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A127666 Odd infinitary abundant numbers. 11
945, 10395, 12285, 15015, 16065, 17955, 19305, 19635, 21735, 21945, 23205, 23625, 25245, 25935, 26565, 27405, 28215, 28875, 29295, 29835, 31395, 33345, 33495, 33915, 34125, 34155, 34965, 35805, 37125, 38745, 39585, 40635, 41055, 42315 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This is also the sequence of odd integers whose infinitary aliquot sequences initially increase. Based on empirical evidence (up to 10 million), this applies to only about 0.1% of odd integers.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., Vol. 54, No. 189, (1990), 395-411.

J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]

J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]

J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]

FORMULA

Odd values of n for which A126168(n)>n.

EXAMPLE

a(5)=16065 because 16065 is the fifth odd number that is exceeded by the sum of its proper infinitary divisors.

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; Select[Range[1, 50000, 2], properinfinitarydivisorsum[ # ]># &] (* 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[1, 50000, 2], isigma[#] > 2 # &] (* Amiram Eldar, Jun 09 2019 *)

PROG

(PARI) A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)))}

isok(k) = A049417(k)>2*k&&k%2==1; \\ Jinyuan Wang, Jun 09 2019

CROSSREFS

Cf. A005231, A126168, A127661, A129656.

Sequence in context: A275449 A293186 A294027 * A274756 A335053 A290034

Adjacent sequences:  A127663 A127664 A127665 * A127667 A127668 A127669

KEYWORD

nonn

AUTHOR

Ant King, Jan 26 2007

STATUS

approved

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Last modified June 1 04:57 EDT 2020. Contains 334758 sequences. (Running on oeis4.)