A117048 -id:A117048 - cp's OEIS frontend

A249622 a(n) = number of ways to express A117048(n) as the sum of two positive triangular numbers.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 2, 1, 2, 1, 1, 6, 2, 1, 1, 1, 1

Offset: 1

Zak Seidov, Nov 03 2014

nonn

			a(6) = 2 because A117048(6) = 31 and 31 = 3 + 28 = 10 + 21 (first case of two-way expression).
a(22) = 3 because A117048(22) = 181 and 181 = A000217(i) + A000217(k), for {i,k} = {{4, 18}, {7, 17}, {9, 16}} (first case of three-way expression): 181 = 10 + 171 = 28 + 153 = 45 + 136.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000040, A000217, A002383.

A117112 Primes expressible as the sum of two triangular numbers (including zero).

Original entry on oeis.org

2, 3, 7, 11, 13, 29, 31, 37, 43, 61, 67, 73, 79, 83, 97, 101, 127, 137, 139, 151, 157, 163, 181, 191, 193, 199, 211, 227, 241, 263, 277, 281, 307, 331, 353, 367, 373, 379, 389, 409, 421, 433, 443, 461, 463, 487, 499, 541, 571, 577, 587, 601, 619, 631, 659, 661

Offset: 1

Greg Huber, Apr 18 2006

See A117048 for the primes that are the sum of two positive triangular numbers. The only difference is that the prime 3 occurs here.

			2 = 1 + 1
3 = 0 + 3
7 = 1 + 6
and so on.

Cf. A000217, A117048.

tri = Table[n (n + 1)/2, {n, 0, 40}]; Select[Union[Flatten[Outer[Plus, tri, tri]]], # <= tri[[-1]]+1 && PrimeQ[#] &] (* T. D. Noe, Apr 07 2011 *)
Select[Total/@Tuples[Accumulate[Range[0,40]],2],PrimeQ]//Union (* Harvey P. Dale, Apr 21 2019 *)

A000040 intersect A020756. - Jonathan Vos Post, Apr 17 2006

A117314 Twin-prime pairs expressible as the sum of two triangular numbers.

Original entry on oeis.org

11, 13, 29, 31, 137, 139, 191, 193, 461, 463, 659, 661, 821, 823, 1091, 1093, 1721, 1723, 2027, 2029, 2081, 2083, 2711, 2713, 3359, 3361, 3539, 3541, 3917, 3919, 6131, 6133, 6761, 6763, 7589, 7591, 7877, 7879, 7949, 7951, 8219, 8221, 9461, 9463, 9857

Offset: 1

Greg Huber, Apr 24 2006

			a(1) = 11 = 1 + 10; a(2) = 13 = 3 + 10.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A117048, A117112, A118638.

s = Select[Union@ Flatten@ Table[i(i + 1)/2 + j(j + 1)/2, {i, 200}, {j, 0, i}], PrimeQ@ # &]; t = Select[Range@Length@s - 1, s[[ # ]] + 2 == s[[ # + 1]] &]; Sort@Join[s[[t]], s[[t + 1]]] (* Robert G. Wilson v, Apr 27 2006 *)

More terms from Robert G. Wilson v, Apr 27 2006

A118638 Lesser of a twin-prime pair where both are expressible as the sum of two triangular numbers.

Original entry on oeis.org

11, 29, 137, 191, 461, 659, 821, 1091, 1721, 2027, 2081, 2711, 3359, 3539, 3917, 6131, 6761, 7589, 7877, 7949, 8219, 9461, 9857, 11351, 12107, 12377, 13691, 13997, 14447, 16139, 16229, 17291, 17417, 17579, 18911, 19469, 19541, 19991, 20549, 20639, 20747, 20981

Offset: 1

Greg Huber, May 09 2006

nonn

A bisection of A117314. Thought more interesting by Robert G. Wilson v.

			a(1) = 11 = 1 + 10;
a(2) = 29 = 1 + 28.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A117314, A117313, A117048, A117112, A001097.

a(40) inserted by Amiram Eldar, Dec 27 2019

A117313 Average of twin-prime pairs for pairs that are expressible as the sum of two triangular numbers.

Original entry on oeis.org

12, 30, 138, 192, 462, 660, 822, 1092, 1722, 2028, 2082, 2712, 3360, 3540, 3918, 6132, 6762, 7590, 7878, 7950, 8220, 9462, 9858, 11352, 12108, 12378, 13692, 13998, 14448, 16140, 16230, 17292, 17418, 17580, 18912, 19470, 19542, 19992, 20550, 20748

Offset: 1

Greg Huber, Apr 24 2006

			a(1) = 12 as witnessed by 11 = 1 + 10 and 13 = 3 + 10;
a(5) = 462 as witnessed by 461 = 55 + 406 and 463 = 28 + 435.

Cf. A001097, A117048, A117112.

s = Select[Union@ Flatten@ Table[i(i + 1)/2 + j(j + 1)/2, {i, 200}, {j, 0, i}], PrimeQ@ # &]; t = Select[Range@ Length@s - 1, s[[ # ]] + 2 == s[[ # + 1]] &]; s[[t]] + 1 (* Robert G. Wilson v, Apr 27 2006 *)

More terms from Robert G. Wilson v, Apr 27 2006

A119961 Semiprimes that are the sum of two triangular numbers.

Original entry on oeis.org

4, 6, 9, 10, 15, 21, 22, 25, 34, 38, 39, 46, 49, 51, 55, 57, 58, 65, 69, 87, 91, 93, 94, 106, 111, 115, 119, 121, 123, 133, 141, 142, 146, 159, 169, 177, 183, 202, 205, 213, 214, 218, 219, 226, 235, 237, 249, 253, 254, 259, 262, 265, 267, 274, 289, 291, 295

Offset: 1

Jonathan Vos Post, Aug 02 2006

Semiprime analog of A117048 Prime numbers that are expressible as the sum of two triangular numbers.

Cf. A001358, A020756, A117048.

With[{nn=60},Take[Union[Select[Total/@Tuples[Accumulate[Range[0,nn]],2],PrimeOmega[ #] ==2&]],nn]] (* Harvey P. Dale, Nov 04 2020 *)

A020756 intersection A001358.

Missing a(2) and a(19)-a(53) from Giovanni Resta, Jun 13 2016

cp's OEIS Frontend

A249622 a(n) = number of ways to express A117048(n) as the sum of two positive triangular numbers.

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A117112 Primes expressible as the sum of two triangular numbers (including zero).

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A117314 Twin-prime pairs expressible as the sum of two triangular numbers.

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A118638 Lesser of a twin-prime pair where both are expressible as the sum of two triangular numbers.

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A117313 Average of twin-prime pairs for pairs that are expressible as the sum of two triangular numbers.

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A119961 Semiprimes that are the sum of two triangular numbers.

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Formula

Extensions