A065577 -id:A065577 - cp's OEIS frontend

A061358 Number of ways of writing n = p+q with p, q primes and p >= q.

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

Offset: 0

Amarnath Murthy, Apr 28 2001

For an odd number n, a(n) = 0 if n-2 is not a prime, otherwise a(n) = 1.
For n > 1, a(2n) is at least 1, according to Goldbach's conjecture.
a(A014092(n)) = 0; a(A014091(n)) > 0; a(A067187(n)) = 1. - Reinhard Zumkeller, Nov 22 2004
Number of partitions of n into two primes.
Number of unordered ways of writing n as the sum of two primes.
a(2*n) = A068307(2*n+2). - Reinhard Zumkeller, Aug 08 2009
4*a(n) is the total number of divisors of all primes p and q such that n = p+q and p >= q. - Wesley Ivan Hurt, Mar 05 2016
Indices where a(n) = 0 correspond to A164376 UNION A025584. - Bill McEachen, Jan 31 2024

			a(22) = 3 because 22 can be written as 3+19, 5+17 and 11+11.

a(2n) is A045917.
Cf. A067187, A067188, A067189, A067190, A067191, A063610, A073610, A107318.
Column k=2 of A117278.

[#RestrictedPartitions(n,2,{p:p in PrimesUpTo(1000)}):n in [0..100] ] // Marius A. Burtea, Jan 19 2019

g:=sum(sum(x^(ithprime(i)+ithprime(j)),i=1..j),j=1..30): gser:=series(g,x=0,110): seq(coeff(gser,x,n),n=0..105); # Emeric Deutsch, Apr 03 2006

a[n_] := Length[Select[n - Prime[Range[PrimePi[n/2]]], PrimeQ]]; Table[a[n], {n, 0, 100}] (* Paul Abbott, Jan 11 2005 *)
With[{nn=110},CoefficientList[Series[Sum[x^(Prime[i]+Prime[j]),{j,nn},{i,j}],{x,0,nn}],x]] (* Harvey P. Dale, Aug 17 2017 *)
Table[Count[IntegerPartitions[n,{2}],?(AllTrue[#,PrimeQ]&)],{n,0,110}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jul 03 2021 *)

a(n)=my(s);forprime(q=2,n\2,s+=isprime(n-q));s \\ Charles R Greathouse IV, Mar 21 2013

from sympy import primerange, isprime, floor
def a(n):
    s=0
    for q in primerange(2, n//2 + 1): s+=isprime(n - q)
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 30 2017

G.f.: Sum_{j>0} Sum_{i=1..j} x^(p(i)+p(j)), where p(k) is the k-th prime. - Emeric Deutsch, Apr 03 2006
A065577(n) = a(10^n).
From Wesley Ivan Hurt, Jan 04 2013: (Start)
a(n) = Sum_{i=1..floor(n/2)} A010051(i) * A010051(n-i).
a(n) = Sum_{i=1..floor(n/2)} floor((A010051(i) + A010051(n-i))/2). (End)
a(n) + A062610(n) + A062602(n) = A004526(n). - R. J. Mathar, Sep 10 2021
a(n) = Sum_{k=floor((n-1)^2/4)+1..floor(n^2/4)} c(A339399(2k-1)) * c(A339399(2k)), where c = A010051. - Wesley Ivan Hurt, Jan 19 2022

More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001

Comments edited by Zak Seidov, May 28 2014

A073610 Number of primes of the form n-p where p is a prime.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Aug 05 2002

nonn

a(p) = 2 if p-2 is a prime else a(p) = 0. If n = 2p, p is a prime then a(n) is odd else a(n) is even. As p is counted only once and if q and n-q both are prime then the count is increased by 2. ( Analogous to the fact that perfect squares have odd number of divisors).
a(2k+1) = 2 if (2k-1) is prime, else a(2k+1)=0 (for any k). This sequence can be used to re-describe a couple of conjectures: the Goldbach conjecture == a(2n) > 0 for all n>=2; twin primes conjecture == for any n, there is a prime p>n s.t. a(p)>0.
Number of ordered ways of writing n as the sum of two primes.

			a(16) = 4 as there are 4 primes 3,5,11 and 13 such that 16-3,16-5,16-11and 16-13 are primes.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A061358, A065577, A107318, A098238

for i from 1 to 500 do a[i] := 0:j := 1:while(ithprime(j)

nn=20;a[x]:=Sum[x^i,{i,Table[Prime[n],{n,1,nn}]}];Drop[CoefficientList[a[x]^2,x],1]  (* Geoffrey Critzer, Nov 22 2012 *)

Vec(sum(i=1,100,x^prime(i),O(x^prime(101)))^2) \\ Charles R Greathouse IV, Jan 21 2015

G.f.: (Sum_{k>0} x^prime(k))^2. - Vladeta Jovovic, Mar 12 2005

Self-convolution of characteristic function of primes (A010051). - Graeme McRae, Jul 18 2006

Corrected and extended by Vladeta Jovovic and Sascha Kurz, Aug 06 2002

A107318 Number of ordered Goldbach partitions of 10^n.

