A036497 -id:A036497 - cp's OEIS frontend

A341980 Number of partitions of n into 9 distinct primes (counting 1 as a prime).

1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 3, 0, 1, 0, 3, 0, 5, 0, 4, 1, 6, 0, 10, 0, 6, 1, 11, 1, 16, 1, 11, 2, 19, 2, 25, 1, 18, 5, 32, 4, 36, 2, 32, 9, 47, 7, 55, 7, 49, 14, 69, 10, 80, 12, 74, 22, 98, 19, 117, 22, 106, 34, 140, 31, 158, 32, 149, 54, 194, 48, 215, 50

Offset: 78

Ilya Gutkovskiy, Feb 24 2021

nonn

Cf. A008578, A036497, A219203, A341719, A341973, A341974, A341975, A341976, A341977, A341978, A341979, A341981.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i-1)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 10)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 9):
seq(a(n), n=78..151);  # Alois P. Heinz, Feb 24 2021

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i - 1]]][
     If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 10}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 9];
Table[a[n], {n, 78, 151}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)

A341981 Number of partitions of n into 10 distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 5, 0, 4, 0, 2, 0, 9, 0, 7, 1, 7, 1, 14, 0, 10, 0, 12, 2, 22, 0, 19, 2, 22, 3, 34, 1, 31, 4, 32, 5, 54, 3, 48, 7, 50, 9, 78, 7, 70, 11, 76, 16, 113, 9, 100, 19, 114, 26, 155, 17, 147, 32, 164, 37, 212, 26

Offset: 101

Ilya Gutkovskiy, Feb 24 2021

nonn

Cf. A008578, A036497, A219204, A341972, A341973, A341974, A341975, A341976, A341977, A341978, A341979, A341980.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i-1)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 11)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 10):
seq(a(n), n=101..174);  # Alois P. Heinz, Feb 24 2021

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i - 1]]][
     If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 11}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 10];
Table[a[n], {n, 101, 174}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)

A331835 Replace 2^k in binary expansion of n with k-th prime number for any k > 0 (and keep 2^0).

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 5, 6, 7, 8, 8, 9, 10, 11, 7, 8, 9, 10, 10, 11, 12, 13, 12, 13, 14, 15, 15, 16, 17, 18, 11, 12, 13, 14, 14, 15, 16, 17, 16, 17, 18, 19, 19, 20, 21, 22, 18, 19, 20, 21, 21, 22, 23, 24, 23, 24, 25, 26, 26, 27, 28, 29, 13, 14, 15, 16, 16

Offset: 0

Rémy Sigrist, Jan 28 2020

Every nonnegative integer appears in this sequence as A008578 is a complete sequence.

For any m >= 0, m appears A036497(m) times, the first and last occurrences being at indices A345297(m) and A200947(m), respectively. - Rémy Sigrist, Jun 13 2021

			For n = 43:
- 43 = 2^0 + 2^1 + 2^3 + 2^5,
- so a(43) = 2^0 + prime(1) + prime(3) + prime(5) = 1 + 2 + 5 + 11 = 19.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Eric Weisstein's World of Mathematics, Complete Sequence
Index entries for sequences related to binary expansion of n

Cf. A008578, A036497, A048672, A089625, A200947, A345297.

Array[Total@ Map[If[# == 0, 1, Prime[#]] &, Position[Reverse@ IntegerDigits[#, 2], 1][[All, 1]] - 1] &, 68] (* Michael De Vlieger, Jan 29 2020 *)

a(n) = my (b=Vecrev(binary(n\2))); n%2 + sum(k=1, #b, if (b[k], prime(k), 0))

from sympy import prime
def p(n): return prime(n) if n >= 1 else 1
def a(n): return sum(p(i)*int(b) for i, b in enumerate(bin(n)[:1:-1]))
print([a(n) for n in range(69)]) # Michael S. Branicky, Jun 13 2021

a(2*n) = A089625(n) for any n > 0.
a(2*n+1) = A089625(n) + 1 for any n > 0.
G.f.: x/(1 - x^2) + (1/(1 - x)) * Sum_{k>=1} prime(k) * x^(2^k) / (1 + x^(2^k)). - Ilya Gutkovskiy, May 24 2024

A298604 Number of partitions of n into distinct odd prime parts (including 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 6, 5, 5, 6, 6, 7, 7, 8, 9, 8, 9, 10, 11, 12, 11, 12, 14, 14, 15, 16, 17, 17, 17, 20, 22, 21, 22, 24, 25, 27, 28, 30, 31, 31, 33, 36, 39, 40, 40, 42, 46, 47, 49, 53, 54, 55, 58, 63, 67, 68, 70, 73, 77, 81, 84

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

nonn

			a(16) = 3 because we have [13, 3], [11, 5] and [7, 5, 3, 1].

