A068307 -id:A068307 - cp's OEIS frontend

A230493 Number of ways to write n = (2-(n mod 2))p + q + r with p <= q <= r such that p, q, r, 2p^2 - 1, 2q^2 - 1, 2r^2 - 1 are all prime.

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 3, 3, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 2, 2, 1, 1, 2, 2, 1, 3, 3, 1, 3, 2, 4, 1, 2, 2, 4, 3, 3, 2, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 2, 2, 2, 3, 3, 2, 4, 3, 2, 3, 5, 1, 4, 3, 3, 2, 4, 4, 3, 4, 5, 2, 4, 5, 4, 3, 2, 4, 4, 3, 2

Offset: 1

Zhi-Wei Sun, Oct 20 2013

nonn

Conjecture:  a(n) > 0 for all n > 6.
This is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013. It also implies that there are infinitely many primes p with 2*p^2 - 1 also prime.
We have verified the conjecture for n up to 10^6.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 22 2023
See also A230351, A230494 and A230502 for similar conjectures.

			a(14) = 1 since 14 = 2*2 + 3 + 7 with 2, 3, 7, 2*2^2 - 1 = 7, 2*3^2 - 1 = 17, 2*7^2 - 1 = 97 all prime.
a(19) = 1 since 19 = 3 + 3 + 13, and 3, 13, 2*3^2 - 1 = 17 and 2*13^2 - 1 = 337 are all prime.
a(53) = 1 since 53 = 3 + 7 + 43, and all the six numbers 3, 7, 43, 2*3^2 - 1 = 17, 2*7^2 - 1 = 97, 2*43^2 - 1 = 3697 are prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A000040, A068307, A106483, A230219, A230351, A230494, A230502.

pp[n_]:=PrimeQ[2n^2-1]
pq[n_]:=PrimeQ[n]&&pp[n]
a[n_]:=Sum[If[pp[Prime[i]]&&pp[Prime[j]]&&pq[n-(2-Mod[n,2])Prime[i]-Prime[j]],1,0],{i,1,PrimePi[n/(4-Mod[n,2])]},{j,i,PrimePi[(n-(2-Mod[n,2])Prime[i])/2]}]
Table[a[n],{n,1,100}]

A236832 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q and r terms of A234695.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 31 2014

nonn

Conjecture: a(n) > 0 for all n > 3.

This is stronger than Goldbach's weak conjecture which was finally proved by H. A. Helfgott in 2013.

			a(4) = 1 since 2*4 - 1 = 2 + 2 + 3 with 2 and 3 terms of A234695.
a(5) = 2 since 2*5 - 1 = 2 + 2 + 5 = 3 + 3 + 3 with 2, 3, 5 terms of A234695.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252 [math.NT], 2012-2013.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Cf. A000040, A068307, A230219, A234695, A235189.

p[n_]:=PrimeQ[Prime[n]-n+1]
q[n_]:=PrimeQ[n]&&p[n]
a[n_]:=Sum[If[p[Prime[i]]&&p[Prime[j]]&&q[2n-1-Prime[i]-Prime[j]],1,0],{i,1,PrimePi[(2n-1)/3]},{j,i,PrimePi[(2n-1-Prime[i])/2]}]
Table[a[n],{n,1,100}]

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

Original entry on oeis.org

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

A210681 Number of ways to write 2n = p+q+r (p<=q) with p, q, r-1, r+1 all prime and p-1, p+1, q-1, q+1, r all practical.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 29 2013

nonn

Conjecture: a(n) > 0 for all n > 4.
This conjecture involves two kinds of sandwiches introduced by the author, and it is much stronger than the Goldbach conjecture for odd numbers. We have verified the conjecture for n up to 10^7.
Zhi-Wei Sun also made the following conjectures:
(1) Any even number greater than 10 can be written as the sum of four elements in the set
    S = {prime p: p-1 and p+1 are both practical}.
Also, every n=3,4,5,... can be represented as the sum of a prime in S and two triangular numbers.
(2) Each integer n>7 can be written as p + q + x^2 (or p + q + x(x+1)/2), where p is a prime with p-1 and p+1 both practical, and q is a practical number with q-1 and q+1 both prime.
(3) Every n=3,4,... can be written as the sum of three elements in the set
    T = {x: 6x is practical with 6x-1 and 6x+1 both prime}.
(4) Any integer n>6 can be represented as the sum of two elements of the set S and one element of the set T.
(5) Each odd number greater than 11 can be written in the form 2p+q+r, where p and q belong to S, and r is a practical number with r-1 and r+1 both prime.

