A344447 -id:A344447 - cp's OEIS frontend

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

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

A072931 Number of ways to write n as a sum of 2 semiprimes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 2, 2, 2, 1, 0, 1, 2, 2, 1, 1, 2, 2, 3, 3, 2, 0, 1, 3, 3, 2, 1, 3, 3, 2, 3, 4, 4, 2, 1, 4, 5, 3, 3, 1, 3, 3, 2, 5, 3, 2, 2, 5, 6, 6, 1, 3, 5, 3, 4, 4, 5, 3, 3, 6, 7, 5, 3, 3, 4, 4, 4, 5, 5, 3, 2, 7, 7, 2, 4, 4, 5, 4, 6, 8, 6, 3, 3, 8, 7, 7, 4, 6, 8, 6, 5, 7, 7, 2

Offset: 0

Benoit Cloitre, Aug 13 2002

Sequence is probably > 0 for n > 33.

The graph of this sequence is compelling evidence that 33 is the last term of sequence A072966. - T. D. Noe, Apr 10 2007

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 10000 terms from T. D. Noe)

Cf. A001222, A001358, A072966, A101048.

Column k=2 of A344447.

lim = 10000;
s = Select[Range[lim], PrimeOmega[#] == 2 &];
c = Tally[ Sort[ Map[ Total, Union[Subsets[s, {2}],
      Table[{s[[i]], s[[i]]}, {i, 1, Length[s]}]]]]];
a = Table[0, lim];
i=1; While[c[[i]] [[1]]<=lim, a[[c[[i]] [[1]]]]=c[[i]] [[2]]; i++];
a (* Robert Price, Mar 30 2019 *)

a(n)=sum(i=1, n, sum(j=1, i, if(abs(bigomega(i)-2) + abs(bigomega(j)-2) + abs(n-i-j),0,1)))

a(n)=my(s); forprime(p=2,n\4, forprime(q=2,min(n\(2*p),p), if(bigomega(n-p*q)==2, s++))); s \\ Charles R Greathouse IV, Dec 07 2014

From Reinhard Zumkeller, Jan 21 2010: (Start)
a(A100592(n)) = n;
a(m) < n for m < A100592(n);
A171963(n) = a(A001358(n)). (End)
a(n) = Sum_{i=1..floor(n/2)} [Omega(i) == 2] * [Omega(n-i) == 2], where Omega = A001222 and [] is the Iverson Bracket. - Wesley Ivan Hurt, Apr 04 2018
a(n) = [x^n y^2] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 21 2021

A344257 Number of partitions of n into 10 semiprime parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 4, 3, 5, 3, 5, 5, 8, 8, 10, 8, 13, 13, 18, 17, 20, 19, 25, 28, 33, 33, 38, 40, 50, 52, 59, 63, 71, 75, 86, 94, 105, 110, 124, 131, 150, 159, 174, 189, 205, 217, 242, 264, 288, 303, 327, 354, 388, 414, 443, 476, 511, 547, 594, 641

Offset: 40

Wesley Ivan Hurt, May 13 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 40..10000
Index entries for sequences related to partitions

Cf. A001222 (Omega), A001358.

Column k=10 of A344447.

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; series(`if`(n=0, 1, `if`(i<1, 0,
     `if`(i>n, 0, x*b(n-i, h(min(n-i, i))))+b(n, h(i-1)))), x, 11)
    end:
a:= n-> coeff(b(n, h(n)), x, 10):
seq(a(n), n=40..120);  # Alois P. Heinz, May 26 2021

h[n_] := h[n] = If[n == 0, 0, If[PrimeOmega[n] == 2, n, h[n-1]]];
b[n_, i_] := b[n, i] = Series[If[n == 0, 1, If[i < 1, 0, If[i > n, 0, x*b[n-i, h[Min[n-i, i]]]]+b[n, h[i-1]]]], {x, 0, 11}];
a[n_] := SeriesCoefficient[b[n, h[n]], {x, 0, 10}];
Table[a[n], {n, 40, 120}] (* Jean-François Alcover, May 15 2025, after Alois P. Heinz *)

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} [Omega(r) = Omega(q) = Omega(p) = Omega(o) = Omega(m) = Omega(l) = Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k-l-m-o-p-q-r) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.

a(n) = [x^n y^10] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 19 2021

a(83)-a(103) from Alois P. Heinz, May 18 2021

A340756 Number of partitions of n into 4 semiprime parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 3, 3, 4, 2, 3, 4, 5, 6, 5, 4, 7, 7, 9, 9, 9, 7, 9, 12, 13, 11, 11, 13, 16, 17, 17, 18, 18, 17, 20, 25, 25, 23, 24, 26, 32, 29, 31, 33, 31, 33, 35, 43, 43, 40, 39, 45, 48, 52, 50, 52, 53, 52, 61, 69, 67, 61, 61, 70, 79, 76, 76, 80, 81, 85, 88, 101

Offset: 16

Wesley Ivan Hurt, Jan 19 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 16..10000
Index entries for sequences related to partitions

Cf. A001222 (Omega), A001358.

