A002095 -id:A002095 - cp's OEIS frontend

A051898 Erroneous version of A002095.

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

Offset: 1

dead

A096258 Number of partitions of n into distinct nonprime parts.

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, 22, 23, 27, 31, 32, 36, 43, 47, 52, 57, 61, 70, 80, 84, 93, 105, 114, 127, 141, 150, 167, 188, 202, 220, 244, 264, 291, 322, 346, 377, 416, 450, 493, 540, 580, 633, 696, 750, 814, 888

Offset: 0

Vladeta Jovovic, Jul 31 2004

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

Cf. A002095.

Cf. A204389.

a096258 = p a018252_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Jan 15 2012

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+ `if`(i>n or isprime(i), 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, May 29 2013

CoefficientList[ Series[ Product[(1 + x^n)/(1 + x^Prime[n]), {n, 70}], {x, 0, 67}], x] (* Robert G. Wilson v, Aug 02 2004 *)

G.f.: Product_{i>0} (1+x^i)/(1+x^prime(i)).

More terms from Robert G. Wilson v, Aug 02 2004

A023895 Number of partitions of n into composite parts.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 4, 1, 4, 2, 7, 2, 9, 3, 12, 6, 15, 6, 23, 11, 26, 15, 37, 19, 48, 26, 61, 39, 78, 47, 105, 65, 126, 88, 167, 111, 211, 146, 264, 196, 331, 241, 426, 318, 519, 408, 657, 511, 820, 651, 1010, 833, 1252, 1028, 1564, 1301, 1900

Offset: 0

Olivier Gérard

nonn

First differences of A002095. - Emeric Deutsch, Apr 03 2006

a(n+1) > a(n) for n > 108. - Reinhard Zumkeller, Aug 22 2007

			a(12) = 4 because 12 = 4 + 4 + 4 = 6 + 6 = 4 + 8 = 12 (itself a composite number).

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (terms n = 0..150 from Reinhard Zumkeller)

Cf. A002808.
Cf. A002095.
Cf. A132456.
Cf. A204389.

a023895 = p a002808_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jan 15 2012

g:=(1-x)*product((1-x^ithprime(j))/(1-x^j),j=1..80): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..62); # Emeric Deutsch, Apr 03 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
       b(n, i-1)+ `if`(i>n or isprime(i), 0, b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, May 29 2013

Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; CoefficientList[ Series[1/Product[1 - x^Composite[i], {i, 1, 50}], {x, 0, 75}], x]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<2, 0, b[n, i-1] + If[i>n || PrimeQ[i], 0, b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

G.f.: (1-x)*Product_{j>=1} (1-x^prime(j))/(1-x^j). - Emeric Deutsch, Apr 03 2006

More terms from Reinhard Zumkeller, Aug 22 2007

A101048 Number of partitions of n into semiprimes (a(0) = 1 by convention).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 3, 2, 3, 1, 5, 3, 5, 4, 7, 4, 9, 7, 10, 8, 13, 10, 17, 13, 18, 17, 25, 21, 29, 25, 34, 34, 43, 37, 51, 49, 61, 59, 73, 69, 89, 87, 103, 103, 124, 122, 148, 149, 172, 176, 206, 208, 244, 248, 281, 293, 337, 344, 391, 405, 456, 479, 537, 553

Offset: 0

Reinhard Zumkeller, Nov 28 2004

nonn

Semiprime analog of A000607. a(n) <= A002095(n). - Jonathan Vos Post, Oct 01 2007

Das, Robles, Zaharescu, & Zeindler give an asymptotic formula, see Links. - Charles R Greathouse IV, Jan 20 2023

			a(12) = #{6 + 6, 4 + 4 + 4} = #{2 * (2*3), 3 * (2*2)} = 2.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Madhuparna Das, Nicolas Robles, Alexandru Zaharescu, and Dirk Zeindler, Partitions into semiprimes, arXiv preprint (2022). arXiv:2212.12489 [math.NT]

Cf. A000041, A000607, A101049, A001358, A064911, A002095.
Cf. A112020, A112021.
Cf. A002100.
Row sums of A344447.

a101048 = p a001358_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Mar 21 2014

g:=1/product(product(1-x^(ithprime(i)*ithprime(j)),i=1..j),j=1..30): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=1..71); # Emeric Deutsch, Apr 04 2006
# second Maple program:
h:= proc(n) option remember; `if`(n=0, 0,
     `if`(numtheory[bigomega](n)=2, n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
     `if`(i>n, 0, b(n-i, h(min(n-i, i))))+b(n, h(i-1))))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, May 19 2021

terms = 100; CoefficientList[1/Product[1 - x^(Prime[i] Prime[j]), {i, 1, PrimePi[Ceiling[terms/2]]}, {j, 1, i}] + O[x]^terms, x] (* Jean-François Alcover, Aug 01 2018 *)

issemi(n)=if(n<4, return(0)); forprime(p=2,97, if(n%p==0, return(isprime(n/p)))); bigomega(n)==2
allsemi(v)=for(i=1,#v, if(!issemi(v[i]), return(0))); 1
a(n)=my(s); if(n<4, return(n==0)); forpart(k=n, if(allsemi(k), s++),[4,n]); s \\ Charles R Greathouse IV, Jan 20 2023

