A231847 -id:A231847 - cp's OEIS frontend

A231988 Primes of the form triangular(p) + 1, where p is a prime.

7, 29, 67, 191, 277, 947, 1129, 2557, 5779, 8647, 13367, 14029, 16111, 22367, 24977, 31627, 36857, 47279, 64621, 87991, 93097, 96581, 98347, 109279, 126757, 163307, 239087, 282377, 329267, 342379, 352381, 372817, 411779, 448879, 468029, 471907, 491537, 519691, 531997

Offset: 1

Alex Ratushnyak, Nov 16 2013

nonn

Triangular(p) = p*(p+1)/2.

Cf. A000040, A000217, A231847.

t = Select[Prime[Range[300]], PrimeQ[# (# + 1)/2 + 1] &]; t*(t + 1)/2 + 1 (* T. D. Noe, Nov 19 2013 *)

a(n) = A000217(A231847(n))+1.

A343151 Numbers k such that 1+A083266(k) is prime.

Original entry on oeis.org

1, 3, 4, 7, 8, 11, 19, 23, 25, 27, 36, 43, 47, 50, 64, 71, 107, 131, 163, 167, 179, 211, 223, 225, 242, 243, 251, 271, 307, 343, 359, 419, 431, 439, 443, 467, 503, 571, 691, 751, 800, 811, 827, 839, 863, 900, 907, 947, 967, 971, 991, 1019, 1031, 1058, 1063, 1091, 1103, 1187, 1279, 1296, 1331

Offset: 1

J. M. Bergot and Robert Israel, Apr 07 2021

nonn

Numbers k such that the sum of divisors of k + the sum of totatives of k is prime.

			a(4) = 7 is a term because 1+A083266(n) = 29 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A083266, A343148. Includes A231847.

f:= n -> numtheory:-sigma(n) + n*numtheory:-phi(n)/2: f(1):= 2:
select(t -> isprime(f(t)), [$1..2000]);

{1}~Join~Select[Range[1331], PrimeQ[DivisorSigma[1, #] + # EulerPhi[#]/2] &] (* Michael De Vlieger, Apr 07 2021 *)

A357218 Primes p such that T(p) - 2 is prime, where T(p) is the triangular number (A000217) with index p.

Original entry on oeis.org

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 149, 157, 193, 197, 233, 257, 269, 277, 281, 313, 337, 389, 401, 409, 457, 509, 521, 541, 613, 641, 673, 701, 797, 857, 877, 881, 929, 953, 997, 1009, 1033, 1093, 1109, 1117, 1129, 1153, 1193, 1297, 1301, 1373, 1381, 1433, 1481, 1493

Offset: 1

Bernard Schott, Sep 18 2022

nonn

T(p) must be odd, so these primes p satisfy p == 1 (mod 4) (A002144).
Corresponding values of T(p)-2 are in A357219.
The first eleven primes == 1 (mod 4) are terms. The smallest Pythagorean prime that is not a term is A002144(12) = 101 because T(101) = 5151 and 5151 - 2 = 5149 = 19 * 271 (see Wells reference).

			T(5) - 2 = 5*6/2 - 2 = 13, hence 5 is a term.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 496, page 142.

Cf. A000217, A231847, A357219.

Subsequence of A002144.

filter := p -> isprime(p) and irem(p-1, 4) = 0 and isprime(p*(p+1)/2 -2) : select(filter, [$1 .. 1500]);

Select[Prime[Range[240]], PrimeQ[#*(# + 1)/2 - 2] &] (* Amiram Eldar, Sep 18 2022 *)

A231989 Least prime p such that f(0),...,f(n) are all primes, where f(0) = p, then f(i+1) = triangular(f(i))+1.

Original entry on oeis.org

3, 3, 43, 236367611, 31542795419

Offset: 1

Alex Ratushnyak, Nov 16 2013

Triangular(p) = p*(p+1)/2 (see A034953).

			a(2) = 3 because 3 is the least prime such that the following are two primes:
  p1 = 3 * 4 / 2 + 1 = 7.
  p2 = 7 * 8 / 2 + 1 = 29.
a(3) = 43 because 43 is the least prime such that the following are three primes:
  p1 = 43 * 44 / 2 + 1 = 947.
  p2 = 947 * 948 / 2 + 1 = 448879.
  p3 = 448879 * 448880 / 2 + 1 = 100746402761.

Cf. A000040, A000217, A034953, A231847, A231988.

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A231988 Primes of the form triangular(p) + 1, where p is a prime.

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A343151 Numbers k such that 1+A083266(k) is prime.

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A357218 Primes p such that T(p) - 2 is prime, where T(p) is the triangular number (A000217) with index p.

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A231989 Least prime p such that f(0),...,f(n) are all primes, where f(0) = p, then f(i+1) = triangular(f(i))+1.

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