A061358 -id:A061358 - cp's OEIS frontend

A083338 Number of partitions of odd numbers into three primes and of even numbers into two primes.

0, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 2, 4, 2, 3, 2, 5, 3, 5, 3, 5, 3, 7, 2, 7, 3, 6, 2, 9, 4, 8, 4, 9, 2, 10, 3, 11, 4, 10, 3, 12, 4, 13, 5, 12, 4, 15, 3, 16, 5, 14, 3, 17, 4, 16, 6, 16, 3, 19, 5, 21, 6, 20, 2, 20, 5, 22, 6, 21, 5, 22, 5, 28, 7, 24, 4, 25, 5, 29, 8, 27, 5, 29, 4, 33, 9, 29, 4

Offset: 1

Reinhard Zumkeller, Apr 24 2003

a(n) > 0 for all n iff Goldbach's conjectures hold.

Antti Karttunen, Table of n, a(n) for n = 1..1001
Antti Karttunen, Scheme-program for computing A083338 and A083339
Eric Weisstein's World of Mathematics, Goldbach Conjecture.
Index entries for sequences related to Goldbach conjecture

Cf. A000607, A061358, A068307, A071335, A083339.

f[n_] := Length@ IntegerPartitions[n, If[ OddQ@ n, {3}, {2}], Prime@ Range@ PrimePi@ n]; Array[f, 92] (* Robert G. Wilson v, Nov 28 2012 *)

a(n) = if n is even then A045917(n/2) else A054860((n-1)/2).

For even n: a(n) = A061358(n); for odd n: a(n) = A068307(n). - Antti Karttunen, Sep 14 2017

A137791 Number of ways to write n as sum of two positive numbers having no prime gaps in their factorization.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 7, 8, 7, 7, 6, 8, 7, 8, 7, 8, 7, 9, 9, 10, 9, 10, 9, 11, 9, 11, 9, 10, 9, 11, 10, 10, 8, 10, 10, 13, 11, 12, 9, 11, 10, 14, 11, 12, 8, 10, 10, 14, 12, 13, 9, 12, 12, 14, 11, 12, 8, 11, 10, 16, 12, 11, 10, 14, 14, 16, 14, 14, 9, 13, 14, 18

Offset: 1

Reinhard Zumkeller, Feb 11 2008

nonn

			a(20) = #{1+19,2+18,3+17,4+16,5+15,7+13,8+12,9+11} = 8;
a(21) = #{2+19,3+18,4+17,5+16,6+15,8+13,9+12} = 7;
a(22) = #{3+19,4+18,5+17,6+16,7+15,9+13,11+11} = 7;
a(23) = #{4+19,5+18,6+17,7+16,8+15,11+12} = 6;
a(24) = #{1+23,5+19,6+18,7+17,8+16,9+15,11+13,12+12} = 8.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A073491, A002375, A061358, A071330, A137792, A137793.

A180007 Number of Goldbach partitions of 6^n.

Original entry on oeis.org

1, 4, 13, 49, 161, 656, 2751, 12505, 58482, 280348, 1374563, 6864809

Offset: 1

Jonathan Vos Post, Aug 06 2010

nonn

Number of ways of writing 6^n as the sum of two odd primes, when the order does not matter. Number of ways writing 6^n as unordered sums of 2 primes. This is to 6 as A006307 is to 2 and as A065577 is to 10. This is the 6th row of the array A[k,n] = Number of ways writing k^n as unordered sums of 2 primes.

A061358(4^n) starts 1, 2, 5, 8, 22, 53, 151, 435, for n=1,2,... (bisection of A006307). A061358(8^n) starts 1, 5, 11, 53, 244, 1314, 7471, (tri-section of A006307). A061358(10^n) = A065577(n). A061358(12^n) = 1, 11, 53, 348, 2523, 20564... A061358(14^n) = 2, 9, 50, 330, 2924, 27225,... - R. J. Mathar, Aug 07 2010

			a(1) = 1 because 6^1 = 6 = 3+3.
a(2) = 4 because 6^2 = 36 = 5+31 = 7+29 = 13+23 = 17+19.
a(3) = 13 because 6^3 = 216 = 5+211 = 17+199 = 19+197 = 23+193 = 37+179 = 43+173 = 53+163 = 59+157 = 67+149 = 79+137 = 89+127 = 103+113 = 107+109.

