A002375 -id:A002375 - cp's OEIS frontend

A219025 Number of primes p

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

Offset: 1

Zhi-Wei Sun, Nov 10 2012

nonn

Conjecture: a(n)>0 for all n=6,7,...
This has been verified for n up to 10^8.
Zhi-Wei Sun also made the following general conjecture:
Let P(x) be any non-constant integer-valued polynomial with positive leading coefficient. If n is large enough, then there is a prime pSee also A219023 for similar conjectures.

			a(11)=2 since the 5 and 7 are the only primes p<11 with 66-p and 66+p both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588, 2012.

Cf. A000040, A002375, A218754, A218585, A218654, A218656, A218825.

a[n_]:=a[n]=Sum[If[PrimeQ[6n-Prime[k]]==True&&PrimeQ[6n+Prime[k]]==True,1,0],{k,1,PrimePi[n-1]}]
Do[Print[n," ",a[n]],{n,1,20000}]

A219966 Number of ways to write n=p+q+(n mod 2)q with q<=n/2 and p, q, q+6 all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 02 2012

nonn

Conjecture: a(n)>0 for all n>11.

This conjecture is stronger than Goldbach's conjecture and Lemoine's conjecture. It can be further strengthened; see A219055 and the comments there.

			a(19)=1 since 19=5+2*7 with 5, 7, 7+6 all prime.
a(20)=1 since 20=13+7 with 13, 7, 7+6 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A219055, A023201, A002375, A046927.

a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[n-(1+Mod[n,2])Prime[k]]==True,1,0],{k,1,PrimePi[n/2]}]
Do[Print[n," ",a[n]],{n,1,10000}]

A230762 List of commonest number of decompositions of 2k into an unordered sum of two odd primes in range 3 <= k <= m, integer m >= 3, where m is explained below.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 18, 27, 44, 48, 52, 58, 61, 75, 77, 98, 141, 165, 200, 231, 337, 360, 378, 384, 466, 517, 697, 880, 1061, 1400, 1503, 1615, 1700, 1896, 2082, 2163, 3242, 3929, 4232, 5373

Offset: 1

Lei Zhou, Oct 29 2013

If making a statistical bar chart with x-axis denoting the number of decompositions of an even number, and y-axis denoting the number of hits of an x-axis value for all 3 <= k <= m, there are one or more commonest x value presenting with highest y value.  Such commonest x values increase when m increases, and fall on the x values listed in this sequence.
Hypothesis: With the increase of m, the commonest number of decompositions of 2n into an unordered sum of two odd primes in the range of 3 <= k <= m ascends.
This hypothesis derives that the corresponding smallest m to the terms of this sequence makes an ascending sequence.  Or say, when testing with m ascending, once a number a(n) enters this sequence, no number smaller than a(n) will be able to enter this sequence if they had not enter previous.

			When m=3, k has only one value 3, 2k=6=3+3.  Only one possible decomposition, making a decomposition statistics {{x,y}}={{1,1}}.  So a(1)=1;
When m=4, k gets another value 4, 2k=8=3+5. The decomposition statistics {{x,y}}={{1,2}};...
Thereafter, k=5 makes 2k=10=5+5=3+7, {{x,y}}={{1,2},{2,1}}, the commonest value is still 1.
k=6, 2k=12=5+7, {{x,y}}={{1,3},{2,1}}, commonest x is still 1.
k=7, 2k=14=3+11=7+7, {{x,y}}={{1,3},{2,2}}, commonest x is still 1.
k=8, 2k=16=3+13=5+11, {{x,y}}={{1,3},{2,3}}, except for 1, 2 is now eligible to be the new possible commonest x, so a(2)=2.
...
Counting up to k=28, the decomposition statistics is {{1,3},{2,8},{3,8},{4,5},{5,2}}, 2 and 3 are now the commonest decompositions. It is the first time for 3 to appear.  So a(3)=3.

