A229166 -id:A229166 - cp's OEIS frontend

A230494 Number of ways to write n = x^2 + y (x, y >= 0) with 2*y^2 - 1 prime.

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

Offset: 1

Zhi-Wei Sun, Oct 20 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1. Moreover, if n > 1 is not among 2, 69, 76, then there are positive integers x and y such that x^2 + y is equal to n and 2*y^2 - 1 is prime.
(ii) Any integer n > 1 can be written as x*(x+1)/2 + y with 2*y^2 - 1 prime, where x and y are nonnegative integers. Moreover, if n is not equal to 2 or 15, then we may require additionally that x and y are both positive.
We have verified the conjecture for n up to 2*10^7.
Both conjectures verified for n up to 10^9. - Mauro Fiorentini, Aug 08 2023
See also A230351 and A230493 for similar conjectures.

			a(9) = 1 since 9 = 1^2 + 8 with 2*8^2 - 1 = 127 prime.
a(69) = 1 since 69 = 0^2 + 69 with 2*69^2 - 1 = 9521 prime.
a(76) = 1 since 76 = 0^2 + 76 with 2*76^2 - 1 = 11551 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.

Cf. A000040, A000290, A066049, A220272, A229166, A230351, A230493.

a[n_]:=Sum[If[PrimeQ[2(n-x^2)^2-1],1,0],{x,0,Sqrt[n]}]
Table[a[n],{n,1,100}]

A230596 Number of ways to write n = x + y + z with 0 < x <= y <= z such that xyz is a triangular number, and that x is a triangular number of the form (p^2 - 1)/8 with p an odd prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 24 2013

nonn

Conjecture: (i) a(n) > 0 except for n = 1, 2, 4, 7.
(ii) For any integer n > 7, there are positive integers x, y, z with x + y + z = n such that x*y*z is a triangular number and x is among 1, 2, 3, 4, 5, 6.
Note that a(3k) and a(3k+2) are positive for every k = 1, 2, 3, .... In fact, 3k = 1 + k + (2k-1) with 1*k*(2k-1) = 2k*(2k-1)/2 a triangular number, and  3k+2 = 1 + k + (2k+1) with 1*k*(2k+1) = 2k(2k+1)/2 a triangular number.

			a(10) = 1 since 10 = 3 + 3 + 4, and 3 = (5^2-1)/8 with 5 an odd prime, and 3*3*4 = 8*9/2 is a triangular number.
a(31) = 1 since 31 = 3 + 11 + 17, and 3 = (5^2-1)/8 with 5 an odd prime, and 3*11*17 = 33*34/2 is a triangular number.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000217, A000040, A132399, A229166, A230121, A230451.

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
a[n_]:=Sum[If[TQ[(Prime[i]^2-1)/8*y*(n-(Prime[i]^2-1)/8-y)],1,0],{i,2,PrimePi[Sqrt[8n/3+1]]},{y,(Prime[i]^2-1)/8,(n-(Prime[i]^2-1)/8)/2}]
Table[a[n],{n,1,100}]

A227877 Number of ways to write n = x + y + z (x, y, z > 0) such that xy and xz are triangular numbers, and 6y-1 and 6z+1 are both prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 25 2013

nonn

Conjecture: a(n) > 0 for all n > 4.
For n = 4*k - 1, we have n = (2k-1) + k + k with (2k-1)*k = 2k*(2k-1)/2 a triangular number. For n = 4*k + 1, we have n = (2k+1) + k + k with (2k+1)*k = 2k*(2k+1)/2 a triangular number.  For n = 4*k + 2, we have n = (2k+1) + k + (k+1), and (2k+1)*k = 2k*(2k+1)/2 and (2k+1)*(k+1) = (2k+1)(2k+2)/2 are both triangular numbers.
For n = 5*k, we have n = k + (2k-1) + (2k+1), and k*(2k-1) = 2k*(2k-1)/2 and k*(2k+1) = 2k*(2k+1)/2 are both triangular numbers. For n = 5*k - 2, we have n = k + (2k-1) + (2k-1) with k*(2k-1) = 2k*(2k-1)/2 a triangular number. For n = 5*k + 2, we have n = k + (2k+1) + (2k+1) with k*(2k+1) = 2k*(2k+1)/2 a triangular number.

			a(77) = 1 since 77 = 1 + 10 + 66, and 1*10 = 4*5/2 and 1*66 = 11*12/2 are triangular numbers, and 6*10 - 1 = 59 and 6*66 + 1 = 397 are both prime.
a(90) = 1 since 90 = 45 + 22 + 23, and 45*22 = 44*45/2 and 45*23 = 45*46/2 are triangular numbers, and 6*22 - 1 = 131 and 6*23 + 1 = 139 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, A new conjecture on triangular numbers, a message to Number Theory List, Oct. 25, 2013.

Cf. A000040, A000217, A132399, A229166, A230121, A230451, A230596.

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
a[n_]:=Sum[If[PrimeQ[6j-1]&&PrimeQ[6(n-i-j)+1]&&TQ[i*j]&&TQ[i(n-i-j)],1,0],{i,1,n-2},{j,1,n-1-i}]
Table[a[n],{n,1,100}]

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}]

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A230494 Number of ways to write n = x^2 + y (x, y >= 0) with 2*y^2 - 1 prime.

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A230596 Number of ways to write n = x + y + z with 0 < x <= y <= z such that x*y*z is a triangular number, and that x is a triangular number of the form (p^2 - 1)/8 with p an odd prime.

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A227877 Number of ways to write n = x + y + z (x, y, z > 0) such that x*y and x*z are triangular numbers, and 6*y-1 and 6*z+1 are both prime.

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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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A230596 Number of ways to write n = x + y + z with 0 < x <= y <= z such that xyz is a triangular number, and that x is a triangular number of the form (p^2 - 1)/8 with p an odd prime.

A227877 Number of ways to write n = x + y + z (x, y, z > 0) such that xy and xz are triangular numbers, and 6y-1 and 6z+1 are both 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.