Manfred Scheucher, Sage Script

Cf. A000400, A001031, A061358, A065577, A195295.

A061358 := proc(n) local a,p ; a := 0 ; p := nextprime(floor((n-1)/2)) ; while p <= n do if isprime(n-p) then a := a+1 ; end if; p := nextprime(p) ; end do ; return a; end proc:
A180007 := proc(n) A061358(6^n) ; end proc:
for n from 1 do printf("%d,\n",A180007(n)) ; end do:
# R. J. Mathar, Aug 07 2010

Table[Count[Sort@ IntegerPartitions[6^n, {2}], {u_, v_} /; And[PrimeQ@ u, u != 2, PrimeQ@ v]], {n, 6}] (* Michael De Vlieger, Jun 02 2015 *)

a(n)=my(t=6^n,s); forprime(p=2,t\2, if(isprime(t-p), s++)); s \\ Charles R Greathouse IV, Jun 02 2015

a(n) = A061358(6^n) = A061358(A000400(n)).

a(5) corrected, 4 terms added by R. J. Mathar, Aug 07 2010

a(10)-a(12) from Manfred Scheucher, Jun 01 2015

A307726 Number of partitions of n into 2 prime powers (not including 1).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 2, 4, 3, 4, 4, 4, 2, 4, 3, 4, 4, 4, 3, 5, 3, 6, 4, 7, 4, 7, 2, 5, 4, 6, 3, 5, 3, 5, 5, 6, 2, 7, 3, 7, 4, 6, 2, 8, 3, 7, 4, 6, 2, 7, 3, 6, 4, 7, 2, 9, 2, 7, 5, 7, 2, 9, 3, 7, 6, 7, 3, 9, 2, 8, 4, 6, 4, 10, 3, 9, 4, 7, 3, 11, 4, 8, 3, 7, 2, 10, 2, 8, 3, 8

Offset: 0

Ilya Gutkovskiy, Apr 24 2019

			a(10) = 3 because we have [8, 2], [7, 3] and [5, 5].

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions

Cf. A000961, A023894, A061358, A071068, A071330, A071331, A246655, A280242, A307727.

# note that this requires A246655 to be pre-computed
f:= proc(n, k, pmax) option remember;
  local t, p, j;
  if n = 0 then return `if`(k=0, 1, 0) fi;
  if k = 0 then return 0 fi;
  if n > k*pmax then return 0 fi;
  t:= 0:
  for p in A246655 do
    if p > pmax then return t fi;
    t:= t + add(procname(n-j*p, k-j, min(p-1, n-j*p)), j=1..min(k, floor(n/p)))
  od;
  t
end proc:
map(f, [$0..100]); # Robert Israel, Apr 29 2019

Array[Count[IntegerPartitions[#, {2}], _?(AllTrue[#, PrimePowerQ] &)] &, 101, 0]

a(n) = [x^n y^2] Product_{k>=1} 1/(1 - y*x^A246655(k)).

A347552 Number of partitions of n into at most 2 prime parts.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

nonn

Cf. A000040, A000607, A061358, A071335, A117278, A347578, A347609, A347610.

a(n) = Sum_{k=1..2} A117278(n,k) for n >= 2. - Alois P. Heinz, Sep 08 2021

A113631 Number of distinct representations of (2n)^2 as the sum of two primes.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 11, 9, 8, 20, 14, 14, 26, 17, 18, 48, 22, 22, 49, 28, 36, 69, 33, 37, 68, 47, 43, 83, 49, 47, 125, 50, 53, 118, 56, 94, 126, 63, 63, 153, 98, 71, 186, 79, 94, 230, 89, 91, 197, 127, 127, 215, 112, 105, 220, 172, 147

Offset: 0

Jonathan Vos Post, Mar 31 2006

From Halberstam and Richert: A045917(2n)<(8+0(1))*c(n)*n/log(n)^2 where c(n)=prod(p>2,(1-1/(p-1)^2))*prod(p|n,p>2,(p-1)/(p-2)). Hence a(n) = A045917(2n) < (8+0(1))*c(2n)*2n/log(2n)^2 where c(k)=prod(p>2,(1-1/(p-1)^2))*prod(p|k,p>2,(p-1)/(p-2)). See also: A045917 From Goldbach problem: number of decompositions of 2n into unordered sums of two primes. A016742 Even squares: (2n)^2.