Lei Zhou, Table of n, a(n) for n = 1..45

Cf. A002375.

check=0;posts={};mpos=0;res={};sres=0;s={};size=0;k=2;
While[k++;k2=2*k;p2=k-1;ct=0;
  While[p2=NextPrime[p2];p2size, Do[AppendTo[s,0],{i,size+1,ct}]; size=ct];
  (*and construct statistics in array s*)
  s[[ct]]++;m=Max[s];aa=Position[s,m];la=Length[aa];
  Do[a=aa[[pos,1]];
    If[a>sres,
      While[sres

Lei Zhou, Nov 08 2013, uploaded a b-file extending the known elements of this sequence to the 45th.

A237353 For n=g+h, a(n) is the minimum value of omega(g)+omega(h).

Original entry on oeis.org

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

Offset: 2

Lei Zhou, Feb 06 2014

nonn

omega(g) is defined in A001221.
If Goldbach's conjecture is true, all items with even index of this sequence is less than or equal to 2.
This sequence is defined for n >= 2.
It is conjectured that the maximum value of this sequence is 3.
2=1+1 makes the only zero term of this sequence a(2)=0.
This sequence gets a(n)=1 when n=1+p^k, where p is a prime number and k >= 1.

			For n=2, 2=1+1. 1 does not have prime factor. So a(2)=0+0=0;
For n=6, 6=1+5.  1 does not have prime factor where 5 has one. Another case 6=3+3 yields sum of prime factors of g and h 1+1=2.  Since 1 < 2, according to the definition, we chose the smaller one. So a(6)=1;
For n=7, 7=2+5.  Both 2 and 5 have one prime factor.  So a(7)=1+1=2;
For n=331, one of the case is 331=2+329=2+7*47.  In which 2 has one prime factor, and 329 has two.  So a(331)=1+2=3.

Lei Zhou, Table of n, a(n) for n = 2..10000

Cf. A002375, A001221

Table[ct = n; Do[h = n - g; c = Length[FactorInteger[g]] + Length[FactorInteger[h]]; If[g == 1, c--]; If[h == 1, c--]; If[c < ct, ct = c], {g, 1, Floor[n/2]}]; ct, {n, 2, 88}]
Table[ Min@Table[PrimeNu[ n - k ] + PrimeNu[  k  ], {k, n - 1}], {n, 2, 88}]

def a(n): return min(A001221(a)+A001221(n-a) for a in range(1,floor(n/2)+1)) # Ralf Stephan, Feb 23 2014

A237638 a(n) is the number of prime sets such that each set contains enough prime numbers to decompose every even number from 6 to 2n into the sum of two of its elements (reuse allowed), while none of the sets is a subset of another such set.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 9, 11, 11, 11, 13, 16, 23, 25, 31, 47, 57, 63, 70, 74, 79, 82, 122, 131, 129, 180, 215, 219, 323, 367, 446, 501, 531, 661, 867, 897, 1311, 1471, 1691, 1695, 2130, 2288, 2833, 3363, 3891, 5435, 8068, 8867, 13476, 15451, 15897

Offset: 3

Lei Zhou, May 02 2014

			n=4, 2n=8. There is only one set of primes {3,5} such that 6=3+3, 8=3+5. So a(4)=1.
...
n=8, 2n=16. We can find two sets, {3,5,7,11} and {3,5,7,13} that have such features. So a(8)=2. Here any set with more primes either contains an unused prime number or one of these two sets is a subset of them, like {3,5,7,11,13}, and thus is not considered. So a(8)=2.
...
n=13, 2n=26. Five such sets are found: {3,5,7,11,13}, {3,5,7,13,17},{3,5,7,13,19}, {3,5,7,11,17,19}, {3,5,7,11,17,23}. So a(13)=5.

Lei Zhou, Table of n, a(n) for n = 3..79

Cf. A000040, A002375, A240708, A237628.

a = {{{3}}}; Table[n2 = 2*n; na = {}; la = Last[a]; lo = Length[la]; Do[ok = 0; Do[p1 = la[[i, j]]; p2 = n2 - p1; If[MemberQ[la[[i]], p2], ok = 1], {j, 1, Length[la[[i]]]}];
  If[ok == 1, na = Sort[Append[na, la[[i]]]], Do[p1 = la[[i, j]]; p2 = n2 - p1; If[PrimeQ[p2], ng = Sort[Append[la[[i]], p2]]; big = 0; If[Length[na] > 0, Do[If[Intersection[na[[k]], ng] == na[[k]], big = 1], {k, 1, Length[na]}]]; If[big == 0, na = Sort[Append[na, ng]]]], {j, 1, Length[la[[i]]]}]], {i, 1, lo}]; AppendTo[a, na]; Length[na], {n, 4, 60}](* Program lists the 4th item and beyond *)

