

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
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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. 617623. 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. 123133.


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



