A204879 -id:A204879 - 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 *)

A185351 Sums of distinct perfect numbers.

Original entry on oeis.org

0, 6, 28, 34, 496, 502, 524, 530, 8128, 8134, 8156, 8162, 8624, 8630, 8652, 8658, 33550336, 33550342, 33550364, 33550370, 33550832, 33550838, 33550860, 33550866, 33558464, 33558470, 33558492, 33558498, 33558960, 33558966, 33558988, 33558994, 8589869056

Offset: 1

Charles R Greathouse IV, Feb 09 2012

nonn

The first 131072 terms of this sequence are even. Conjecturally, all terms are even.

Numbers in the range of the sum of perfect divisors function (A187794). - Timothy L. Tiffin, Jul 13 2016

			502 = 496 + 6, where 496 and 6 are perfect.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A083865, A204879, A187794.

With[{perf = Select[Range[10000], DivisorSigma[1, #] == 2# &]}, Rest[Union[Total/@Subsets[perf]]]] (* Harvey P. Dale, Feb 07 2012 *)

vecsum(v)=sum(i=1,#v,v[i]);
v=apply(n->binomial(n+1,2), select(k->ispseudoprime(k), vector(15,n,2^prime(n)-1))); u=List();for(i=0,2^#v-1,listput(u,vecsum(vecextract(v,i))));vecsort(Vec(u)) \\ Charles R Greathouse IV, Feb 09 2012

A204878 Numbers that cannot be written as sum of perfect numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105

Offset: 1

Reinhard Zumkeller, Jan 20 2012

nonn

Complement of A204879; A097796(a(n)) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Perfect Number
Wikipedia, Perfect number

Cf. A000396 (perfect numbers).

import Data.List (elemIndices)
a204878 n = a204878_list !! (n-1)
a204878_list = map (+ 1) $ elemIndices 0 a097796_list

a(38+k) = 49+2k for all k>0 (up to occurrence of an odd perfect number, known to be > 10^300, if it exists). - M. F. Hasler, Feb 09 2012

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A097796 Number of partitions of n into perfect numbers.

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A185351 Sums of distinct perfect numbers.

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A204878 Numbers that cannot be written as sum of perfect numbers.

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