a(n)=A061358(4n^2). - Emeric Deutsch, Apr 03 2006

			a(1) = 1 because (2*1)^2 = 4 = 2 + 2 uniquely.
a(2) = 2 because (2*2)^2 = 16 = 3 + 13 = 5 + 11.
a(3) = 4 because (2*3)^2 = 36 = 5 + 31 = 7 + 29 = 13 + 23 = 17 + 19.
a(4) = 5 because (2*4)^2 = 64 = 3 + 61 = 5 + 59 = 11 + 53 = 17 + 47 = 23 + 41.
a(5) = 6 because (2*5)^2 = 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53.
a(6) = 11 because (2*6)^2 = 144 = 5 + 139 = 7 + 137 = 13 + 131 = 17 + 127 = 31 + 113 = 37 + 107 = 41 + 103 = 43 + 101 = 47 + 97 = 61 + 83 = 71 + 73.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
H. Halberstam and H. E. Richert, 1974, "Sieve methods", Academic press, London, New York, San Francisco.

Marius A. Burtea, Table of n, a(n) for n = 0..500

Cf. A000040, A045917, A016742.

Cf. A061358.

[#RestrictedPartitions(4*n^2,2,{p:p in PrimesUpTo(20000)}):n in [0..56] ] // Marius A. Burtea, Jan 19 2019

g:=sum(sum(x^(ithprime(i)+ithprime(j)),i=1..j),j=1..1500): gser:=series(g,x=0,12560): 0,seq(coeff(gser,x^(4*n^2)),n=1..56); # Emeric Deutsch, Apr 03 2006

a(n) = A045917(2n). a(n) = #{p(i) + p(j) = A016742(n) for p(k) = A000040(k) and i >= j}. a(n) = #{p(i) + p(j) = (2*n)^2 for p(k) = A000040(k) and i >= j}.

Corrected and extended by Emeric Deutsch, Apr 03 2006

A156642 Number of decompositions of 4n+2 into unordered sums of two primes of the form 4k+3.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 1, 3, 3, 3, 3, 4, 3, 4, 6, 3, 2, 4, 3, 4, 5, 3, 2, 5, 4, 4, 5, 4, 4, 7, 4, 4, 5, 3, 6, 7, 3, 5, 7, 4, 4, 7, 4, 5, 10, 5, 4, 7, 3, 7, 9, 5, 6, 8, 5, 5, 9, 5, 5, 11, 6, 5, 9, 5, 6, 10, 5, 6, 8, 6, 6, 9, 5, 5, 12, 6, 5, 9

Offset: 0

Vladimir Shevelev, Feb 12 2009

nonn

Conjecture. For n >= 1, a(n) > 0. This conjecture does not follow from the validity of the Goldbach binary conjecture because numbers of the form 4n+2, generally speaking, also have decompositions into sums of two primes of the form 4k+1.

			From _Lei Zhou_, Mar 19 2013: (Start)
n=1: 4n+2=6, 6=3+3; this is the only case that matches the definition, so a(1)=1;
n=3: 4n+2=14, 14=3+11=7+7; two instances found, so a(3)=2. (End)

Lei Zhou, Table of n, a(n) for n = 0..10000

Cf. A002145, A002375, A061358, A002372, A045917, A214834, A217696.

Table[m = 4*n + 2; p1 = m + 1; ct = 0; While[p1 = p1 - 4; p2 = m - p1; p1 >= p2, If[PrimeQ[p1] && PrimeQ[p2], ct++]]; ct, {n, 1, 100}] (* Lei Zhou, Mar 19 2013 *)

A002126 Number of solutions to n=p+q where p and q are primes or zero.

Original entry on oeis.org

1, 0, 2, 2, 1, 4, 1, 4, 2, 2, 3, 2, 2, 4, 3, 2, 4, 2, 4, 4, 4, 2, 5, 2, 6, 2, 5, 0, 4, 2, 6, 4, 4, 2, 7, 0, 8, 2, 3, 2, 6, 2, 8, 4, 6, 2, 7, 2, 10, 2, 8, 0, 6, 2, 10, 2, 6, 0, 7, 2, 12, 4, 5, 2, 10, 0, 12, 2, 4, 2, 10, 2, 12, 4, 9, 2, 10, 0, 14, 2, 8, 2, 9, 2, 16, 2, 9, 0, 8, 2, 18, 2, 8, 0, 9, 0, 14

Offset: 0

N. J. A. Sloane

nonn

Arises in studying the Goldbach conjecture.