Original entry on oeis.org

3, 12, 56, 254, 1620, 10804, 77614, 582800, 4548410, 36400976, 298182320, 2487444740, 21066301710

Offset: 1

N. J. A. Sloane, based on email from R. J. Mathar, Nov 07 2006

nonn

Number of ways of writing 10^n as the sum of two odd primes, when the order matters.

Apart from a(1), equals 2*A065577(n).

See A065577 for further information.

a(13) from Donovan Johnson, Nov 16 2009

A124450 Lesser of a pair of not necessarily distinct closest primes that add up to 10^n.

Original entry on oeis.org

5, 47, 491, 4919, 49877, 499943, 4999913, 49999757, 499999931, 4999999937, 49999999811, 499999999769, 4999999998431, 49999999999619, 499999999999769, 4999999999998557, 49999999999998887, 499999999999999679, 4999999999999999661, 49999999999999998647

Offset: 1

Zak Seidov, Nov 02 2006

nonn

a(n) is always an n digit number.

Note that if distinct primes are required, the only change is that a(1) = 3.

			10^1=5+5; 10^2=47+53; 10^3=491+509;
10^4=4919+5081; 10^5=49877=50123; 10^6=499943+500057;
10^7=4999913+5000087; 10^8=49999757+50000243;
10^9=499999931+500000069;
10^10=4999999937+5000000063; etc.

Hans Havermann, Table of n, a(n) for n = 1..50

Cf. A065577 (number of Goldbach partitions of 10^n).

Cf. A124013.

Table[ h =10^n/2; c=0; While[ PrimeQ[ h-c ]==False || PrimeQ[ h+c ]==False, c++ ]; h-c, {n, 1, 50} ] (* Hans Havermann, Nov 02 2006 *)

10^n - a(n) is prime.

Edited by N. J. A. Sloane May 15 2008 at the suggestion of R. J. Mathar.

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

A124013 Lesser of pair of most widely separated primes whose sum is 10^n.

Original entry on oeis.org

3, 3, 3, 59, 11, 17, 29, 11, 71, 71, 23, 11, 29, 29, 11, 83, 3, 11, 281, 11, 101, 71, 23, 257, 401, 293, 107, 293, 53, 11, 113, 251, 47, 587, 23, 179, 389, 59, 173, 17, 1427, 83, 431, 53, 563, 593, 41, 47, 239, 383, 431, 1181, 701, 971, 149, 593, 569, 149, 191, 1973

Offset: 1

Zak Seidov, Nov 02 2006

nonn

			10^1 = 3 + 7, 10^2 = 3 + 97, 10^3 = 3 + 997, 10^4 = 59 + 9941, 10^5 = 11 + 99989, 10^6 = 17 + 999983, 10^7 = 29 + 9999971, 10^8 = 11 + 99999989, 10^9 = 71 + 999999929, 10^10 = 71 + 9999999929, etc.

Cf. A065577 (Number of Goldbach partitions of 10^n), A124450 (Lesser of pair of closest primes summed to 10^n).

Table[DeleteCases[Map[{#, 10^n - #} &, Prime@ Range@ PrimePi@ Floor[10^n/2]] /. {, k} /; ! PrimeQ@ k -> 0, 0][[1, 1]], {n, 8}] (* or *)
Table[First@ SelectFirst[Map[{#, 10^n - #} &, Prime@ Range@ PrimePi@ Floor[10^n/2]], PrimeQ@ Last@ # &], {n, 9}] (* Version 10, Michael De Vlieger, Aug 01 2016 *)
lp[n_]:=Module[{p=3,x=10^n},While[CompositeQ[x-p],p=NextPrime[p]];p]; Array[lp,60] (* Harvey P. Dale, Jun 11 2022 *)

10^n - a(n) is prime and 10^n - k is composite for 0 <= k < a(n). - corrected by David A. Corneth, Aug 18 2016

a(1) corrected and a(2) inserted by Gionata Neri, Aug 01 2016

A124049 a(n) = c is least number such that 10^n/2 -/+ c are primes.

Original entry on oeis.org

0, 3, 9, 81, 123, 57, 87, 243, 69, 63, 189, 231, 1569, 381, 231, 1443, 1113, 321, 339, 1353, 363, 519, 1323, 1503, 741, 1221, 957, 1053, 339, 5931, 2121, 2301, 2031, 4773, 4737, 10281, 1317, 129, 3873, 1443, 387, 11769, 8271, 5337, 2883, 7137, 8193, 8493

Offset: 1

Hans Havermann and Zak Seidov, Nov 03 2006

nonn

Related to Goldbach pairs of 10^n: a(n)=10^n/2 -A124450(n) Lesser of pair of closest primes whose sum is 10^n. Cf. A124013 Lesser of pair of most widely separated primes whose sum is 10^n, A065577 Number of Goldbach partitions of 10^n

All terms are divisible by 3 - see A108163.

