OFFSET
1,1
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.
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.
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 29 2019
STATUS
approved