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A307222 Deficient numbers n with at least one divisor being the sum of other distinct divisors of n. 0


%S 45,63,99,105,110,117,130,135,154,165,170,182,189,195,225,231,238,255,

%T 266,273,285,286,297,315,322,345,351,357,374,385,399,405,418,429,441,

%U 455,459,475,483,494,495,506,513,525,561,567,585,595,598,609,621,627,646

%N Deficient numbers n with at least one divisor being the sum of other distinct divisors of n.

%C 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.

%C Differs from A051773 from n >= 12.

%H S. J. Benkoski and P. Erdős, <a href="https://doi.org/10.1090/S0025-5718-1974-0347726-9">On weird and pseudoperfect numbers</a>, Math. Comp., 28 (1974), pp. 617-623. <a href="http://www.renyi.hu/~p_erdos/1974-24.pdf">Alternate link</a>; <a href="https://doi.org/10.1090/S0025-5718-1975-0360452-6">1975 corrigendum</a>

%H Paul Erdős, <a href="http://www.renyi.hu/~p_erdos/1970-21.pdf">Some extremal problems in combinatorial number theory</a>, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio (1970), pp. 123-133.

%e 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.

%t 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]

%Y Cf. A000005, A005100, A051773, A119347.

%K nonn

%O 1,1

%A _Amiram Eldar_, Mar 29 2019

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Last modified June 16 21:55 EDT 2021. Contains 345080 sequences. (Running on oeis4.)