Column k=4 of A344447.

Table[Sum[Sum[Sum[KroneckerDelta[PrimeOmega[k], PrimeOmega[j], PrimeOmega[i], PrimeOmega[n - i - j - k], 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 16, 100}]
Table[Count[IntegerPartitions[n,{4}],?(PrimeOmega[#]=={2,2,2,2}&)],{n,16,95}] (* _Harvey P. Dale, May 14 2022 *)

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} [Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.

a(n) = [x^n y^4] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 21 2021

A281617 Expansion of Sum_{i = pq, p prime, q prime} x^i/(1 - x^i) / Product_{j = pq, p prime, q prime} (1 - x^j).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 3, 0, 5, 2, 6, 3, 9, 3, 14, 7, 16, 10, 23, 12, 32, 20, 37, 28, 52, 35, 69, 49, 80, 68, 110, 83, 137, 112, 166, 150, 215, 178, 268, 239, 324, 303, 406, 365, 504, 472, 604, 584, 747, 708, 917, 888, 1089, 1085, 1337, 1311, 1618, 1606, 1916, 1954, 2332, 2334, 2782, 2829, 3300, 3407, 3963

Offset: 0

Ilya Gutkovskiy, Jan 25 2017

nonn

Total number of parts in all partitions of n into semiprimes (A001358).

Convolution of A086971 and A101048.

			a(12) = 5 because we have [6, 6], [4, 4, 4] and 2 + 3 = 5.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A001358, A084993, A086971, A101048, A344447.

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, 0], `if`(i<1, [0$2],
     `if`(i>n, 0, (p-> p+[0, p[1]])(b(n-i, h(min(n-i, i)))))+b(n, h(i-1))))
    end:
a:= n-> b(n, h(n))[2]:
seq(a(n), n=0..70);  # Alois P. Heinz, May 19 2021

nmax = 70; Rest[CoefficientList[Series[Sum[Floor[PrimeOmega[i]/2] Floor[2/PrimeOmega[i]] x^i/(1 - x^i), {i, 2, nmax}]/Product[1 - Floor[PrimeOmega[j]/2] Floor[2/PrimeOmega[j]] x^j, {j, 2, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{i = p*q, p prime, q prime} x^i/(1 - x^i) / Product_{j = p*q, p prime, q prime} (1 - x^j).

a(n) = Sum_{k>0} k * A344447(n,k). - Alois P. Heinz, May 19 2021

A344245 Number of partitions of n into 5 semiprime parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 4, 3, 4, 3, 4, 4, 6, 7, 7, 6, 8, 9, 13, 11, 11, 12, 15, 16, 18, 18, 19, 19, 23, 26, 28, 27, 27, 32, 36, 37, 39, 42, 45, 44, 51, 55, 58, 55, 57, 66, 71, 75, 76, 82, 84, 87, 93, 104, 103, 103, 105, 119, 131, 130, 134, 141, 145, 151, 163, 173, 176, 173

Offset: 20

Wesley Ivan Hurt, May 12 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 20..10000
Index entries for sequences related to partitions

Cf. A001222 (Omega), A001358 (semiprimes).
Cf. A340756.
Column k=5 of A344447.

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-1)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} [Omega(l) = Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k-l) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.

a(n) = [x^n y^5] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 21 2021

A344246 Number of partitions of n into 6 semiprime parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 4, 3, 5, 3, 4, 5, 7, 7, 8, 7, 10, 11, 14, 13, 15, 14, 17, 21, 24, 22, 25, 27, 32, 33, 36, 38, 41, 43, 47, 54, 58, 57, 63, 68, 77, 78, 83, 89, 94, 97, 106, 118, 123, 125, 131, 146, 156, 162, 166, 179, 187, 198, 211, 226, 236, 236, 251, 274, 290, 296

Offset: 24

Wesley Ivan Hurt, May 12 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 24..10000
Index entries for sequences related to partitions

Cf. A001222 (Omega), A001358.

