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Search: id:A049417
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| A049417 |
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a(n) = isigma(n): sum of infinitary divisors of n. |
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+0
16
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| 1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 51, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144, 68, 90
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.
Multiplicative: If e = sum_{k >= 0} d_k 2^k (binary representation, then a(p^e) = prod_{k >= 0} (p^(2^k*{d_k+1}) - 1)/(p^(2^k) - 1). Christian G. Bower (bowerc(AT)usa.net) and Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu) May 20, 2005.
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LINKS
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S. R. Finch, Unitarism and infinitarism.
J. O. M. Pedersen, Tables of Aliquot Cycles
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EXAMPLE
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If n = 8: 8 = 2^3 = 2^"11" (writing 3 in binary) so the infinitary divisors are 2^"00" = 1, 2^"01" = 2, 2^"10" = 4 and 2^"11" = 8; so a(8) = 1+2+4+8 = 15.
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MATHEMATICA
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Table[Plus@@((Times @@ (First[it]^(#1 /. z -> List)) & ) /@
Flatten[Outer[z, Sequence @@ bitty /@
Last[it = Transpose[FactorInteger[k]]], 1]]), {k, 2, 120}]
bitty[k_] := Union[Flatten[Outer[Plus, Sequence @@ ({0, #1} & ) /@ Union[2^Range[0, Floor[Log[2, k]]]*Reverse[IntegerDigits[k, 2]]]]]]
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CROSSREFS
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Cf. A037445, A004607.
Adjacent sequences: A049414 A049415 A049416 this_sequence A049418 A049419 A049420
Sequence in context: A103402 A034448 A069184 this_sequence A125139 A107224 A026493
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KEYWORD
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nonn,mult
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AUTHOR
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Yasutoshi Kohmoto (zbi74583(AT)boat.zero.ad.jp)
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EXTENSIONS
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More terms from wouter.meeussen(AT)pandora.be, Sep 02, 2001
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