A097795 -id:A097795 - cp's OEIS frontend

A097796 Number of partitions of n into perfect numbers.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Aug 25 2004

nonn

a(2*n) = A097795(n).

a(A204878(n)) = 0; a(A204879(n)) > 0.

			a(90)=2: 90 = 15*6 = 15*A000396(1) = 3*28 + 1*6 = 3*A000396(2) + 1*A000396(1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Number
Eric Weisstein's World of Mathematics, Partition
Wikipedia, Perfect number

Cf. A000396, A000041, A058696, A080225.

a097796 = p a000396_list where
   p _ 0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jan 20 2012

f[x_] := Product[-(1/(-1 + x^i)), {i, {6, 28, 496, 8128, 33550336}}]; CoefficientList[Series[f[x], {x, 0, 1000}], x] (* Ben Branman, Jan 07 2012 *)

A097797 Number of partitions of n into deficient numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 65, 85, 111, 143, 184, 234, 297, 374, 469, 585, 727, 899, 1108, 1360, 1664, 2028, 2464, 2985, 3606, 4343, 5218, 6252, 7474, 8913, 10605, 12591, 14918, 17639, 20816, 24519, 28829, 33836, 39646, 46377, 54165, 63162

Offset: 1

Reinhard Zumkeller, Aug 25 2004

nonn

			n=10: 6 is the only non-deficient number <= 10 and five partitions of 10 contain 6 as part: 6 + 4 = 6 + 3 + 1 = 6 + 2 + 2 = 6 + 2 + 1 + 1 = 6 + 1 + 1 + 1 + 1, therefore a(10) = A000041(10) - 5 = 42 - 5 = 37.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Deficient Number
Eric Weisstein's World of Mathematics, Partition

Cf. A005100, A000041, A097799, A097798, A097796, A097795.

n = 50; d = Select[Range[n], DivisorSigma[1, #] < 2 # &]; Rest@CoefficientList[ Series[1/Product[1 - x^d[[i]], {i, 1, Length[d]}], {x, 0, n}], x] (* Amiram Eldar, Aug 02 2019 *)

A097798 Number of partitions of n into abundant numbers.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Aug 25 2004

nonn

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

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

Cf. A005101, A005231, A000041, A097800, A097797, A097796, A097795.

Cf. also A066874, A282568, A283550.

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

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 *)

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
```

a(0) = 1 prepended by David A. Corneth, Sep 08 2018

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A097796 Number of partitions of n into perfect numbers.

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A097797 Number of partitions of n into deficient numbers.

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A097798 Number of partitions of n into abundant numbers.

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