A238268 The number of unordered ways that n can be written as the sum of two numbers of the form p or 2p, where p is prime.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4, 3, 3, 5, 4, 4, 4, 5, 4, 4, 3, 4, 6, 4, 3, 7, 4, 3, 5, 6, 5, 5, 5, 6, 7, 4, 4, 9, 5, 5, 7, 6, 5, 5, 4, 5, 7, 4, 3, 10, 4, 4, 8, 8, 7, 7, 5, 6, 8, 5, 4, 10, 5, 5, 9, 8, 7, 8, 5, 7, 9, 5, 4, 13, 8, 6, 8, 8, 7

Offset: 4

Lei Zhou, Feb 21 2014

nonn

p and 2p are terms of A001751.
Sequence defined for n >= 4.
It is conjectured that all terms of this sequence are greater than zero.

			n=4, 4=2+2, one case found. So a(4)=1;
...
n=24, 24 = 2+2*11 = 5+19 = 7+17 = 2*5+2*7 = 11+13, 5 cases found.  So a(24)=5;
...
n=33, 33 = 2+31 = 2*2+29 = 7+2*13 = 2*5+23 = 11+2*11 = 2*7+19, 6 cases found.  So a(33)=6.

Lei Zhou, Table of n, a(n) for n = 4..10000

Cf. A001751, A002375, A103151.

Table[ct = 0; Do[If[((PrimeQ[i]) || (PrimeQ[i/2])) && ((PrimeQ[n - i]) || (PrimeQ[(n - i)/2])), ct++], {i, 2, Floor[n/2]}]; ct, {n, 4, 89}]

isp(i) = isprime(i) || (((i % 2) == 0) && isprime(i/2));
a(n) = sum(i=1, n\2, isp(i) && isp(n-i)); \\ Michel Marcus, Mar 07 2014

A239617 Number of ways to write 2n = p + q with p, q and pi(2p) - pi(p) all prime, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 22 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 4.
(ii) Each integer n > 5 can be written as p + q (q > 0) with p and pi(2*q) - pi(q) both prime.
(iii) Any integer n > 2 not equal to 11 can be written as p + q with p prime and pi(2*q) - pi(q) a square.
Part (i) is a refinement of Goldbach's conjecture. It implies that there are infinitely many primes p with pi(2*p) - pi(p) prime.

			a(5) = 1 since 2*5 = 7 + 3 with 7, 3 and pi(2*7) - pi(7) = 6 - 4 = 2 all prime.
a(6) = 1 since 2*6 = 7 + 5 with 7, 5 and pi(2*7) - pi(7) = 2 all prime.
a(11) = 1 since 2*11 = 11 + 11 with 11 and pi(2*11) - pi(11) = 8 - 5 = 3 both prime.
a(16) = 1 since 2*16 = 13 + 19 with 13, 19 and pi(2*13) - pi(13) = 9 - 6 = 3 all prime.
a(23) = 1 since 2*23 = 23 + 23 with 23 and pi(2*23) - pi(23) = 14 - 9 = 5 both prime.
a(44) = 1 since 2*44 = 59 + 29 with 59, 29 and pi(2*59) - pi(59) = 30 - 17 = 13 all prime.
a(166) = 1 since 2*166 = 103 + 229 with 103, 229 and pi(2*103) - pi(103) = 46 - 27 = 19 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A000720, A002372, A002375, A237284.

p[n_,k_]:=PrimeQ[PrimePi[2*Prime[k]]-k]&&PrimeQ[2n-Prime[k]]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,PrimePi[2n-1]}]
Table[a[n],{n,1,80}]