Index entries for related partition-counting sequences

Cf. A000586, A006005, A008578, A024939, A036497, A298603.

nmax = 78; CoefficientList[Series[(1 + x) Product[(1 + x^Prime[k]), {k, 2, nmax}], {x, 0, nmax}], x]

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

A331926 Number of compositions (ordered partitions) of n into distinct prime parts (counting 1 as a prime).

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 8, 3, 10, 8, 14, 31, 10, 33, 16, 38, 40, 61, 138, 69, 48, 98, 190, 121, 308, 128, 340, 270, 472, 991, 572, 885, 534, 446, 888, 1872, 914, 1927, 1084, 2300, 2058, 4303, 6508, 3759, 2246, 4856, 8238, 6889, 12630, 6368, 8708

Offset: 0

Ilya Gutkovskiy, Feb 01 2020

nonn

			a(6) = 8 because we have [5, 1], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 5], [1, 3, 2] and [1, 2, 3].

Index entries for sequences related to compositions

Cf. A008578, A023360, A034891, A036497, A219107, A280917.

a(n)={subst(serlaplace(y^0*polcoef(prod(k=1, n, 1 + if(k==1 || isprime(k), y*x^k) + O(x*x^n)), n)), y, 1)} \\ Andrew Howroyd, Feb 01 2020

A338826 G.f.: (1/(1 + x)) * Product_{k>=1} 1/(1 + x^prime(k)).

Original entry on oeis.org

1, -1, 0, -1, 2, -2, 2, -3, 4, -4, 5, -7, 8, -9, 11, -13, 15, -18, 21, -24, 28, -32, 37, -43, 49, -55, 63, -72, 81, -92, 104, -117, 131, -147, 166, -185, 206, -231, 257, -285, 317, -353, 391, -432, 478, -528, 583, -643, 708, -778, 855, -940, 1031, -1130, 1238, -1354

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

sign

The difference between the number of partitions of n into an even number of prime parts (including 1) and the number of partitions of n into an odd number of prime parts (including 1).

Convolution inverse of A036497.

Cf. A000586, A000607, A008578, A036497, A046675, A048165, A298602.

nmax = 55; CoefficientList[Series[(1/(1 + x)) Product[1/(1 + x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSum[k, (-1)^(k/#) # &, PrimeQ[#] || # == 1 &] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 55}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * A048165(k).

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[#]&]

A369765 Maximal coefficient of (1 + x) * (1 + x^2) * (1 + x^3) * (1 + x^5) * ... * (1 + x^prime(n-1)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 6, 7, 13, 19, 32, 53, 90, 156, 277, 494, 878, 1566, 2836, 5146, 9401, 17358, 32042, 59434, 110292, 204332, 380548, 713601, 1342448, 2538012, 4808578, 9043605, 17070234, 32268611, 61271738, 116123939, 220993892, 421000142, 802844420, 1534312896

Offset: 0

Ilya Gutkovskiy, Jan 31 2024

nonn

Cf. A008578, A025591, A036497, A350457, A350514, A369708.

Table[Max[CoefficientList[Product[(1 + x^If[k == 1, 1, Prime[k - 1]]), {k, 1, n}], x]], {n, 0, 40}]

from collections import Counter
from sympy import prime
def A369765(n):
    c = {0:1,1:1}
    for k in range(1,n):
        p, d = prime(k), Counter(c)
        for j in c:
            d[j+p] += c[j]
        c = d
    return max(c.values()) # Chai Wah Wu, Feb 01 2024

cp's OEIS Frontend

A341980 Number of partitions of n into 9 distinct primes (counting 1 as a prime).

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A341981 Number of partitions of n into 10 distinct primes (counting 1 as a prime).

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Maple

Mathematica

A331835 Replace 2^k in binary expansion of n with k-th prime number for any k > 0 (and keep 2^0).

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A298604 Number of partitions of n into distinct odd prime parts (including 1).

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A331926 Number of compositions (ordered partitions) of n into distinct prime parts (counting 1 as a prime).

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PARI

A338826 G.f.: (1/(1 + x)) * Product_{k>=1} 1/(1 + x^prime(k)).

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Mathematica

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A379313 Positive integers whose prime indices are not all composite.

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A369765 Maximal coefficient of (1 + x) * (1 + x^2) * (1 + x^3) * (1 + x^5) * ... * (1 + x^prime(n-1)).

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