A129657 Infinitary deficient numbers: integers for which A126168(n) < n, or equivalently for which A049417(n) < 2n.

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

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. 395-411.

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 * A377313 A249407 A360305

Adjacent sequences: A129654 A129655 A129656 * A129658 A129659 A129660

Keyword

easy,nonn

Author

Ant King, Apr 29 2007

Status

approved