			a(5)=1 since 2*5=3+3+4 with 3 and 5 both prime, and 2 and 4 both practical.
a(6)=2 since 2*6=3+3+6=3+5+4 with 3,5,7 all prime and 2,4,6 all practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56(1996), 205-210.
Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A005153, A068307, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320, A210479, A210528, A210531, A210533, A258838.

f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
pp[k_]:=pp[k]=pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True
pq[n_]:=pq[n]=PrimeQ[n-1]==True&&PrimeQ[n+1]==True&&pr[n]==True
a[n_]:=a[n]=Sum[If[pp[j]==True&&pp[k]==True&&pq[2n-Prime[j]-Prime[k]]==True,1,0],{j,1,PrimePi[n-1]},{k,j,PrimePi[2n-Prime[j]]}]
Do[Print[n," ",a[n]],{n,1,100}]

A237291 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q, r, pi(p), pi(q), pi(r) all prime, where pi(x) denotes the number of primes not exceeding x (A000720).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 0, 1, 2, 0, 2, 3, 1, 0, 2, 2, 1, 3, 2, 1, 1, 1, 1, 2, 3, 2, 2, 0, 3, 4, 2, 2, 3, 2, 1, 3, 4, 1, 5, 2, 1, 2, 3, 4, 3, 1, 1, 3, 2, 2, 4, 3, 2, 3, 3, 1, 5, 5, 1, 3, 4, 2, 3, 4, 4, 2, 4, 2, 3, 4, 2, 2

Offset: 1

Zhi-Wei Sun, Feb 06 2014

nonn

Conjecture: a(n) > 0 for all n > 36.

This is stronger than Goldbach's weak conjecture finally proved by H. A. Helfgott in 2013.

			a(16) = 1 since 2*16 - 1 = 3 + 11 + 17 with 3, 11, 17, pi(3) = 2, pi(11) = 5 and pi(17) = 7 all prime.
a(179) = 1 since 2*179 - 1 = 83 + 83 + 191 with 83, 191, pi(83) = 23 and pi(191) = 43 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252, 2012.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897, 2013.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A000720, A006450, A068307, A230219, A236832, A237284.

p[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
a[n_]:=Sum[If[p[2n-1-Prime[Prime[i]]-Prime[Prime[j]]],1,0],{i,1,PrimePi[PrimePi[(2n-1)/3]]},{j,i,PrimePi[PrimePi[(2n-1-Prime[Prime[i]])/2]]}]
Table[a[n],{n,1,80}]

A355196 Sum of the largest parts of the partitions of n into exactly 3 prime parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 3, 3, 8, 5, 12, 12, 12, 7, 23, 18, 38, 24, 31, 24, 59, 30, 73, 47, 71, 49, 113, 55, 115, 40, 102, 59, 171, 48, 168, 100, 191, 102, 220, 50, 265, 89, 246, 120, 322, 109, 383, 136, 348, 181, 477, 158, 516, 117, 468, 199, 605, 133, 574, 170, 600, 252, 751, 133

Offset: 0

Wesley Ivan Hurt, Jun 23 2022

nonn

			a(9) = 8; since 9 can be written as the sum of 3 primes in two ways: 2+2+5 = 3+3+3 and the sum of the largest parts of these partitions is 5+3 = 8, we have a(9) = 8.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions

Cf. A010051, A068307, A355197, A355198, A355199.

Table[Sum[Sum[(n - i - j) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[n - i - j] - PrimePi[n - i - j - 1]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
Table[Total[Select[IntegerPartitions[n,{3}],AllTrue[#,PrimeQ]&][[;;,1]]],{n,0,70}] (* Harvey P. Dale, Aug 24 2025 *)

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} c(i) * c(j) * c(n-i-j) * (n-i-j), where c = A010051.

a(n) = A355199(n) - A355197(n) - A355198(n).

A164023 Smallest of largest parts in partitions of n into exactly three primes.

