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A127666 Odd infinitary abundant numbers. 17
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.
The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 1, 77, 473, 5703, 53569, 561610, 5525461, 54979537, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0005... . - Amiram Eldar, Sep 09 2022
LINKS
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
Sequence in context: A275449 A293186 A294027 * A274756 A335053 A290034
KEYWORD
nonn
AUTHOR
Ant King, Jan 26 2007
STATUS
approved

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Last modified March 29 10:22 EDT 2024. Contains 371268 sequences. (Running on oeis4.)