A096258 -id:A096258 - cp's OEIS frontend

A300893 L.g.f.: log(Product_{k>=1} (1 + x^k)/(1 + x^prime(k))) = Sum_{n>=1} a(n)*x^n/n.

1, -1, 1, 3, 1, 5, 1, 3, 10, 9, 1, 9, 1, 13, 16, 3, 1, 14, 1, 13, 22, 21, 1, 9, 26, 25, 37, 17, 1, 30, 1, 3, 34, 33, 36, 18, 1, 37, 40, 13, 1, 40, 1, 25, 70, 45, 1, 9, 50, 34, 52, 29, 1, 41, 56, 17, 58, 57, 1, 34, 1, 61, 94, 3, 66, 60, 1, 37, 70, 58, 1, 18, 1, 73, 116, 41, 78, 70, 1, 13, 118, 81, 1, 44, 86

Offset: 1

Ilya Gutkovskiy, Mar 14 2018

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			L.g.f.: L(x) = x - x^2/2 + x^3/3 + 3*x^4/4 + x^5/5 + 5*x^6/6 + x^7/7 + 3*x^8/8 + 10*x^9/9 + 9*x^10/10 + ...
exp(L(x)) = 1 + x + x^4 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 3*x^10 + ... + A096258(n)*x^n + ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A006005, A018252, A023890, A096258, A300852.

nmax = 85; Rest[CoefficientList[Series[Log[Product[(1 + x^k)/(1 + x^Prime[k]), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 85; Rest[CoefficientList[Series[Sum[Boole[!PrimeQ[k]] k x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, !PrimeQ[#] &], {n, 85}]

G.f.: Sum_{k>=1} A018252(k)*x^A018252(k)/(1 + x^A018252(k)).

a(n) = 1 if n is an odd prime or 1 (A006005).

A302234 Expansion of Product_{k>=1} (1 - x^k)/(1 - x^prime(k)).

Original entry on oeis.org

1, -1, 0, 0, -1, 1, -1, 1, -1, 0, 1, 0, 0, 1, 0, -1, 1, 0, 0, 0, 0, -1, 1, -1, 1, -2, 1, -1, 0, 1, -1, 1, -2, 2, -1, -1, 2, -1, -1, 2, -2, 2, -1, 1, 0, -1, 1, 0, 1, -2, 2, 0, 0, 2, -1, 0, 0, 1, 0, 0, 1, -2, 0, -1, 0, 0, -2, 2, -3, 0, 2, -2, 1, -1, 1, -2, 1, -1, -1, 1

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

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The difference between the number of partitions of n into an even number of distinct nonprime parts and the number of partitions of n into an odd number of distinct nonprime parts.

Convolution of the sequences A000607 and A010815.

Index entries for sequences related to partitions

Cf. A000607, A002095, A010815, A018252, A046675, A096258, A302236.

nmax = 80; CoefficientList[Series[Product[(1 - x^k)/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 80; CoefficientList[Series[Product[(1 - Boole[!PrimeQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^A018252(k)).

A341468 Number of partitions of n into 9 distinct nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 6, 7, 11, 12, 18, 20, 25, 30, 38, 45, 57, 67, 81, 95, 114, 133, 162, 187, 219, 255, 297, 343, 401, 462, 529, 607, 696, 793, 910, 1032, 1168, 1324, 1497, 1689, 1905, 2142, 2400, 2692, 3009, 3362, 3754, 4182, 4643, 5165

Offset: 79

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A005171, A018252, A096258, A219203, A302479, A341457, A341461, A341462, A341464, A341465, A341466, A341467, A341469.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i b(n$2, 9):
seq(a(n), n=79..130);  # Alois P. Heinz, Feb 12 2021

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[PrimeQ[i], 0, b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 9];
Table[a[n], {n, 79, 130}] (* Jean-François Alcover, Feb 28 2022, after Alois P. Heinz *)

A341469 Number of partitions of n into 10 distinct nonprime parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 3, 6, 7, 10, 11, 17, 17, 25, 28, 38, 44, 57, 64, 82, 95, 117, 136, 168, 189, 231, 264, 317, 366, 433, 490, 579, 660, 770, 877, 1019, 1146, 1327, 1497, 1720, 1940, 2215, 2481, 2825, 3165, 3583, 4008, 4523, 5033, 5664

Offset: 95

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A005171, A018252, A096258, A219204, A302479, A341460, A341461, A341462, A341464, A341465, A341466, A341467, A341468.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i b(n$2, 10):
seq(a(n), n=95..145);  # Alois P. Heinz, Feb 12 2021

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < t || t < 1, 0, b[n, i - 1, t] +
     If[PrimeQ[i], 0, b[n - i, Min[n - i, i - 1], t - 1]]]];
a[n_] := b[n, n, 10];
Table[a[n], {n, 95, 145}] (* Jean-François Alcover, Feb 28 2022, after Alois P. Heinz *)

A358639 Number of partitions of n into at most 3 distinct nonprime parts.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 3, 2, 2, 3, 4, 5, 5, 4, 6, 7, 7, 8, 10, 9, 11, 13, 14, 14, 16, 15, 20, 20, 21, 21, 27, 26, 30, 29, 32, 33, 39, 35, 43, 42, 46, 46, 53, 49, 58, 58, 63, 61, 69, 64, 77, 75, 81, 78, 90, 85, 98, 95, 102, 100, 114, 106, 122, 116, 126, 124, 140

Offset: 0

Ilya Gutkovskiy, Nov 24 2022

nonn

Cf. A018252, A096258, A307857, A341461, A347796, A358638, A358640.