Original entry on oeis.org

2, 3, 3, 3, 5, 5, 5, 5, 7, 5, 7, 7, 11, 7, 11, 7, 13, 11, 11, 11, 13, 11, 13, 11, 17, 13, 17, 11, 19, 13, 17, 13, 19, 13, 19, 17, 23, 17, 23, 17, 31, 17, 23, 19, 29, 17, 31, 19, 29, 19, 31, 19, 37, 23, 29, 23, 31, 23, 31, 23, 41, 29, 37, 23, 37, 29, 41, 31, 41, 29, 37, 29, 47, 31

Offset: 6

Reinhard Zumkeller, Aug 08 2009

nonn

a(n) >= floor(n/3); a(A001748(n)) = A000040(n).

			a(16) = min{max(2,3,11),max(2,7,7)} = min{11,7} = 7;
a(17) = min{max(2,2,13),max(2,3,11),max(3,7,7),max(5,5,7)} = min{13,11,7,7} = 7.

R. Zumkeller, Table of n, a(n) for n = 6..2500
Index entries for sequences related to Goldbach conjecture

A164024, A068307.

A164024 Largest of smallest parts in partitions of n into exactly three primes.

Original entry on oeis.org

2, 2, 2, 3, 2, 3, 2, 3, 2, 5, 2, 5, 2, 5, 2, 7, 2, 5, 2, 7, 2, 7, 2, 7, 2, 7, 2, 11, 2, 11, 2, 11, 2, 13, 2, 11, 2, 13, 2, 13, 2, 13, 2, 13, 2, 17, 2, 17, 2, 17, 2, 19, 2, 17, 2, 19, 2, 17, 2, 19, 2, 19, 2, 23, 2, 19, 2, 19, 2, 23, 2, 23, 2, 19, 2, 23, 2, 23, 2, 23, 2, 29, 2, 29, 2, 29, 2, 31, 2

Offset: 6

Reinhard Zumkeller, Aug 08 2009

nonn

a(even) = 2.

a(n)<=n/3; equality occurs at n= 3*prime, and then a(n) are new records 2,3,5,7,11,.. - Zak Seidov, Aug 09 2009

			a(16) = max{min(2,3,11),min(2,7,7)} = max{2,2} = 2;
a(17) = max{min(2,2,13),min(3,3,11),min(3,7,7),min(5,5,7)} = max{2,2,3,5} = 5.

Index entries for sequences related to Goldbach conjecture

A164023, A068307.

A230451 Number of ways to write n = x + y + z (x, y, z > 0) such that 2x + 1, 2y + 3, 2z + 5 are all prime and xy*z is a triangular number.

Original entry on oeis.org

0, 0, 1, 0, 2, 3, 0, 4, 3, 1, 7, 3, 2, 3, 7, 4, 5, 6, 3, 4, 8, 5, 8, 3, 6, 8, 9, 9, 5, 12, 2, 11, 4, 4, 4, 13, 5, 9, 13, 8, 14, 8, 3, 15, 7, 8, 10, 6, 5, 17, 15, 4, 6, 9, 8, 10, 15, 9, 7, 15, 11, 5, 6, 11, 14, 14, 7, 11, 3, 12, 23, 16, 5, 20, 14, 4, 9, 14, 5, 19, 19, 4, 3, 12, 7, 16, 5, 12, 6, 11, 12, 12, 23, 14, 23, 12, 5, 17, 14, 5

Offset: 1

Zhi-Wei Sun, Oct 19 2013

nonn

Conjecture: (i) a(n) > 0 except for n = 1, 2, 4, 7.
(ii) Any integer n > 7 can be written as x + y + z (x, y, z > 0) such that 2*x + 1, 2*y + 1, 2*x*y + 1 are primes and x*y*z is a triangular number.
(iii) Each integer n > 4 not equal to 7 or 14 can be expressed as p + q + r (p, q, r > 0) with p and 2*q + 1 both primes, and p*q*r a triangular number.
(iv) Any integer n > 6 not among 16, 20, 60 can be written as x + y + z (x, y, z > 0) such that x*y + x*z + y*z is a triangular number.
Part (i) is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013.
See also A227877 and A230596 for some related conjectures.

			a(6) = 3 since 6 = 1 + 2 + 3 = 2 + 1 + 3 = 3 + 2 + 1, and 2*1 + 1 = 3, 2*2 + 3 = 7, 2*3 + 5 = 11, 2*2 + 1 = 5, 2*1 + 3 = 5, 2*3 + 1 = 7, 2*1 + 5 = 7 are all prime.
a(10) = 1 since 10 = 3 + 4 + 3, and 2*3 + 1 = 7, 2*4 + 3 = 11, 2*3 + 5 = 11 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..6000
Zhi-Wei Sun, A new conjecture on triangular numbers, a message to Number Theory List, Oct. 25, 2013.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A000040, A000217, A068307, A132399, A227877, A229166, A230121, A230141, A230219, A230596.