G.f.: 1/product(product(1-x^(p(i)p(j)), i = 1..j),j = 1..infinity), p(k) is the k-th prime. - Emeric Deutsch, Apr 04 2006

a(0) set to 1 by N. J. A. Sloane, Nov 23 2007

A052284 Number of compositions of n into nonprime numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 7, 11, 17, 27, 40, 61, 92, 142, 217, 333, 506, 774, 1181, 1807, 2758, 4215, 6434, 9833, 15019, 22948, 35047, 53541, 81780, 124936, 190841, 291532, 445320, 680274, 1039155, 1587405, 2424849, 3704148, 5658321, 8643530

Offset: 0

Robert G. Wilson v, May 16 2002

nonn

Starting at n=1, appears to be row sums of triangle A157424. - Gary W. Adamson & Mats Granvik, Feb 28 2009

			a(6) = 5 because 1+1+1+1+1+1 = 1+1+4 = 1+4+1 = 4+1+1 = 6.

Alois P. Heinz, Table of n, a(n) for n = 0..5437 (first 501 terms from T. D. Noe)

Cf. A002095 (Number of partitions of n into nonprime parts).
Cf. A000041 & A023360.
Column k=0 of A224344.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(isprime(j), 0, a(n-j)), j=1..n))
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Aug 06 2019

nn=50; np=Select[Range[nn], !PrimeQ[ # ] &]; CoefficientList[Series[1/(1-Sum[x^k, {k, np}]), {x, 0, nn}], x] (* T. D. Noe, Aug 20 2010 *)

G.f.: 1/( 1 - (Sum_{m nonprime} x^m) ).

Definition and g.f. corrected by N. J. A. Sloane, Aug 19 2010, who thanks Vladimir Kruchinin for pointing out the errors.

A379315 Number of strict integer partitions of n with a unique 1 or prime part.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 1, 3, 1, 3, 2, 7, 3, 7, 4, 10, 7, 15, 7, 17, 13, 23, 16, 31, 20, 37, 31, 48, 38, 62, 48, 76, 68, 93, 80, 119, 105, 147, 137, 175, 166, 226, 208, 267, 263, 326, 322, 407, 391, 481, 492, 586, 591, 714, 714, 849, 884, 1020, 1050, 1232, 1263

Offset: 0

Gus Wiseman, Dec 28 2024

nonn

The "old" primes are listed by A008578.

			The a(10) = 2 through a(15) = 10 partitions:
  (8,2)  (11)     (9,3)    (13)     (9,5)    (8,7)
  (9,1)  (6,5)    (10,2)   (7,6)    (12,2)   (10,5)
         (7,4)    (6,4,2)  (8,5)    (8,4,2)  (11,4)
         (8,3)             (10,3)   (9,4,1)  (12,3)
         (9,2)             (12,1)            (14,1)
         (10,1)            (6,4,3)           (6,5,4)
         (6,4,1)           (8,4,1)           (8,4,3)
                                             (8,6,1)
                                             (9,4,2)
                                             (10,4,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

For all prime parts we have A000586, non-strict A000607 (ranks A076610).
For no prime parts we have A096258, non-strict A002095 (ranks A320628).
For a unique composite part we have A379303, non-strict A379302 (ranks A379301).
Considering 1 nonprime gives A379305, non-strict A379304 (ranks A331915).
For squarefree instead of old prime we have A379309, non-strict A379308 (ranks A379316).
Ranked by A379312 /\ A005117 = squarefree positions of 1 in A379311.
The non-strict version is A379314.
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A376682 gives k-th differences of old primes.
Cf. A000070, A023895, A034891, A036497, A038348, A095195, A175804, A204389, A257994, A302540.

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

seq(n)={Vec(sum(k=1, n, if(isprime(k) || k==1, x^k)) * prod(k=4, n, 1 + if(!isprime(k), x^k), 1 + O(x^n)), -n-1)} \\ Andrew Howroyd, Dec 28 2024

A379301 Positive integers whose prime indices include a unique composite number.

Original entry on oeis.org

7, 13, 14, 19, 21, 23, 26, 28, 29, 35, 37, 38, 39, 42, 43, 46, 47, 52, 53, 56, 57, 58, 61, 63, 65, 69, 70, 71, 73, 74, 76, 77, 78, 79, 84, 86, 87, 89, 92, 94, 95, 97, 101, 103, 104, 105, 106, 107, 111, 112, 113, 114, 115, 116, 117, 119, 122, 126, 129, 130, 131

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 70 are {1,3,4}, so 70 is in the sequence.
The prime indices of 98 are {1,4,4}, so 98 is not in the sequence.

For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of one in A379300.
Partitions of this type are counted by A379302 (strict A379303).
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.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
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 A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
- A379311 old prime, see A379312-A379315.
Cf. A113646, A376680.

