A077608 -id:A077608 - cp's OEIS frontend

A002124 Number of compositions of n into a sum of odd primes.

1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 4, 3, 7, 7, 8, 14, 15, 21, 28, 33, 47, 58, 76, 103, 125, 169, 220, 277, 373, 476, 616, 810, 1037, 1361, 1763, 2279, 2984, 3846, 5006, 6521, 8428, 10983, 14249, 18480, 24048, 31178, 40520, 52635, 68281, 88765, 115211, 149593, 194381, 252280, 327696, 425587, 552527, 717721

Offset: 0

N. J. A. Sloane

nonn

Arises in studying the Goldbach conjecture.

The g.f. -(z-1)*(z+1)*(z**2+z+1)*(z**2-z+1)/(1-z**6-z**3-z**5-z**7+z**9) conjectured by Simon Plouffe in his 1992 dissertation is wrong.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..1000
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 300
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, II, pp. 354-382] [The sequence i_n]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for sequences related to Goldbach conjecture

Cf. A002125, A023360, A024939, A077608.

Cf. A065091.

import Data.List (genericIndex)
a002124 n = genericIndex a002124_list n
a002124_list = 1 : f 1 [] a065091_list where
   f x qs ps'@(p:ps)
     | p <= x    = f x (p:qs) ps
     | otherwise = sum (map (a002124 . (x -)) qs) : f (x + 1) qs ps'
-- Reinhard Zumkeller, Mar 21 2014

A002124 := proc(n) coeff(series(1/(1-add(z^numtheory[ithprime](j),j=2..n)),z=0,n+1),z,n) end;
M:=120; a:=array(0..M); a[0]:=1; a[1]:=0; a[2]:=0; for n from 3 to M do t1:=0; for k from 2 to n do p := ithprime(k); if p <= n then t1 := t1 + a[n-p]; fi; od: a[n]:=t1; od: [seq(a[n],n=0..M)]; # N. J. A. Sloane, after MacMahon, Dec 03 2006; used in A002125

a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = (s = 0; p = 3; While[p <= n, s = s + a[n-p]; p = NextPrime[p]]; s); a /@ Range[0, 58] (* Jean-François Alcover, Jun 28 2011, after P. A. MacMahon *)

a(0)=1, a(1)=a(2)=0; for n >= 3, a(n) = Sum_{ primes p with 3 <= p <= n} a(n-p). [MacMahon]

G.f.: 1/( 1 - Sum_{k>=2} x^A000040(k) ). [Joerg Arndt, Sep 30 2012]

Better description and more terms from Philippe Flajolet, Nov 11 2002

Edited by N. J. A. Sloane, Dec 03 2006

A283875 Number of partitions of n into twin primes (A001097).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 6, 7, 7, 9, 9, 11, 12, 13, 15, 16, 19, 20, 23, 25, 27, 31, 33, 37, 40, 44, 49, 52, 59, 63, 69, 76, 81, 90, 96, 106, 114, 123, 135, 144, 157, 169, 183, 197, 212, 230, 246, 266, 286, 307, 330, 353, 381, 406, 436, 468, 499, 536, 572, 613, 654, 698, 746, 795, 849, 904, 964

Offset: 0

Ilya Gutkovskiy, Mar 17 2017

nonn

Conjecture: every number > 7 is the sum of at most 4 twin primes (automatically implies the truth of the first version of the twin prime conjecture). For example: 8 = 5 + 3; 9 = 3 + 3 + 3; 10 = 5 + 5; 11 = 5 + 3 + 3; 12 = 7 + 5, etc.

			a(16) = 4 because we have [13, 3], [11, 5], [7, 3, 3, 3] and [5, 5, 3, 3].

Eric Weisstein's World of Mathematics, Twin Primes
Index entries for related partition-counting sequences

Cf. A000607, A001097, A077608, A129363, A283876.

nmax = 79; CoefficientList[Series[Product[1/(1 - Boole[PrimeQ[k] && (PrimeQ[k - 2] || PrimeQ[k + 2])] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Vec(prod(k=1, 79, 1/(1 - (isprime(k) && (isprime(k - 2) || isprime(k + 2)))*x^k)) + O(x^80)) \\ Indranil Ghosh, Mar 17 2017

G.f.: Product_{k>=1} 1/(1 - x^A001097(k)).

A283876 Number of partitions of n into distinct twin primes (A001097).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 4, 2, 4, 4, 3, 4, 4, 5, 4, 4, 5, 5, 5, 5, 6, 6, 5, 7, 6, 8, 7, 7, 9, 7, 9, 8, 9, 9, 9, 9, 11, 11, 11, 12, 11, 14, 12, 13, 14, 14, 13, 15, 15, 17, 16, 16, 19, 17, 20, 19, 21, 21, 21, 21, 23, 23, 23, 23, 24, 26, 25, 28, 28, 30, 29, 30, 32

Offset: 0

Ilya Gutkovskiy, Mar 17 2017

nonn

			a(29) = 4 because we have [29], [19, 7, 3], [17, 7, 5] and [13, 11, 5].

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Twin Primes
Index entries for related partition-counting sequences

Cf. A000586, A001097, A077608, A129363, A283875.

nmax = 95; CoefficientList[Series[Product[1 + Boole[PrimeQ[k] && (PrimeQ[k - 2] || PrimeQ[k + 2])] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

listA001097(lim)=my(v=List([3]),p=5); forprime(q=7,lim, if(q-p==2, listput(v,p); listput(v,q)); p=q); if(p+2>lim && isprime(p+2), listput(v,p)); Vec(v)
first(n)=my(v=listA001097(n),x=O('x^(n+1))+'x); Vec(prod(i=1,#v, 1+x^v[i]))[1..n+1] \\ Charles R Greathouse IV, Mar 17 2017

Vec(prod(k=1, 95, (1 + (isprime(k) && (isprime(k - 2) || isprime(k + 2)))*x^k)) + O(x^96)) \\ Indranil Ghosh, Mar 17 2017

G.f.: Product_{k>=1} (1 + x^A001097(k)).