SQ[n_]:=IntegerQ[Sqrt[n]]
TQ[n_]:=SQ[8n+1]
a[n_]:=Sum[If[PrimeQ[2i+1]&&PrimeQ[2j+3]&&PrimeQ[2(n-i-j)+5]&&TQ[i*j(n-i-j)],1,0],{i,1,n-2},{j,1,n-1-i}]
Table[a[n],{n,1,100}]

A307815 Number of partitions of n into 3 squarefree parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 5, 4, 5, 5, 7, 7, 9, 8, 11, 11, 13, 11, 15, 14, 18, 15, 20, 19, 23, 20, 24, 24, 27, 24, 30, 29, 34, 30, 37, 36, 42, 36, 45, 44, 50, 44, 54, 54, 59, 52, 62, 63, 68, 57, 69, 70, 78, 65, 78, 78, 88, 74, 86, 87, 98, 84, 98, 98, 107, 93, 109, 108, 120, 102, 124, 123

Offset: 0

Ilya Gutkovskiy, Apr 30 2019

nonn

			a(10) = 4 because we have [7, 2, 1], [6, 3, 1], [6, 2, 2] and [5, 3, 2].

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Rémy Sigrist)
Index entries for sequences related to partitions

Cf. A005117, A008683, A068307, A071068, A073576, A098235, A280210, A307727.

b:= proc(n, i) option remember; `if`(n=0, [1, 0$3], `if`(i<1, 0, b(n, i-1)+
      `if`(numtheory[issqrfree](i), [0, b(n-i, min(i, n-i))[1..3][]], 0)))
    end:
a:= n-> b(n$2)[4]:
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 30 2019

Array[Count[IntegerPartitions[#, {3}], _?(AllTrue[#, SquareFreeQ] &)] &, 75, 0]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i < 1, {0, 0, 0, 0}, b[n, i - 1] + If[SquareFreeQ[i], {0, Sequence @@ b[n - i, Min[i, n - i]][[1 ;; 3]]}, {0, 0, 0, 0}]]];
a[n_] := b[n, n][[4]];
a /@ Range[0, 80] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)

a(n) = [x^n y^3] Product_{k>=1} 1/(1 - mu(k)^2*y*x^k).

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} mu(i)^2 * mu(k)^2 * mu(n-i-k)^2, where mu is the Mobius function. - Wesley Ivan Hurt, May 09 2019

cp's OEIS Frontend

A230493 Number of ways to write n = (2-(n mod 2))*p + q + r with p <= q <= r such that p, q, r, 2*p^2 - 1, 2*q^2 - 1, 2*r^2 - 1 are all prime.

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A236832 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q and r terms of A234695.

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

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A210681 Number of ways to write 2n = p+q+r (p<=q) with p, q, r-1, r+1 all prime and p-1, p+1, q-1, q+1, r all practical.

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A237291 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q, r, pi(p), pi(q), pi(r) all prime, where pi(x) denotes the number of primes not exceeding x (A000720).

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A355196 Sum of the largest parts of the partitions of n into exactly 3 prime parts.

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A164023 Smallest of largest parts in partitions of n into exactly three primes.

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A164024 Largest of smallest parts in partitions of n into exactly three primes.

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A230451 Number of ways to write n = x + y + z (x, y, z > 0) such that 2*x + 1, 2*y + 3, 2*z + 5 are all prime and x*y*z is a triangular number.

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A230493 Number of ways to write n = (2-(n mod 2))p + q + r with p <= q <= r such that p, q, r, 2p^2 - 1, 2q^2 - 1, 2r^2 - 1 are all prime.

A230451 Number of ways to write n = x + y + z (x, y, z > 0) such that 2x + 1, 2y + 3, 2z + 5 are all prime and xy*z is a triangular number.