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 A127666 Odd infinitary abundant numbers. 15
 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 January 22 13:23 EST 2022. Contains 350481 sequences. (Running on oeis4.)