A204389 -id:A204389 - cp's OEIS frontend

A379313 Positive integers whose prime indices are not all composite.

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))).

A381251 a(n) is the number of ways to write prime(n) as a sum of distinct composites.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 4, 10, 14, 27, 40, 52, 74, 133, 229, 276, 457, 626, 744, 1189, 1599, 2498, 4450, 5862, 6752, 8835, 10139, 13189, 32481, 41614, 60099, 67900, 122825, 138101, 195147, 274193, 342783, 477381, 661502, 736865, 1252245, 1390615, 1711496, 1897886

Offset: 1

Felix Huber, Apr 19 2025

nonn

Subsequence of A204389.

All sums have an odd number of odd composite summands (A071904).

			a(8) = 3 because prime(8) = 19 can be written in 3 ways as a sum of distinct composites: 19 = 9 + 10 = 9 + 4 + 6 = 15 + 4.

Felix Huber, Table of n, a(n) for n = 1..4000

Cf. A000040, A002808, A053872, A071904, A204389, A383037.

P:= [seq(ithprime(i),i=1..100)]:
C:= {$2..P[-1]} minus convert(P,set):
G:= mul(1+x^c,c=C):
seq(coeff(G,x,P[i]),i=1..100); # Robert Israel, Apr 22 2025

a(n) = A204389(A000040(n)). - R. J. Mathar, Apr 22 2025

A331924 Number of compositions (ordered partitions) of n into distinct composite parts.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 3, 0, 3, 2, 5, 3, 5, 2, 13, 10, 13, 11, 21, 16, 29, 25, 35, 55, 71, 62, 87, 100, 115, 155, 185, 197, 247, 378, 311, 495, 553, 674, 767, 1060, 1047, 1469, 1463, 1846, 2139, 3391, 2713, 4135, 4453, 5930, 6409, 8777

Offset: 0

Ilya Gutkovskiy, Feb 01 2020

nonn

			a(14) = 5 because we have [14], [10, 4], [8, 6], [6, 8] and [4, 10].

Index entries for sequences related to compositions

Cf. A002808, A023895, A052284, A204389, A219107, A280544, A331917.

cp's OEIS Frontend

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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A381251 a(n) is the number of ways to write prime(n) as a sum of distinct composites.

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A331924 Number of compositions (ordered partitions) of n into distinct composite parts.

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