A358640 Number of partitions of n into at most 4 distinct nonprime parts.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 3, 2, 2, 3, 4, 5, 5, 4, 6, 8, 8, 9, 11, 11, 13, 16, 17, 19, 21, 22, 26, 30, 30, 34, 39, 43, 47, 50, 53, 61, 67, 69, 76, 84, 89, 97, 106, 110, 121, 131, 139, 148, 160, 166, 181, 194, 204, 215, 233, 242, 262, 274, 289, 305, 329, 338, 361, 378

Offset: 0

Ilya Gutkovskiy, Nov 24 2022

nonn

Cf. A018252, A096258, A341462, A347586, A347662, A347797, A358638, A358639.

A379313 Positive integers whose prime indices are not all composite.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82

Offset: 1

Gus Wiseman, Dec 28 2024

nonn

Or, positive integers whose prime indices include at least one 1 or prime number.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
     2: {1}
     3: {2}
     4: {1,1}
     5: {3}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    11: {5}
    12: {1,1,2}
    14: {1,4}
    15: {2,3}
    16: {1,1,1,1}
    17: {7}
    18: {1,2,2}
    20: {1,1,3}
    21: {2,4}
    22: {1,5}
    24: {1,1,1,2}

Partitions of this type are counted by A000041 - A023895.
The "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
The complement is A320629, counted by A023895 (strict A204389).
For a unique prime we have A331915, counted by A379304 (strict A379305).
Positions of nonzeros in A379311.
For a unique 1 or prime we have A379312, counted by A379314 (strict A379315).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
A377033 gives k-th differences of composite numbers, see A073445, A377034.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
`Cf. A038550, A087436, A173390, A379300, A379301.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!And@@CompositeQ/@prix[#]&]

A280287 Number of partitions of n into distinct odd composite numbers (A071904).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 0, 2, 0, 1, 2, 1, 0, 3, 2, 1, 2, 1, 0, 3, 2, 1, 3, 2, 1, 5, 2, 1, 4, 3, 2, 4, 2, 1, 6, 4, 2, 6, 4, 3, 7, 4, 3, 6, 5, 4, 9, 5, 4, 10, 8, 4, 10, 6, 6, 12, 9, 5, 13, 9, 8, 14, 11, 7, 17, 13, 9, 16, 12, 11, 21

Offset: 0

Ilya Gutkovskiy, Dec 31 2016

nonn

			a(48) = 3 because we have [39, 9], [33, 15] and [27, 21].

Eric Weisstein's World of Mathematics, Composite Number
Index entries for related partition-counting sequences

Cf. A002095, A002808, A096258, A023895, A071904, A204389, A280285.

nmax = 105; CoefficientList[Series[(1 + x^2)/(1 + x) Product[(1 + x^k)/((1 + x^(2 k)) (1 + x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: ((1 + x^2)/(1 + x))*Product_{k>=1} (1 + x^k)/((1 + x^(2*k))*(1 + x^prime(k))).

A331917 Number of compositions (ordered partitions) of n into distinct nonprime parts.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 1, 2, 1, 3, 5, 8, 3, 10, 11, 17, 13, 16, 19, 54, 49, 55, 59, 90, 89, 129, 127, 183, 307, 358, 351, 456, 553, 649, 889, 1015, 1143, 1490, 2219, 1913, 3021, 3394, 4241, 4944, 6663, 6859, 9337, 9522, 12123, 14895, 22425, 18849, 28341, 31468, 41533

Offset: 0

Ilya Gutkovskiy, Feb 01 2020

nonn

			a(10) = 5 because we have [10], [9, 1], [6, 4], [4, 6] and [1, 9].

Index entries for sequences related to compositions

Cf. A002095, A018252, A052284, A096258, A219107.

A328970 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j) / (1 - x^prime(j)) is zero.

Original entry on oeis.org

2, 3, 9, 11, 12, 14, 17, 18, 19, 20, 28, 44, 47, 51, 52, 55, 56, 58, 59, 62, 64, 65, 69, 80, 81, 82, 83, 87, 91, 92, 94, 96, 99, 105, 106, 107, 113, 118, 119, 126, 127, 131, 147, 155, 157, 160, 161, 162, 164, 178, 179, 180, 215, 218, 224, 227, 257, 259, 269, 295

Offset: 1

Ilya Gutkovskiy, Nov 01 2019

nonn

Numbers k such that number of partitions of k into an even number of distinct nonprime parts equals number of partitions of k into an odd number of distinct nonprime parts.

Positions of 0's in A302234.

Cf. A002095, A018252, A090864, A096258, A302234.

a[j_] := a[j] = If[j == 0, 1, -Sum[Sum[Boole[!PrimeQ[d]] d, {d, Divisors[k]}] a[j - k], {k, 1, j}]/j]; Select[Range[300], a[#] == 0 &]
Flatten[Position[nmax = 300; Rest[CoefficientList[Series[Product[(1 - x^j)/(1 - x^Prime[j]), {j, 1, nmax}], {x, 0, nmax}], x]], 0]]

cp's OEIS Frontend

A300893 L.g.f.: log(Product_{k>=1} (1 + x^k)/(1 + x^prime(k))) = Sum_{n>=1} a(n)*x^n/n.

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A302234 Expansion of Product_{k>=1} (1 - x^k)/(1 - x^prime(k)).

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A341468 Number of partitions of n into 9 distinct nonprime parts.

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A341469 Number of partitions of n into 10 distinct nonprime parts.

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Maple

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A358639 Number of partitions of n into at most 3 distinct nonprime parts.

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A358640 Number of partitions of n into at most 4 distinct nonprime parts.

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A379313 Positive integers whose prime indices are not all composite.

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A280287 Number of partitions of n into distinct odd composite numbers (A071904).

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A331917 Number of compositions (ordered partitions) of n into distinct nonprime parts.

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A328970 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j) / (1 - x^prime(j)) is zero.

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