A330501 Smallest prime of the form m(m+1)/2 - n(n-3)/2 with m >= n, or 0 if no such prime exists.

3, 2, 7, 0, 13, 23, 19, 31, 71, 109, 31, 47, 37, 71, 43, 163, 67, 71, 271, 79, 61, 307, 67, 331, 73, 103, 79, 271, 211, 151, 541, 127, 97, 883, 103, 181, 109, 151, 281, 379, 163, 167, 127, 571, 179, 433, 139, 191, 631, 199, 151, 487, 157, 271, 163, 223, 227, 541, 421, 239, 181, 311, 251, 811

Offset

0,1

Comments

A mathematical game for kids suggested by Peter Luschny on the SeqFan list:

Start with n and then (cumulatively) add m = n, n+1, n+2, ... until a prime p = T(m) - n(n-3)/2 is reached. The only n which never yields a prime is n = 3.

Links

M. F. Hasler, Table of n, a(n) for n = 0..9999

Peter Luschny, Hopping for primes, SeqFan list, Dec 13 2019.

Example

Starting with n = 0, we add 0: sum = 0, not prime, then 1: sum = 1, not prime, then 2: sum = 3, a prime, so a(0) = 3.
Starting with n = 1, we add 1: sum = 2, a prime, so a(1) = 2.
Starting with n = 2, we add 2: sum = 4, not prime, then 3: sum = 7, a prime, so a(2) = 7.
Starting with n = 3 = T(2) = 2(2+1)/2 (triangular number, cf. A000217), we add 3 to get T(2) + 3 = T(3) = 6, then 4 to get T(3) + 4 = T(4) = 10, and so on. A triangular number T(n) = n(n+1)/2 > 3 is never prime, since either product of n and (n+1)/2, or product of n/2 and n+1. So a(3) = 0.

Mathematica

Array[If[# == 3, 0, Block[{m = #, p}, While[! PrimeQ[Set[p, m (m + 1)/2 - # (# - 3)/2]], m++]; p]] &, 64, 0] (* Michael De Vlieger, Dec 16 2019 *)

Prog

(PARI) apply( {A330501(n, p=n)=if(n!=3, while(!isprime(p+=n), n++); p)}, [0..399])

Crossrefs

Cf. A000217 (triangular numbers n(n+1)/2), A000096 (n(n+3)/2).

Cf. A329946 (primes not in this sequence), A330502 (corresponding m).

Sequence in context: A271610 A019951 A019971 * A010606 A348493 A317919

Adjacent sequences: A330498 A330499 A330500 * A330502 A330503 A330504

Keyword

nonn

Author

M. F. Hasler, following an idea of Peter Luschny, Dec 16 2019

Status

approved