

A129657


Infinitary deficient numbers: integers for which A126168(n) < n, or equivalently for which A049417(n) < 2n.


6



1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84
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OFFSET

1,2


COMMENTS

For large n, the distribution of a(n) is approximately linear and asymptotically satisfies a(n)~1.144n. It follows that the density of the infinitary deficient numbers is 1/1.144, which is about 0.874.


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, On an Integer's Infinitary Divisors, Mathematics of Computation, Vol. 54, No. 189, (1990), pp. 395411.
Eric Weisstein's World of Mathematics, Infinitary Divisor.


EXAMPLE

The sixth integer that exceeds its proper infinitary divisor sum is 7. Hence a(6)=7.


MATHEMATICA

ExponentList[n_Integer, factors_List]:={#, IntegerExponent[n, # ]}&/@factors; InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f, g}, BitOr[f, g]==g][ #, Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #, factors]&/@d]], _?(And@@#&), {1}]] ]] ] Null; properinfinitarydivisorsum[k_]:=Plus@@InfinitaryDivisors[k]k; InfinitaryDeficientNumberQ[k_]:=If[properinfinitarydivisorsum[k]<k, True, False]; Select[Range[100], InfinitaryDeficientNumberQ[ # ] &] (* end of program *)
fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m  j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[100], isigma[#] < 2 # &] (* Amiram Eldar, Jun 09 2019 *)


CROSSREFS

Cf. A126168, A049417, A127666, A129656, A007357.
Sequence in context: A272978 A080907 A127161 * A249407 A103679 A029916
Adjacent sequences: A129654 A129655 A129656 * A129658 A129659 A129660


KEYWORD

easy,nonn


AUTHOR

Ant King, Apr 29 2007


STATUS

approved