A299196 Number of partitions of n into distinct parts that are lesser of twin primes (A001359).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Feb 04 2018

nonn

For n > 0 let b(n) be the inverse Euler transform of a(n). It appears that, if p is the lesser of twin primes, then b(p) = 1 and b(2*p) = -1; otherwise b(n) = 0. - Georg Fischer, Aug 15 2020

			a(46) = 2 because we have [41, 5] and [29, 17].

Robert Israel, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Scatter plot of a(n) - A299197(n)
Eric Weisstein's World of Mathematics, Twin Primes
Index entries for related partition-counting sequences

Cf. A000586, A000607, A001097, A001359, A077608, A129363, A283875, A283876, A299197.

P:= select(isprime,{seq(i,i=3..201,2)}):
TP:= P intersect map(`-`,P,2):
G:= mul(1+x^p,p=TP):
seq(coeff(G,x,i),i=0..200); # Robert Israel, Dec 15 2024

nmax = 105; CoefficientList[Series[Product[1 + Boole[PrimeQ[k] && PrimeQ[k + 2]] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A001359(k)).

A282971 Number of compositions (ordered partitions) of n into primes of form x^2 + y^2 (A002313).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 4, 6, 7, 9, 11, 15, 18, 24, 29, 37, 48, 58, 78, 92, 124, 149, 195, 243, 308, 393, 490, 629, 786, 1004, 1263, 1603, 2024, 2564, 3239, 4106, 5184, 6571, 8301, 10508, 13298, 16807, 21296, 26895, 34082, 43060, 54528, 68952, 87245, 110392, 139622, 176696, 223484, 282798, 357731

Offset: 0

Ilya Gutkovskiy, Feb 25 2017

nonn

Number of compositions (ordered partitions) of n into primes congruent to 1 or 2 mod 4.

Conjecture: every number > 16 is the sum of at most 4 primes of form x^2 + y^2.

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

Index entries for sequences related to compositions

Cf. A002124, A002313, A023360, A077608, A280917, A282906, A282970.

nmax = 60; CoefficientList[Series[1/(1 - Sum[Boole[SquaresR[2, k] != 0 && PrimeQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Vec(1/(1 - sum(k=1, 60, (isprime(k) && k%4<3)*x^k)) + O(x^61)) \\ Indranil Ghosh, Mar 15 2017

G.f.: 1/(1 - Sum_{k>=1} x^A002313(k)).

A299197 Number of partitions of n into distinct parts that are greater of twin primes (A006512).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 2, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 2, 0, 0, 1, 1, 3, 1, 1, 0, 0, 2, 3, 2, 0, 0, 1, 2, 3, 2, 1, 0, 0, 2, 3, 3, 0, 0, 0, 1, 4, 3, 1, 0, 0, 2, 3, 3, 2, 0, 0, 1, 4, 4

Offset: 0

Ilya Gutkovskiy, Feb 04 2018

nonn

			a(31) = 2 because we have [31] and [19, 7, 5].

Eric Weisstein's World of Mathematics, Twin Primes
Index entries for related partition-counting sequences

Cf. A000586, A000607, A001097, A006512, A077608, A129363, A283875, A283876, A299196.

nmax = 105; CoefficientList[Series[Product[1 + Boole[PrimeQ[k] && PrimeQ[k - 2]] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A006512(k)).

A332033 Number of compositions (ordered partitions) of n into distinct twin primes.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 2, 1, 2, 1, 2, 6, 4, 1, 4, 7, 4, 12, 4, 12, 6, 12, 26, 18, 26, 19, 4, 19, 52, 18, 52, 24, 54, 24, 74, 144, 98, 25, 76, 145, 100, 258, 102, 150, 104, 156, 124, 396, 146, 282, 148, 396, 890, 510, 890, 403, 198, 403, 940, 636, 988, 642

Offset: 0

Ilya Gutkovskiy, Feb 05 2020

nonn

			a(15) = 6 because we have [7, 5, 3], [7, 3, 5], [5, 7, 3], [5, 3, 7], [3, 7, 5] and [3, 5, 7].

Eric Weisstein's World of Mathematics, Twin Primes
Index entries for sequences related to compositions

Cf. A001097, A077608, A219107, A283875, A283876, A331926, A331981.

A372510 Number of ordered factorizations of 2*n-1 into twin primes.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, May 04 2024

nonn

			a(23) = 3 because we have 23 * 2 - 1 = 45 = 3 * 3 * 5 = 3 * 5 * 3 = 5 * 3 * 3.

Cf. A001097, A008480, A050328, A050363, A062505, A077608, A164292.

If 2*n-1 = Product A001097(k)^e(k) then a(n) = A008480(2*n-1), otherwise 0.

cp's OEIS Frontend

A002124 Number of compositions of n into a sum of odd primes.

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A283875 Number of partitions of n into twin primes (A001097).

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A283876 Number of partitions of n into distinct twin primes (A001097).

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A299196 Number of partitions of n into distinct parts that are lesser of twin primes (A001359).

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A282971 Number of compositions (ordered partitions) of n into primes of form x^2 + y^2 (A002313).

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A299197 Number of partitions of n into distinct parts that are greater of twin primes (A006512).

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A332033 Number of compositions (ordered partitions) of n into distinct twin primes.

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