A023895 -id:A023895 - cp's OEIS frontend

A379303 Number of strict integer partitions of n with a unique composite part.

0, 0, 0, 0, 1, 1, 2, 3, 3, 6, 6, 8, 10, 10, 13, 15, 17, 20, 22, 24, 28, 31, 36, 40, 44, 50, 55, 62, 70, 75, 83, 89, 97, 108, 115, 128, 136, 146, 161, 172, 188, 203, 215, 233, 249, 269, 291, 309, 331, 353, 376, 405, 433, 459, 490, 518, 554, 592, 629, 670, 705

Offset: 0

Gus Wiseman, Dec 25 2024

nonn

			The a(4) = 1 through a(11) = 8 partitions:
  (4)  (4,1)  (6)    (4,3)    (8)      (9)      (10)       (6,5)
              (4,2)  (6,1)    (6,2)    (5,4)    (8,2)      (7,4)
                     (4,2,1)  (4,3,1)  (6,3)    (9,1)      (8,3)
                                       (8,1)    (5,4,1)    (9,2)
                                       (4,3,2)  (6,3,1)    (10,1)
                                       (6,2,1)  (4,3,2,1)  (5,4,2)
                                                           (6,3,2)
                                                           (8,2,1)

If no parts are composite we have A036497,  non-strict A034891 (ranks A302540).
If all parts are composite we have A204389, non-strict A023895 (ranks A320629).
The non-strict version is A379302, ranks A379301 (positions of 1 in A379300).
For a unique prime we have A379305, non-strict A379304 (ranks A331915).
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Cf. A000070, A000586, A000607, A002095, A038348, A096258, A113646, A376680, A379308, A379309, A379314, A379315.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?CompositeQ]==1&]],{n,0,30}]

A379306 Number of squarefree prime indices of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 0, 3, 2, 2, 1, 3, 1, 1, 2, 4, 1, 3, 0, 3, 1, 2, 0, 4, 2, 2, 3, 2, 1, 3, 1, 5, 2, 2, 1, 4, 0, 1, 2, 4, 1, 2, 1, 3, 3, 1, 1, 5, 0, 3, 2, 3, 0, 4, 2, 3, 1, 2, 1, 4, 0, 2, 2, 6, 2, 3, 1, 3, 1, 2, 0, 5, 1, 1, 3, 2, 1, 3, 1, 5, 4, 2, 1, 3, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 25 2024

nonn

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 prime indices of 39 are {2,6}, so a(39) = 2.
The prime indices of 70 are {1,3,4}, so a(70) = 2.
The prime indices of 98 are {1,4,4}, so a(98) = 1.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 2.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 3.

Positions of first appearances are A000079.
Positions of zero are A379307, counted by A114374 (strict A256012).
Positions of one are A379316, counted by A379308 (strict A379309).
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061398 counts squarefree numbers between primes, zeros A068360.
A377038 gives k-th differences of squarefree numbers.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379310 nonsquarefree, see A302478.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A057627, A068361, A070321, A071403, A072284, A112925, A112929, A120327, A377430, A378086.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],SquareFreeQ]],{n,100}]

Totally additive with a(prime(k)) = A008966(k).

A379317 Positive integers with a unique even prime index.

Original entry on oeis.org

3, 6, 7, 12, 13, 14, 15, 19, 24, 26, 28, 29, 30, 33, 35, 37, 38, 43, 48, 51, 52, 53, 56, 58, 60, 61, 65, 66, 69, 70, 71, 74, 75, 76, 77, 79, 86, 89, 93, 95, 96, 101, 102, 104, 106, 107, 112, 113, 116, 119, 120, 122, 123, 130, 131, 132, 138, 139, 140, 141, 142

Offset: 1

Gus Wiseman, Dec 29 2024

nonn

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:
   3: {2}
   6: {1,2}
   7: {4}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  19: {8}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  29: {10}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  37: {12}
  38: {1,8}
  43: {14}
  48: {1,1,1,1,2}

Partitions of this type are counted by A038348 (strict A096911).
For all even parts we have A066207, counted by A035363 (strict A000700).
For no even parts we have A066208, counted by A000009 (strict A035457).
Positions of 1 in A257992.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A349158.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A038550, A087436.

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

A353188 Number of partitions of n that contain at least one composite part.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 4, 8, 12, 19, 27, 41, 56, 80, 109, 150, 199, 268, 350, 461, 596, 771, 984, 1258, 1589, 2007, 2514, 3145, 3905, 4846, 5973, 7356, 9010, 11020, 13418, 16315, 19756, 23890, 28788, 34639, 41548, 49767, 59441, 70899, 84354, 100221, 118803, 140645, 166153, 196035, 230853, 271512

Offset: 0

Omar E. Pol, Jun 22 2022

nonn

			For n = 6 the partitions of 6 that contain at least one composite parts are [6], [4, 2] and [4, 1, 1]. There are three of these partitions so a(6) = 3.

Cf. A000041, A002096, A002808, A023895, A034891, A047967, A085642, A086543, A116449, A144300, A204389.

a(n) = my(nb=0); forpart(p=n, if (#select(x->((x>1) && !isprime(x)), Vec(p)) >=1, nb++);); nb; \\ Michel Marcus, Jun 23 2022

a(n) = A000041(n) - A034891(n).