P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, Vol. II, pp. 354-382] [The sequence N_{n,2}]
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).

T. D. Noe, Table of n, a(n) for n = 0..10000
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. 354-380.

Cf. A002375, A045917, A061358, A073610

(a(n) = sum(k=0, n, zp(k)*zp(n-k))); {zp(n) = if( n==0, 1, isprime(n))}; /* Michael Somos, Jul 26 1999 */

G.f.: (1 + Sum_i x^prime(i))^2. [Corrected by T. D. Noe, Dec 05 2006]

a(54) corrected by Paul Zimmermann, Mar 15 1996

Better description from Michael Somos, Jul 26 1999

A062302 Number of ways writing n-th prime as a sum of a prime and a nonprime.

Original entry on oeis.org

0, 1, 0, 1, 4, 3, 6, 5, 8, 9, 8, 11, 12, 11, 14, 15, 16, 15, 18, 19, 18, 21, 22, 23, 24, 25, 24, 27, 26, 29, 30, 31, 32, 31, 34, 33, 36, 37, 38, 39, 40, 39, 42, 41, 44, 43, 46, 47, 48, 47, 50, 51, 50, 53, 54, 55, 56, 55, 58, 59, 58, 61, 62, 63, 62, 65, 66, 67, 68, 67, 70, 71, 72

Offset: 1

Labos Elemer, Jul 05 2001

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A061358, A062602, A000040, A014092, A025584.

Table[c = 0; Do[i = Prime[k]; If[i + j == Prime[n] && ! PrimeQ[j], c = c + 1], {k, n - 1}, {j, Prime[n] - 1}]; c, {n, 73}] (* Jayanta Basu, Apr 22 2013 *)
nn = 100; mx = Prime[nn]; ps = Prime[Range[nn]]; notPs = Complement[Range[mx], ps]; t2 = Table[0, {Range[mx]}]; Do[s = i + j; If[s <= mx, t2[[s]]++], {i, ps}, {j, notPs}];  t2[[ps]] (* T. D. Noe, Apr 23 2013 *)

a(n) = A062602(A000040(n)) = number of [nonprime+prime] partitions of prime(n)

A062305 Number of ways writing 2^n as a sum of a prime and a nonprime.

Original entry on oeis.org

0, 0, 1, 2, 2, 7, 8, 25, 38, 75, 128, 259, 458, 876, 1598, 3024, 5672, 10753, 20372, 38656, 73547, 140669, 268537, 514307, 986379, 1896755, 3650109, 7036061, 13580371, 26241380, 50765806, 98317489, 190597373, 369832498, 718266991, 1396138085, 2715823187, 5287080080

Offset: 0

Labos Elemer, Jul 05 2001

nonn

			For n = 5: 2^5 = 32 = 31+1 = 2+30 = 5+27 = 7+25 = 11+21 = 17+15 = 23+9 so a(5) = 7.

Amiram Eldar, Table of n, a(n) for n = 0..40

Cf. A061358, A062602, A062306, A006307, A062610.

a(n) = {my(c = 0, m = 1 << n); forprime(p = 2, m-1, if(!isprime(m - p), c++)); c;} \\ Amiram Eldar, Jul 17 2024

a(n) = A062602(2^n) = number of prime+nonprime partitions of 2^n.

a(n) = 2^(n-1) - A006307(n) - A062306(n) for n >= 1. - Amiram Eldar, Jul 17 2024

More terms from Dean Hickerson, Jul 23 2001
a(28)-a(32) from Sean A. Irvine, Mar 25 2023
a(33)-a(37) from Amiram Eldar, Jul 17 2024

cp's OEIS Frontend

A083338 Number of partitions of odd numbers into three primes and of even numbers into two primes.

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A137791 Number of ways to write n as sum of two positive numbers having no prime gaps in their factorization.

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A180007 Number of Goldbach partitions of 6^n.

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A307726 Number of partitions of n into 2 prime powers (not including 1).

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A347552 Number of partitions of n into at most 2 prime parts.

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A113631 Number of distinct representations of (2n)^2 as the sum of two primes.

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Magma

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A156642 Number of decompositions of 4n+2 into unordered sums of two primes of the form 4k+3.

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A002126 Number of solutions to n=p+q where p and q are primes or zero.

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