			Next terms up to n = 101: 14637, 9897,
6471, 183, 8043, 6921,6699, 29127, 3663, 12537, 3777,
6741, 2253, 561, 3783, 26979, 16491, 6543, 10683,
1749, 6417, 38871, 22767, 62403, 8631, 4497, 20739,
453, 16731, 25293, 4341, 37467,
55323,4587,37083,24717,6687,8763,22551,29367,37881,14301,8637,34101,22179,26811,7059,1647

Cf. A065577, A124013, A124450.

lnc[n_]:=Module[{c=0,t=10^n/2},While[!AllTrue[t+{c,-c},PrimeQ],c++];c]; Array[ lnc,50] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 21 2014 *)

A359120 Number of primes p with 10^(n-1) < p < 10^n such that 10^n-p is also prime.

Original entry on oeis.org

3, 11, 47, 221, 1433, 9579, 69044, 519260, 4056919, 32504975, 266490184, 2224590493, 18850792161

Offset: 1

N. J. A. Sloane, Dec 17 2022

The terms of A358310 come in decreasing blocks; a(n) is the length of the n-th block.

			For n = 1, there are three primes p with 1 < p < 10 such that 10-p is also prime, 3, 5, and 7, so a(1) = 3.

A107318 and A065577 are very similar.

Cf. A006879, A358310.

a(n) = {if(n==1,return(3)); my(res=0, pow10=10^n); forprime(p=2, 10^(n-1), if(isprime(pow10-p), res++)); forprime(p=10^(n-1), pow10>>1, if(isprime(pow10-p), res+=2)); res} \\ David A. Corneth, Dec 17 2022

from sympy import isprime, primerange
def a(n):
    lb, ub = 10**(n-1), 10**n
    s1 = sum(1 for p in primerange(1, lb) if isprime(ub-p))
    s2 = sum(2 for p in primerange(lb, 5*lb) if isprime(ub-p))
    return s1 + s2 + int(n == 1)
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Dec 17 2022

a(7)-a(9) from Michael S. Branicky, Dec 17 2022
a(10)-a(11) from David A. Corneth, Dec 17 2022
a(12) from N. J. A. Sloane, Dec 17 2022, found using Corneth's PARI program.
a(13) from Martin Ehrenstein, Dec 18 2022, found using Walisch's primesieve library.

A180041 Number of Goldbach partitions of (2n)^n.

Original entry on oeis.org

0, 2, 13, 53, 810, 20564, 274904, 6341424, 419586990

Offset: 1

Jonathan Vos Post, Aug 07 2010

This is the main diagonal of the array mentioned in A180007, only considering even rows (as odd numbers cannot be the sums of two odd primes), namely A(2n, n) = number of ways of writing (2n)^n as the sum of two odd primes, when the order does not matter.

			a(1) = 0 because 2*1 = 2 is too small to be the sum of two primes.
a(2) = 2 because 4^2 = 16 = 3+13 = 5+11.
a(3) = 13 because 6^3 = 216 and A180007(3) = Number of Goldbach partitions of 6^3 = 13.
a(4) = 53 because 8^4 = 2^12 and A006307(12) = Number of ways writing 2^12 as unordered sums of 2 primes.

Cf. A001031, A061358, A065577, A180007.

A180041 := proc(n) local a,m,p: if(n=1)then return 0:fi: a:=0: m:=(2*n)^n: p:=prevprime(ceil((m-1)/2)): while p > 2 do if isprime(m-p) then a:=a+1: fi: p := prevprime(p): od: return a: end: seq(A180041(n),n=1..5); # Nathaniel Johnston, May 08 2011

f[n_] := Block[{c = 0, p = 3, m = (2 n)^n}, lmt = Floor[m/2] + 1; While[p < lmt, If[ PrimeQ[m - p], c++ ]; p = NextPrime@p]; c]; Do[ Print[{n, f@n // Timing}], {n, 8}] (* Robert G. Wilson v, Aug 10 2010 *)

a(n) = A061358((2*n)^n) = A061358(A062971(n)).

a(6)-a(8) from Robert G. Wilson v, Aug 10 2010

a(9) from Giovanni Resta, Apr 15 2019

A195295 Number of Goldbach partitions of 4^n.

Original entry on oeis.org

0, 1, 2, 5, 8, 22, 53, 151, 435, 1314, 4239, 13705, 45746, 153850, 525236, 1817111, 6341424, 22336060, 79287664, 283277225, 1018369893

Offset: 0

Kausthub Gudipati, Sep 16 2011

Bisection of A006307.

Cf. A061358, A065577, A180007.

cp's OEIS Frontend

A061358 Number of ways of writing n = p+q with p, q primes and p >= q.

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A073610 Number of primes of the form n-p where p is a prime.

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A107318 Number of ordered Goldbach partitions of 10^n.

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A124450 Lesser of a pair of not necessarily distinct closest primes that add up to 10^n.

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A180007 Number of Goldbach partitions of 6^n.

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A124013 Lesser of pair of most widely separated primes whose sum is 10^n.

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A124049 a(n) = c is least number such that 10^n/2 -/+ c are primes.

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