A129755 -id:A129755 - cp's OEIS frontend

A129545 Triangular numbers T such that T+1 is a prime.

1, 6, 10, 28, 36, 66, 78, 136, 190, 210, 276, 378, 630, 820, 946, 990, 1128, 1326, 1596, 1830, 2016, 2080, 2346, 2556, 2850, 2926, 3570, 3916, 4560, 4656, 4950, 5050, 5778, 6216, 6328, 8646, 8778, 9180, 9870, 11026, 12720, 13366, 14028, 14196, 14878

Offset: 1

Zak Seidov, May 30 2007

The only triangular numbers T such that T-1 is a (positive) prime are 3 and 6.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A000217, A055469, A129755, A130178, A345350.

Select[Accumulate[Range[200]],PrimeQ[#+1]&] (* Harvey P. Dale, Nov 08 2011 *)

from sympy import isprime
def T(n): return n*(n+1)//2
def ok(T): return isprime(T+1)
print(list(filter(ok, (T(n) for n in range(175))))) # Michael S. Branicky, Jun 18 2021

a(n) = A000217(A067186(n)). - R. J. Mathar, Dec 10 2007

a(n) = A055469(n) - 1. - Joerg Arndt, Jun 19 2021

A129634 Least nonnegative m such that T(n) + T(m) is prime, where T(n) = n*(n+1)/2.

Original entry on oeis.org

2, 1, 0, 1, 1, 7, 4, 1, 1, 7, 3, 1, 1, 3, 16, 13, 1, 4, 4, 1, 1, 4, 4, 1, 46, 3, 7, 1, 2, 7, 16, 2, 13, 4, 3, 1, 13, 3, 4, 22, 1, 16, 16, 1, 1, 7, 3, 1, 10, 3, 7, 1, 2, 7, 16, 2, 1, 4, 4, 13, 1, 4, 16, 1, 1, 16, 4, 2, 1, 16, 8, 1, 10, 3, 7, 1, 1, 31, 7, 2, 13, 4, 4, 10, 1, 8, 7, 13, 1, 43, 16, 5, 25, 16

Offset: 0

Jonathan Vos Post, May 31 2007

What is the simplest proof that this is defined for all nonzero n?

It appears that a(n)A130504 provides evidence that a(n) exists for all n. - T. D. Noe, Jun 04 2007

			a(6) = 4 because T(4) = 10 is the least triangular number whose sum with T(6) = 21 is prime, since {21+0 = 3*7, 21+3 = 2^3*3, 21+6 = 3^3} are all composite, but 21+10 = 31 is prime.

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A000040, A000217, A129755, A130334.

Cf. A069003 (for square numbers).

nn=100; tri=Range[0,nn]Range[nn+1]/2; Table[k=1; While[k<=Length[tri] && !PrimeQ[tri[[k]]+tri[[n]]], k++ ]; If[k<=Length[tri], k-1,0], {n,Length[tri]}] (* T. D. Noe, Jun 04 2007 *)

a(n) = Min{m: m*(m+1)/2 + n*(n+1)/2 is prime}. a(n) = Min{m: A000217(m) + A000217(n) is an element of A000040}.

Corrected and extended by T. D. Noe, Jun 04 2007

A129546 Numbers k such that T(k)+10 is the next prime after T(k), where T(k) = A000217(k).

Original entry on oeis.org

58, 61, 98, 138, 193, 217, 222, 233, 253, 266, 338, 358, 373, 393, 398, 402, 453, 461, 466, 477, 481, 542, 553, 557, 586, 597, 602, 618, 633, 646, 662, 761, 822, 838, 853, 857, 877, 898, 901, 913, 918, 926, 941, 986, 1006, 1041, 1061, 1077, 1126, 1157, 1161

Offset: 1

Zak Seidov, May 30 2007

nonn

			T(58)=1711 and 1711+10=1721 is the least prime > 1711;
T(61)=1891 and 1891+10=1901 is the least prime > 1891.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000217, A129755, A130178.

nptQ[n_]:=Module[{tr=(n(n+1))/2},NextPrime[tr]-tr==10]; Select[ Range[ 1200], nptQ] (* Harvey P. Dale, Dec 19 2017 *)

isok(n) = t = n*(n+1)/2; nextprime(t+1) == (t + 10); \\ Michel Marcus, Oct 13 2013

A129540 Triangular numbers T such that T+10 is the next prime after T.

Original entry on oeis.org

1711, 1891, 4851, 9591, 18721, 23653, 24753, 27261, 32131, 35511, 57291, 64261, 69751, 77421, 79401, 81003, 102831, 106491, 108811, 114003, 115921, 147153, 153181, 155403, 171991, 178503, 181503, 191271, 200661, 208981, 219453

Offset: 1

Zak Seidov, May 29 2007

nonn

Cf. A000217, A129755.

a := proc (n) if nextprime((1/2)*n*(n+1)) = 10+(1/2)*n*(n+1) then (1/2)*n*(n+1) else end if end proc: seq(a(n), n = 1 .. 700); # Emeric Deutsch, Jun 20 2007

Select[Accumulate[Range[0, 700]],NextPrime[#]==#+10&] (* James C. McMahon, Jan 22 2025 *)

More terms from Emeric Deutsch, Jun 20 2007

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A129545 Triangular numbers T such that T+1 is a prime.

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A129634 Least nonnegative m such that T(n) + T(m) is prime, where T(n) = n*(n+1)/2.

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A129546 Numbers k such that T(k)+10 is the next prime after T(k), where T(k) = A000217(k).

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A129540 Triangular numbers T such that T+10 is the next prime after T.

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