A002966 -id:A002966 - cp's OEIS frontend

A316902 Heinz numbers of aperiodic integer partitions whose reciprocal sum is an integer.

2, 18, 72, 147, 162, 195, 250, 288, 294, 390, 500, 588, 648, 780, 1125, 1152, 1176, 1323, 1458, 1560, 1755, 2000, 2250, 2352, 2592, 2646, 3120, 3185, 3510, 4000, 4500, 4608, 4704, 4802, 5292, 6240, 6370, 6475, 6591, 7020, 7581, 8450, 9000, 9408, 10101, 10125

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A partition is aperiodic if its multiplicities are relatively prime.

			The sequence of partitions whose Heinz numbers belong to this sequence begins: (1), (221), (22111), (442), (22221), (632), (3331), (2211111), (4421), (6321), (33311), (44211), (2222111).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Select[Range[2,20000],And[GCD@@FactorInteger[#][[All,2]]==1,IntegerQ[Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]]]&]

A316903 Heinz numbers of aperiodic integer partitions whose reciprocal sum is the reciprocal of an integer.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 147, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 195, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A partition is aperiodic if its multiplicities are relatively prime.

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Select[Range[2,1000],And[GCD@@FactorInteger[#][[All,2]]==1,IntegerQ[1/Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]]]&]

A325701 Nonprime Heinz numbers of integer partitions whose reciprocal factorial sum is the reciprocal of an integer.

Original entry on oeis.org

1, 9, 25, 49, 77, 121, 125, 169, 221, 245, 289, 323, 343, 361, 375, 437, 529, 841, 899, 961, 1331, 1369, 1517, 1681, 1763, 1849, 1859, 2021, 2197, 2209, 2401, 2773, 2809, 2873, 3127, 3481, 3721, 3757, 4087, 4489, 4757, 4913, 5041, 5183, 5329, 5929, 6137, 6241

Offset: 1

Gus Wiseman, May 17 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

			The sequence of terms together with their prime indices begins:
     1: {}
     9: {2,2}
    25: {3,3}
    49: {4,4}
    77: {4,5}
   121: {5,5}
   125: {3,3,3}
   169: {6,6}
   221: {6,7}
   245: {3,4,4}
   289: {7,7}
   323: {7,8}
   343: {4,4,4}
   361: {8,8}
   375: {2,3,3,3}
   437: {8,9}
   529: {9,9}
   841: {10,10}
   899: {10,11}
   961: {11,11}
For example, the sequence contains 245 because the prime indices of 245 are {3,4,4}, with reciprocal sum 1/6 + 1/24 + 1/24 = 1/4.

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.

Reciprocal factorial sum: A002966, A316854, A316857, A325618, A325620, A325622, A325623.

Select[Range[1000],!PrimeQ[#]&&IntegerQ[1/Total[Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]!]]]&]

A379452 Number of compositions (ordered partitions) of 1 into n distinct reciprocals of positive integers.

Original entry on oeis.org

1, 0, 6, 144, 8640, 1670400, 1238655600, 6095673521280

Offset: 1

Ilya Gutkovskiy, Dec 23 2024

			a(3) = 6 because we have 1 = 1/2 + 1/3 + 1/6
                           = 1/2 + 1/6 + 1/3
                           = 1/3 + 1/2 + 1/6
                           = 1/3 + 1/6 + 1/2
                           = 1/6 + 1/2 + 1/3
                           = 1/6 + 1/3 + 1/2.

Cf. A002966, A002967, A006585.

a(n) = n! * A006585(n).

A381724 a(n) = pos(M(n)), where M(n) is the n X n matrix with every term = 4, and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

Original entry on oeis.org

4, 16, 192, 3072, 61440, 1474560, 41287680, 1321205760, 47563407360, 1902536294400, 83711596953600, 4018156653772800, 208944145996185600, 11700872175786393600, 702052330547183616000, 44931349155019751424000, 3055331742541343096832000

Offset: 1

Clark Kimberling, Mar 05 2025

nonn

For a matrix M with determinant |M|, the numbers pos(M) and neg(M) are the positive and negative parts of |M|, as defined in A380661. The definition implies that (pos(M)+neg(M))/2 = |M| and (pos(M)-neg(M))/2 = permanent of M. Thus, M is singular if and only if pos(M) = - neg(M).
Guide to sequences pos(M(n)), where M(n) is the n X n matrix with every term = c, a constant:
  c = 1:  A001710
  c = 2:  A002866
  c = 3:  A032108
  c = 4:  this sequence
For each c >=1, let s(n) = pos(M(n)); then s(1) = c, s(2) = c^2, and s(n) = c*n*s(n-1) for n >= 3.

Cf. A001710, A002966, A032108, A380661.

Essentially the same as A051711.

c = 4; d = Table[Det[ConstantArray[c, {n, n}]], {n, 1, 18}]
p = Table[Permanent[ConstantArray[c, {n, n}]], {n, 1, 18}]
neg = (d - p)/2
pos = (d + p)/2

s(1) = 4, s(2) = 16, and s(n) = 4*n*s(n-1) for n >= 3.

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A316902 Heinz numbers of aperiodic integer partitions whose reciprocal sum is an integer.

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A316903 Heinz numbers of aperiodic integer partitions whose reciprocal sum is the reciprocal of an integer.

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A325701 Nonprime Heinz numbers of integer partitions whose reciprocal factorial sum is the reciprocal of an integer.

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A379452 Number of compositions (ordered partitions) of 1 into n distinct reciprocals of positive integers.

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A381724 a(n) = pos(M(n)), where M(n) is the n X n matrix with every term = 4, and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

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