A112020 -id:A112020 - 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

A322353 Number of factorizations of n into distinct semiprimes; a(1) = 1 by convention.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 1

Antti Karttunen, Dec 06 2018

nonn

A semiprime (A001358) is a product of any two prime numbers. In the even case, these factorizations have A001222(n)/2 factors. - Gus Wiseman, Dec 31 2020

Records 1, 2, 3, 4, 5, 9, 13, 15, 17, ... occur at 1, 60, 210, 840, 1260, 4620, 27720, 30030, 69300, ...

			a(4) = 1, as there is just one way to factor 4 into distinct semiprimes, namely as {4}.
From _Gus Wiseman_, Dec 31 2020: (Start)
The a(n) factorizations for n = 60, 210, 840, 1260, 4620, 12600, 18480:
  4*15   6*35    4*6*35    4*9*35    4*15*77    4*6*15*35    4*6*10*77
  6*10   10*21   4*10*21   4*15*21   4*21*55    4*6*21*25    4*6*14*55
         14*15   4*14*15   6*10*21   4*33*35    4*9*10*35    4*6*22*35
                 6*10*14   6*14*15   6*10*77    4*9*14*25    4*10*14*33
                           9*10*14   6*14*55    4*10*15*21   4*10*21*22
                                     6*22*35    6*10*14*15   4*14*15*22
                                     10*14*33                6*10*14*22
                                     10*21*22
                                     14*15*22
(End)

Unlabeled multiset partitions of this type are counted by A007717.
The version for partitions is A112020, or A101048 without distinctness.
The non-strict version is A320655.
Positions of zeros include A320892.
Positions of nonzero terms are A320912.
The case of squarefree factors is A339661, or A320656 without distinctness.
Allowing prime factors gives A339839, or A320732 without distinctness.
A322661 counts loop-graphs, ranked by A320461.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A027187 counts partitions of even length, ranked by A028260.
A037143 lists primes and semiprimes.
A338898/A338912/A338913 give the prime indices of semiprimes.
A339846 counts even-length factorizations, with ordered version A174725.
Cf. A001221, A006125, A006129, A028260, A320893, A338915, A339841.

Table[Count[Subsets[Select[Divisors[n], PrimeOmega[#] == 2 &]], ?(Times @@ # == n &)], {n, 105}] (* _Michael De Vlieger, Dec 11 2020 *)

A322353(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((2==bigomega(d)&&(d<=m)), s += A322353(n/d, d-1))); (s)); \\ Antti Karttunen, Dec 10 2020

a(n) = Sum_{d|n} (-1)^A001222(d) * A339839(n/d). - Gus Wiseman, Dec 31 2020

A186337 Smallest integer that can be expressed as the sum of one or more consecutive semiprimes in exactly n ways.

Original entry on oeis.org

4, 10, 39, 185, 417, 589, 8351, 40181, 50809, 69195, 2614377, 4044299, 27691722, 110046335, 473061053

Offset: 1

Alois P. Heinz, Feb 18 2011

			a(1) =   4 = A001358(1) is the first semiprime.
a(2) =  10 = A001358(1)+A001358(2) = 4+6 = A001358(4) = 10.
a(3) =  39 = 6 + 9 + 10 + 14 = 10 + 14 + 15 = 39.
a(4) = 185 = 58 + ... + 65 = 39 + ... + 51 = 4 + ... + 33 = 185.

Cf. A001358, A112020, A186336.

a(11)-a(15) from Donovan Johnson, Feb 21 2011

A112022 Number of partitions of n into distinct Chen primes.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 5, 5, 6, 5, 6, 7, 6, 9, 7, 9, 9, 9, 11, 11, 11, 13, 12, 14, 15, 14, 17, 15, 17, 19, 18, 21, 21, 21, 24, 24, 26, 28, 27, 30, 30, 32, 35, 34, 37, 37, 39, 41, 43, 45, 46, 48, 51, 53, 56, 58, 59, 61, 64, 66, 70, 71, 73

Offset: 0

Reinhard Zumkeller, Aug 26 2005

nonn

a(n) = A000586(n) for n <= 42.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A112021, A112020, A109611.

terms = 81;
gf = Times @@ (1 + x^SequencePosition[ PrimeOmega[ Range[terms]], {1, _, 1|2}][[All, 1]]) + O[x]^terms;
CoefficientList[gf, x] (* Jean-François Alcover, Jul 02 2018 *)

P=1+O(x^1001); forprime(p=2,1e3,if(bigomega(p+2)<3,P*=1+x^p)); Vec(P) \\ Charles R Greathouse IV, May 13 2013

G.f.: Product_{k>=1} (1 + x^A109611(k)). - Andrew Howroyd, Dec 28 2017

A186336 Number of ways of representing n as the sum of one or more consecutive semiprimes.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 0, 2, 0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 0, 0, 0, 2, 1, 1, 1, 0, 1, 3, 0, 0, 0, 2, 0, 0, 1, 1, 1, 1, 1, 2, 0, 0, 1, 1, 0, 1, 3, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 3, 0, 0, 1, 2, 1, 1, 0, 2, 0, 1, 0, 0, 2, 1, 1, 2, 1, 1, 0, 0, 0, 2, 0, 2, 2, 2, 0, 2, 0, 0, 1, 1, 1, 0, 0, 0, 3, 2, 0, 1, 0, 1, 2, 0, 0, 2, 1, 0, 2, 1, 1