Column k=6 of A344447.

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} [Omega(m) = Omega(l) = Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k-l-m) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.

a(n) = [x^n y^6] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 21 2021

A344254 Number of partitions of n into 7 semiprime parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 4, 3, 5, 3, 5, 5, 7, 8, 9, 7, 11, 12, 16, 15, 16, 16, 21, 23, 26, 27, 31, 31, 38, 41, 45, 46, 50, 55, 62, 66, 71, 77, 85, 85, 97, 105, 113, 117, 124, 136, 149, 156, 167, 179, 189, 199, 214, 235

Offset: 28

Wesley Ivan Hurt, May 12 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 28..10000
Index entries for sequences related to partitions

Cf. A001222 (Omega), A001358.

Column k=7 of A344447.

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} [Omega(o) = Omega(m) = Omega(l) = Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k-l-m-o) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.

a(n) = [x^n y^7] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 21 2021

A344255 Number of partitions of n into 8 semiprime parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 4, 3, 5, 3, 5, 5, 8, 8, 9, 8, 12, 12, 17, 16, 18, 18, 22, 25, 30, 29, 33, 36, 44, 45, 51, 54, 59, 63, 71, 78, 87, 90, 99, 106, 120, 124, 136, 147, 157, 166, 182, 199, 216, 223, 238, 259, 280, 298, 314

Offset: 32

Wesley Ivan Hurt, May 12 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 32..10000
Index entries for sequences related to partitions

Cf. A001222 (Omega), A001358.

Column k=8 of A344447.

h[n_] := h[n] = If[n == 0, 0, If[PrimeOmega[n] == 2, n, h[n-1]]];
b[n_, i_] := b[n, i] = Series[If[n == 0, 1, If[i < 1, 0, If[i > n, 0, x*b[n-i, h[Min[n-i, i]]]]+b[n, h[i-1]]]], {x, 0, 9}];
a[n_] := SeriesCoefficient[b[n, h[n]], {x, 0, 8}];
Table[a[n], {n, 32, 120}] (* Jean-François Alcover, May 15 2025, after Alois P. Heinz in A344257 *)

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} [Omega(p) = Omega(o) = Omega(m) = Omega(l) = Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k-l-m-o-p) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.

a(n) = [x^n y^8] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 21 2021

A344256 Number of partitions of n into 9 semiprime parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 4, 3, 5, 3, 5, 5, 8, 8, 10, 8, 12, 13, 18, 16, 19, 19, 24, 27, 31, 31, 37, 38, 46, 50, 57, 58, 65, 71, 80, 86, 96, 102, 115, 119, 134, 146, 160, 167, 181, 197, 217, 232, 252, 269, 290, 306, 333, 364, 387, 407, 434, 474, 512, 541

Offset: 36

Wesley Ivan Hurt, May 13 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 36..10000
Index entries for sequences related to partitions

Cf. A001222 (Omega), A001358.

Column k=9 of A344447.

h[n_] := h[n] = If[n == 0, 0,If[PrimeOmega[n] == 2, n, h[n-1]]];
b[n_, i_] := b[n, i] = Series[If[n == 0, 1, If[i < 1, 0, If[i > n, 0, x*b[n-i, h[Min[n-i, i]]]]+b[n, h[i-1]]]], {x, 0, 10}];
a[n_] := SeriesCoefficient[b[n, h[n]], {x, 0, 9}];
Table[a[n], {n, 36, 120}] (* Jean-François Alcover, May 15 2025, after Alois P. Heinz in A344257 *)

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} [Omega(q) = Omega(p) = Omega(o) = Omega(m) = Omega(l) = Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k-l-m-o-p-q) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.

a(n) = [x^n y^9] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 21 2021

More terms from Alois P. Heinz, May 18 2021

cp's OEIS Frontend

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

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A072931 Number of ways to write n as a sum of 2 semiprimes.

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A344257 Number of partitions of n into 10 semiprime parts.

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A340756 Number of partitions of n into 4 semiprime parts.

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A281617 Expansion of Sum_{i = p*q, p prime, q prime} x^i/(1 - x^i) / Product_{j = p*q, p prime, q prime} (1 - x^j).

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A344245 Number of partitions of n into 5 semiprime parts.

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A344246 Number of partitions of n into 6 semiprime parts.

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A344254 Number of partitions of n into 7 semiprime parts.

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A281617 Expansion of Sum_{i = pq, p prime, q prime} x^i/(1 - x^i) / Product_{j = pq, p prime, q prime} (1 - x^j).