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A097798
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Number of partitions of n into abundant numbers.
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4
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1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 4, 0, 1, 0, 2, 0, 4, 0, 2, 0, 0, 0, 7, 0, 2, 0, 2, 0, 8, 0, 5, 0, 2, 0, 14, 0, 4, 0, 4, 0, 14, 0, 8, 0, 5, 0, 23, 0, 9, 0, 9, 0, 26, 0, 18, 0, 9, 0, 38, 0, 16, 0, 17, 0, 46, 0, 29, 0, 19, 0, 65, 0, 32, 0
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OFFSET
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0,25
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COMMENTS
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n = 977 = 945 + 32 is the first prime for which sequence obtains a nonzero value, as a(977) = a(32) = 1. 945 is the first term in A005231. - Antti Karttunen, Sep 06 2018
a(n) = 0 for 496 values of n, the largest of which is 991 (see A283550). - David A. Corneth, Sep 08 2018
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LINKS
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David A. Corneth, Table of n, a(n) for n = 0..10000 (a(1) through a(532) by Antti Karttunen)
David A. Corneth, PARI program
Eric Weisstein's World of Mathematics, Abundant Number
Eric Weisstein's World of Mathematics, Partition
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MATHEMATICA
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n = 100; d = Select[Range[n], DivisorSigma[1, #] > 2 # &]; CoefficientList[ Series[1/Product[1 - x^d[[i]], {i, 1, Length[d]} ], {x, 0, n}], x] (* Amiram Eldar, Aug 02 2019 *)
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PROG
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(PARI)
abundants_up_to_reversed(n) = { my(s = Set([])); for(k=1, n, if(sigma(k)>(2*k), s = setunion([k], s))); vecsort(s, , 4); };
partitions_into(n, parts, from=1) = if(!n, 1, my(k = #parts, s=0); for(i=from, k, if(parts[i]<=n, s += partitions_into(n-parts[i], parts, i))); (s));
A097798(n) = partitions_into(n, abundants_up_to_reversed(n)); \\ Antti Karttunen, Sep 06 2018
(PARI) \\ see Corneth link
(Magma) v:=[n:n in [1..100]| SumOfDivisors(n) gt 2*n]; [#RestrictedPartitions(n, Set(v)): n in [0..100]]; // Marius A. Burtea, Aug 02 2019
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CROSSREFS
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Cf. A005101, A005231, A000041, A097800, A097797, A097796, A097795.
Cf. also A066874, A282568, A283550.
Sequence in context: A305558 A341624 A341620 * A065205 A036272 A257403
Adjacent sequences: A097795 A097796 A097797 * A097799 A097800 A097801
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KEYWORD
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nonn
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AUTHOR
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Reinhard Zumkeller, Aug 25 2004
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EXTENSIONS
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a(0) = 1 prepended by David A. Corneth, Sep 08 2018
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STATUS
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approved
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