cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Original entry on oeis.org

2, 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

Views

Author

Andrew S. Plewe, Apr 15 2006

Keywords

Comments

If the triangular number 0 is allowed, only one additional prime occurs: 3. In that case, the sequence becomes A117112.
A subsequence of A051533. - Wolfdieter Lang, Jan 11 2017

Examples

			2 = 1 + 1
7 = 1 + 6
11 = 1 + 10
13 = 10 + 3, etc.

Links

Crossrefs

Cf. A000040, A000217, A002243, A002244, A020756, A051533, A053614, A060773, A002636.

Programs

  • Mathematica
    tri = Table[n (n + 1)/2, {n, 40}]; Select[Union[Flatten[Outer[Plus, tri, tri]]], # <= tri[[-1]]+1 && PrimeQ[#] &] (* T. D. Noe, Apr 07 2011 *)
  • PARI
    is(n)=for(k=sqrtint(4*n+1)\2+1,(sqrtint(8*n+1)-1)\2, if(ispolygonal(n-k*(k+1)/2,3), return(n>3 && isprime(n)))); n==2 \\ Charles R Greathouse IV, Nov 07 2014

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

Views

Author

Greg Huber, Apr 24 2006

Keywords

Examples

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

Links

Crossrefs

Cf. A117048, A117112, A118638.

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Greg Huber, May 09 2006

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Extensions

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

Views

Author

Greg Huber, Apr 24 2006

Keywords

Examples

			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.

Crossrefs

Cf. A001097, A117048, A117112.

Programs

  • Mathematica
    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 *)

Extensions

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

A145032 If t(n) is the maximal triangular number not exceeding n, then a(n) is the n-th prime for which a(n)-t(a(n)) is a triangular number.

Original entry on oeis.org

2, 3, 7, 11, 13, 29, 31, 37, 61, 67, 79, 97, 101, 137, 139, 151, 163, 181, 191, 193, 211, 241, 263, 277, 331, 379, 409, 421, 463, 499, 571, 601, 631, 709, 739, 751, 769, 821, 823, 947, 967, 991, 1063, 1087, 1091, 1109, 1117, 1129, 1231, 1303, 1327, 1381, 1399
Offset: 1

Views

Author

Vladimir Shevelev, Sep 30 2008

Keywords

Comments

Primes p for which p-A057944(p) is in A000217. [From R. J. Mathar, Oct 25 2010]

Examples

			E. g., t(181)=171 (see A000217) and 181-171=10 is triangular number. Therefore p=181 is in the sequence

Crossrefs

A000217 A117112 A145016

Programs

  • Maple
    Contribution from R. J. Mathar, Oct 25 2010: (Start)
A057944 := proc(n) for i from 0 do if i*(i+1)/2 > n then return (i-1)*i /2 ; end if; end do: end proc:
isA000217 := proc(n) issqr(8*n+1) ; end proc:
isA145032 := proc(p) if isprime(p) then tres := p-A057944(p) ; isA000217(tres) ; else false; end if; end proc:
for n from 1 to 400 do p := ithprime(n) ; if isA145032(p) then printf("%d,",p) ; end if; end do: (End)

Extensions

More terms from R. J. Mathar, Oct 25 2010

A307950 Primes that are the sum of two prime-indexed triangular numbers.

Original entry on oeis.org

31, 43, 97, 157, 181, 193, 281, 367, 463, 499, 587, 709, 769, 1051, 1381, 1459, 1621, 1831, 1861, 2081, 2281, 2293, 2377, 2473, 2647, 2707, 2713, 2729, 2767, 2837, 3019, 3163, 3251, 3259, 3313, 3709, 3863, 4021, 4447, 4591, 4759, 4943, 4951, 5051, 5179, 5647, 5791, 5861, 5869, 5881, 6217, 6271
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, May 07 2019

Keywords

Comments

Primes of the form A034953(i) + A034953(j).
There are primes with more than one expression of this form; e.g., 18749 = A034953(4) + A034953(44) = A034953(19) + A034953(42).

Examples

			a(3) = 97 is a term because 97 = 6 + 91 is prime where 6=A000217(3) and 91=A000217(13) are in A034953.

Links

Crossrefs

Cf. A000217, A034953, A117112.

Programs

  • Maple
    A034953:= map(t ->t*(t+1)/2, [seq(ithprime(i),i=1..100)]):
A:= select(t -> t <= A034953[-1]+3 and isprime(t), {seq(seq(A034953[i]+A034953[j],j=i+1..100),i=1..99)}):
sort(convert(A,list));

A117382 Primes not expressible as the sum of two triangular numbers.

Original entry on oeis.org

5, 17, 19, 23, 41, 47, 53, 59, 71, 89, 103, 107, 109, 113, 131, 149, 167, 173, 179, 197, 223, 229, 233, 239, 251, 257, 269, 271, 283, 293, 311, 313, 317, 337, 347, 349, 359, 383, 397, 401, 419, 431, 439, 449, 457, 467, 479, 491, 503, 509, 521, 523, 547, 557
Offset: 0

Views

Author

Greg Huber, Apr 24 2006

Keywords

Crossrefs

Cf. A000040, A117112.

Programs

  • Mathematica
    Complement[ Prime@ Range@115, Union@ Flatten@ Table[i(i + 1)/2 + j(j + 1)/2, {i, 35}, {j, 0, i}]] (* Robert G. Wilson v, Apr 27 2006 *)

Extensions

More terms from Robert G. Wilson v, Apr 27 2006
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.