A240713 Number of decompositions of 2n=p1+p2 (prime p1 <= p2), where at least one other such pair 2n=p3+p4 (prime p3 <= p4) exists such that |p1-p3|= 6 or 12.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 0, 3, 4, 2, 2, 4, 2, 3, 5, 3, 3, 5, 2, 4, 6, 2, 4, 5, 0, 4, 5, 4, 4, 7, 3, 4, 6, 3, 4, 9, 3, 5, 7, 3, 5, 7, 3, 5, 7, 5, 6, 10, 5, 6, 12, 2, 5, 10, 2, 6, 7, 4, 4, 4, 5, 7, 9, 6, 5, 11, 0, 6, 10, 3, 7, 8, 4, 4, 13, 8

Offset: 1

Lei Zhou, Apr 10 2014

It is conjectured that a(n)=0 only when n=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 34, 76, 229.
The 2n decompositions counted in this sequence are a subset of 2n decompositions as of in Goldbach conjecture (A002375).
Per definition, all nonzero terms are greater than 1.

			For n = 11, 2n=22. 22 = 3 + 19 = 5 + 17 = 11 + 11. |5-11|=6 so pair 5+17 and 11+11 are counted. So a(11)=2.
...
For n = 17, 2n=34. 34 = 3 + 31 = 5 + 29 = 11 + 23 = 17 + 17. |5-11|=6, so 5+29 and 11+23 are counted. Also since |11-17|=6, 17+17 is also counted (where 11+23 is already counted). In case |5-17|=12, both instances are already counted. So overall three instances are found. a(17)=3.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A000040, A002375, A240712.

Table[s = 2*n; ct = 0; p = 1; While[p = NextPrime[p]; p <= n, If[PrimeQ[s - p], ok = 0; a1 = p - 12; b1 = s - a1; a2 = p - 6; b2 = s - a2; a3 = p + 6; b3 = s - a3; a4 = p + 12; b4 = s - a4; If[a1 > 0, If[PrimeQ[a1] && PrimeQ[b1], ok = 1]]; If[a2 > 0, If[PrimeQ[a2] && PrimeQ[b2], ok = 1]]; If[a3 <= n, If[PrimeQ[a3] && PrimeQ[b3], ok = 1]]; If[a4 <= n, If[PrimeQ[a4] && PrimeQ[b4], ok = 1]]; If[ok == 1, ct++]]]; ct, {n, 1, 85}]

A240718 Number of decompositions of 2n into an unordered sum of two primes, one of the two primes less than sqrt(2n-2).

Original entry on oeis.org

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

Offset: 1

Lear Young, Apr 11 2014

nonn

			For n = 7, the a(7) = 1 solution is 2*7 = 3 + 11 = 7 + 7; one of these pairs, 3 + 11, contains a number less than sqrt(2*7 - 2).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002375.

P:= NULL: A[1]:= 0: nextp:= 2:
for n from 2 to 100 do
 while nextp^2 < 2*n-2 do
   P:= P, nextp;
   nextp:= nextprime(nextp);
 od;
 A[n]:= numboccur(true, map(t -> isprime(2*n-t), [P]))
od:
seq(A[i],i=1..100); # Robert Israel, Apr 30 2019

a(n)=sum(i=2,primepi(floor(sqrt(2*n-2))),isprime(2*n-prime(i))) \\ Lear Young, Apr 11 2014

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A218469 Number of partitions of n into at most three distinct primes (including 1).

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A219025 Number of primes p

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A219966 Number of ways to write n=p+q+(n mod 2)q with q<=n/2 and p, q, q+6 all prime.

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A230762 List of commonest number of decompositions of 2k into an unordered sum of two odd primes in range 3 <= k <= m, integer m >= 3, where m is explained below.

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Extensions

A237353 For n=g+h, a(n) is the minimum value of omega(g)+omega(h).

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Sage

A237638 a(n) is the number of prime sets such that each set contains enough prime numbers to decompose every even number from 6 to 2n into the sum of two of its elements (reuse allowed), while none of the sets is a subset of another such set.

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A238268 The number of unordered ways that n can be written as the sum of two numbers of the form p or 2p, where p is prime.

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PARI

A239617 Number of ways to write 2*n = p + q with p, q and pi(2*p) - pi(p) all prime, where pi(x) denotes the number of primes not exceeding x.

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A239617 Number of ways to write 2n = p + q with p, q and pi(2p) - pi(p) all prime, where pi(x) denotes the number of primes not exceeding x.