. A000005, A005100, A051773, A119347.

Programs

Mathematica

T[n_, k_] := Module[{d = Divisors[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, k}], k]]; seqQ[n_] := DivisorSigma[1, n] < 2n && Max[T[n, #] & /@ Range[DivisorSigma[1, n]]] > 1; Select[Range[1000], seqQ]