A132456 Number of partitions of n into composite parts such that succeeding parts have a common prime factor.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 4, 0, 4, 2, 7, 0, 9, 1, 12, 4, 14, 1, 23, 4, 24, 9, 35, 3, 46, 7, 56, 18, 68, 14, 99, 16, 107, 36, 145, 24, 186, 39, 219, 72, 270, 56, 361, 84, 408, 134, 514, 117, 655, 173, 761, 244, 926, 238, 1194, 312, 1347, 458, 1669, 458, 2073, 588, 2389

Offset: 0

Reinhard Zumkeller, Aug 22 2007

nonn

a(n) <= A023895(n).

			a(19) = #{9+6+4} = 1, A023895(19) = #{15+4, 10+9, 9+6+4} = 3;
a(20) = A023895(20) = 12;
a(21) = #{21, 15+6, 12+9, 9+6+6} = 4, A023895(21) = 6;

Alois P. Heinz, Table of n, a(n) for n = 0..1000

A280285 Number of partitions of n into 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, 1, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 2, 1, 1, 3, 0, 0, 3, 1, 0, 4, 1, 1, 5, 1, 0, 5, 2, 2, 6, 2, 1, 8, 3, 1, 8, 3, 2, 11, 3, 2, 12, 5, 4, 13, 5, 3, 16, 8, 4, 18, 7, 6, 22, 9, 7, 24, 12, 9, 28, 12, 9, 33, 18, 11, 36, 18, 14, 45, 22, 16, 48, 26, 22, 54, 29, 23, 66, 38

Offset: 0

Ilya Gutkovskiy, Dec 31 2016

nonn

			a(36) = 3 because we have [27, 9], [21, 15] and [9, 9, 9, 9].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Composite Number
Index entries for related partition-counting sequences

Cf. A002095, A002808, A023895, A071904, A204389, A280287.

a:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d>1 and d::odd and not isprime(d), d, 0),
       d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 31 2016

nmax = 100; CoefficientList[Series[(1 - x)/(1 - x^2) Product[(1 - x^(2 k)) (1 - x^Prime[k])/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: ((1 - x)/(1 - x^2))*Product_{k>=1} (1 - x^(2*k))*(1 - x^prime(k))/(1 - x^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[#]&]

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))).

A280544 Expansion of 1/(1 - Sum_{k>=2} (1 - floor(2/d(k)))*x^k), where d(k) is the number of divisors (A000005).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 3, 0, 5, 2, 8, 3, 13, 5, 22, 10, 34, 18, 58, 31, 94, 57, 153, 99, 254, 172, 417, 302, 685, 523, 1136, 901, 1872, 1557, 3097, 2673, 5133, 4577, 8505, 7843, 14109, 13380, 23440, 22816, 38953, 38855, 64789, 66053, 107871, 112190, 179664, 190369, 299478, 322683, 499501, 546548

Offset: 0

Ilya Gutkovskiy, Jan 05 2017

nonn

Number of compositions (ordered partitions) of n into composite parts (A002808).

			a(10) = 3 because we have [10], [6, 4] and [4, 6].

Eric Weisstein's World of Mathematics, Composite Number
Index entries for sequences related to compositions

Cf. A000005, A002808, A023360, A023895, A052284.

nmax = 59; CoefficientList[Series[1/(1 - Sum[(1 - Floor[2/DivisorSigma[0, k]]) x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

x='x+O('x^60); Vec(1/(1 - sum(k=2, 59, (1 - 2\numdiv(k))*x^k))) \\ Indranil Ghosh, Apr 03 2017

G.f.: 1/(1 - Sum_{k>=2} (1 - floor(2/d(k)))*x^k).

A303663 Expansion of (1/(1 - x))*Product_{k>=1} (1 - x^prime(k))/(1 - x^k).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 14, 19, 25, 33, 41, 53, 66, 83, 102, 128, 156, 193, 233, 285, 343, 416, 495, 597, 710, 849, 1003, 1194, 1404, 1662, 1946, 2291, 2675, 3137, 3646, 4260, 4939, 5744, 6637, 7697, 8868, 10250, 11778, 13570, 15558, 17877, 20437, 23423, 26727, 30550, 34781, 39669, 45068, 51287, 58157

Offset: 0

Ilya Gutkovskiy, Apr 28 2018

Partial sums of A002095.

Number of partitions of n into nonprime parts if there are two kinds of 1's.

David A. Corneth, Table of n, a(n) for n = 0..9999
Index entries for sequences related to partitions

Cf. A000070, A002095, A018252, A023895, A034891.

b:= proc(n, i) option remember; `if`(n=0 or i=1, n+1,
      b(n, i-1)+`if`(isprime(i), 0, b(n-i, min(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 28 2018

nmax = 55; CoefficientList[Series[1/(1 - x) Product[(1 - x^Prime[k])/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

cp's OEIS Frontend

A379303 Number of strict integer partitions of n with a unique composite part.

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A379306 Number of squarefree prime indices of n.

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A379317 Positive integers with a unique even prime index.

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A353188 Number of partitions of n that contain at least one composite part.

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PARI

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A132456 Number of partitions of n into composite parts such that succeeding parts have a common prime factor.

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A280285 Number of partitions of n into odd composite numbers (A071904).

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Maple

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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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Formula

A280544 Expansion of 1/(1 - Sum_{k>=2} (1 - floor(2/d(k)))*x^k), where d(k) is the number of divisors (A000005).

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