A330501 - OEIS

%I #17 Dec 23 2024 14:53:46

%S 3,2,7,0,13,23,19,31,71,109,31,47,37,71,43,163,67,71,271,79,61,307,67,

%T 331,73,103,79,271,211,151,541,127,97,883,103,181,109,151,281,379,163,

%U 167,127,571,179,433,139,191,631,199,151,487,157,271,163,223,227,541,421,239,181,311,251,811

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

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

%C 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.

%H M. F. Hasler, <a href="/A330501/b330501.txt">Table of n, a(n) for n = 0..9999</a>

%H Peter Luschny, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2019-December/">Hopping for primes</a>, SeqFan list, Dec 13 2019.

%e 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.

%e Starting with n = 1, we add 1: sum = 2, a prime, so a(1) = 2.

%e Starting with n = 2, we add 2: sum = 4, not prime, then 3: sum = 7, a prime, so a(2) = 7.

%e 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.

%t 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 *)

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

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

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

%K nonn

%O 0,1

%A _M. F. Hasler_, following an idea of _Peter Luschny_, Dec 16 2019