Offset: 0

Alois P. Heinz, Feb 18 2011

nonn

			a(4)  = 1:  4 = A001358(1) is the first semiprime.
a(10) = 2: 10 = A001358(1)+A001358(2) = 4+6 = A001358(4) = 10.
a(39) = 3: 39 = 6+9+10+14 = 10+14+15 = 39.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A001358, A112020, A186337.

a186336 n = f $ takeWhile (<= n) a001358_list where
   f []       = 0
   f (sp:sps) = g sp sps + f sps
   g spSum []                    = fromEnum (spSum == n)
   g spSum (sp:sps) | spSum < n  = g (sp + spSum) sps
                    | spSum == n = 1
                    | otherwise  = 0
-- Reinhard Zumkeller, Feb 28 2011

b:= proc(n) option remember; local k;
      if n=0 then 0
    else for k from b(n-1)+1
           while isprime(k) or 2<>add(i[2], i=ifactors(k)[2])
         do od; k
      fi
    end:
pis:= proc(n) option remember; local k;
        if n<4 then 0
      elif n=4 then 1
      else k:= pis(n-1);
           k +`if`(b(k+1)=n, 1 ,0)
        fi
      end:
ssp:= proc(i,j) option remember;
        b(j) + `if`(i=j, 0, ssp(i, j-1))
      end:
a:= proc(n) option remember; local i, j, cnt, s;
      cnt:= 0;
      j:= pis(n);
      i:= j;
      while i>0 do
        s:= ssp(i,j);
        if sn then j:= j-1
      else cnt:= cnt+1;
           i, j:= i-1, j-1
        fi
      od; cnt
    end:
seq(a(n), n=0..200);

nmax = 120;
sp = Select[Range[nmax], PrimeOmega[#] == 2&];
lsp = Length[sp]; Clear[a]; a[_] = 0;
Do[n = Total[sp[[i ;; j]]]; a[n] = a[n]+1, {i, 1, lsp}, {j, i, lsp}];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Mar 13 2019 *)

A280912 Number of partitions of n into odd semiprimes (A046315).

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, 1, 0, 0, 2, 0, 0, 2, 1, 1, 2, 0, 0, 3, 1, 0, 3, 1, 1, 3, 1, 0, 4, 2, 2, 5, 1, 1, 5, 3, 1, 6, 3, 2, 8, 2, 1, 7, 5, 4, 9, 4, 3, 11, 6, 3, 11, 6, 6, 14, 7, 5, 15, 9, 7, 16, 9, 8, 20, 14, 9, 21, 13, 11, 26, 16, 12, 28, 19, 17, 29, 19, 17, 37, 27

Offset: 0

Ilya Gutkovskiy, Jan 10 2017

nonn

			a(39) = 3 because we have [39], [21, 9, 9] and [15, 15, 9].

Amiram Eldar, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Semiprime
Index entries for related partition-counting sequences

Cf. A001222, A001358, A046315, A099773, A101048, A112020.

nmax = 100; CoefficientList[Series[Product[1/(1 - Floor[PrimeOmega[2 k + 1]/2] Floor[2/PrimeOmega[2 k + 1]] x^(2 k + 1)), {k, 1, nmax}], {x, 0, nmax}], x]
Join[{1},Table[Count[IntegerPartitions[n],?(AllTrue[#,OddQ]&&Union[PrimeOmega[#]]=={2}&)],{n,110}]] (* _Harvey P. Dale, Nov 11 2024 *)

G.f.: Product_{k>=1} 1/(1 - floor(bigomega(2*k+1)/2)*floor(2/bigomega(2*k+ 1))*x^(2*k+1)), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).

A357236 Number of compositions (ordered partitions) of n into distinct semiprimes.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 3, 0, 0, 2, 3, 3, 2, 0, 2, 10, 8, 3, 1, 8, 10, 17, 3, 8, 14, 40, 16, 18, 10, 37, 63, 55, 24, 40, 45, 79, 84, 82, 70, 170, 228, 166, 135, 86, 232, 295, 334, 309, 292, 228, 604, 719, 600, 383, 1265, 904, 1020, 840, 867, 1008, 1864, 2569, 2154, 1676, 2414, 3541, 3958

Offset: 0

Ilya Gutkovskiy, Sep 19 2022

nonn

Index entries for sequences related to compositions

Cf. A001358, A101048, A112020, A280238.

cp's OEIS Frontend

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

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A322353 Number of factorizations of n into distinct semiprimes; a(1) = 1 by convention.

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A186337 Smallest integer that can be expressed as the sum of one or more consecutive semiprimes in exactly n ways.

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Extensions

A112022 Number of partitions of n into distinct Chen primes.

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Mathematica

PARI

Formula

A186336 Number of ways of representing n as the sum of one or more consecutive semiprimes.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

A280912 Number of partitions of n into odd semiprimes (A046315).

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Examples

Links

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Programs

Mathematica

Formula

A357236 Number of compositions (ordered partitions) of n into distinct semiprimes.

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