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

A379304 Number of integer partitions of n with a unique prime part.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 6, 7, 9, 11, 17, 20, 26, 31, 41, 47, 62, 72, 93, 108, 136, 156, 199, 226, 279, 321, 398, 452, 555, 630, 767, 873, 1051, 1188, 1433, 1618, 1930, 2185, 2595, 2921, 3458, 3891, 4580, 5155, 6036, 6776, 7926, 8883, 10324, 11577, 13421, 15014

Offset: 0

Gus Wiseman, Dec 27 2024

nonn

			The a(2) = 1 through a(9) = 9 partitions:
  (2)  (3)   (31)   (5)     (42)     (7)       (62)       (54)
       (21)  (211)  (311)   (51)     (43)      (71)       (63)
                    (2111)  (3111)   (421)     (431)      (621)
                            (21111)  (511)     (4211)     (711)
                                     (31111)   (5111)     (4311)
                                     (211111)  (311111)   (42111)
                                               (2111111)  (51111)
                                                          (3111111)
                                                          (21111111)

For all prime parts we have A000607 (strict A000586), ranks A076610.
For no prime parts we have A002095 (strict A096258), ranks A320628.
Ranked by A331915 = positions of one in A257994.
For a unique composite part we have A379302 (strict A379303), ranks A379301.
The strict case is A379305.
For squarefree instead of prime we have A379308 (strict A379309), ranks A379316.
Considering 1 prime gives A379314 (strict A379315), ranks A379312.
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A095195 gives k-th differences of prime numbers.
Cf. A000070, A023895, A034891, A036497, A204389, A302540, A330944.

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

A379305 Number of strict integer partitions of n with a unique prime part.

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 2, 3, 3, 3, 3, 6, 8, 8, 8, 10, 12, 17, 18, 18, 22, 28, 30, 36, 40, 44, 52, 62, 67, 78, 87, 97, 113, 129, 137, 156, 177, 200, 227, 251, 271, 312, 350, 382, 425, 475, 521, 588, 648, 705, 785, 876, 957, 1061, 1164, 1272, 1411, 1558, 1693, 1866

Offset: 0

Gus Wiseman, Dec 27 2024

nonn

			The a(2) = 1 through a(12) = 8 partitions (A=10, B=11):
  (2)  (3)   (31)  (5)  (42)  (7)    (62)   (54)   (82)   (B)    (93)
       (21)             (51)  (43)   (71)   (63)   (541)  (65)   (A2)
                              (421)  (431)  (621)  (631)  (74)   (B1)
                                                          (83)   (642)
                                                          (92)   (651)
                                                          (821)  (741)
                                                                 (831)
                                                                 (921)

For all prime parts we have A000586, non-strict A000607 (ranks A076610).
For no prime parts we have A096258, non-strict A002095 (ranks A320628).
Ranked by A331915 /\ A005117 = squarefree positions of one in A257994.
For a composite instead of prime we have A379303, non-strict A379302 (ranks A379301).
The non-strict version is A379304.
For squarefree instead of prime we have A379309, non-strict A379308 (ranks A379316).
Considering 1 prime gives A379315, non-strict A379314 (ranks A379312).
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A095195 gives k-th differences of prime numbers.
Cf. A000070, A023895, A034891, A036497, A038348, A204389, A302540, A320629, A330944.

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

A379308 Number of integer partitions of n with a unique squarefree part.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 2, 2, 0, 3, 5, 5, 1, 6, 9, 9, 2, 10, 14, 18, 6, 18, 24, 30, 11, 28, 39, 47, 24, 48, 63, 76, 41, 74, 95, 118, 65, 120, 149, 181, 107, 181, 221, 266, 169, 266, 335, 398, 262, 394, 487, 578, 391, 578, 697, 844, 592, 834, 997, 1198, 867

Offset: 0

Gus Wiseman, Dec 26 2024

nonn

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

If all parts are squarefree we have A073576 (strict A087188), ranks A302478.
If no parts are squarefree we have A114374 (strict A256012), ranks A379307.
For composite instead of squarefree we have A379302 (strict A379303), ranks A379301.
For prime instead of squarefree we have A379304, (strict A379305), ranks A331915.
The strict case is A379309.
For old prime instead of squarefree we have A379314, (strict A379315), ranks A379312.
Ranked by A379316, positions of 1 in A379306.
A000041 counts integer partitions, strict A000009.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A377038 gives k-th differences of squarefree numbers.
A379310 counts nonsquarefree prime indices.
Cf. A000586, A000607, A002095, A013928, A023895, A034891, A072284, A073247, A120327, A175804, A376657, A377430.

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

cp's OEIS Frontend

A051898 Erroneous version of A002095.

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A096258 Number of partitions of n into distinct nonprime parts.

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A023895 Number of partitions of n into composite parts.

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A101048 Number of partitions of n into semiprimes (a(0) = 1 by convention).

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A052284 Number of compositions of n into nonprime numbers.

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A379315 Number of strict integer partitions of n with a unique 1 or prime part.

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Mathematica

PARI

A379301 Positive integers whose prime indices include a unique composite number.

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A379304 Number of integer partitions of n with a unique prime part.