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A307222 Deficient numbers n with at least one divisor being the sum of other distinct divisors of n. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Erdős used the term "integers with the property P" for numbers n such that all the 2^d(n) sums formed from the d(n) divisors of n 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.

LINKS

Table of n, a(n) for n=1..53.

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.

EXAMPLE

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.

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]

CROSSREFS

Cf. A000005, A005100, A051773, A119347.

Sequence in context: A046364 A321498 A051773 * A336553 A140278 A046426

Adjacent sequences:  A307219 A307220 A307221 * A307223 A307224 A307225

KEYWORD

nonn

AUTHOR

Amiram Eldar, Mar 29 2019

STATUS

approved

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Last modified May 14 16:48 EDT 2021. Contains 343898 sequences. (Running on oeis4.)