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%I #35 Jan 02 2023 12:30:54
%S 3,5,11,17,29,41,53,59,83,89,101,107,113,131,137,149,173,197,233,257,
%T 269,293,317,353,389,419,443,449,461,467,509,557,563,569,587,593,617,
%U 653,677,761,773,797,809,827,857,929,941,947,977,1013,1049,1097,1109
%N Primes that are not of the form u(u+1)/2 - v(v-3)/2 for any u >= v >= 1.
%C These primes were originally called "hidden primes", but since that term is already in use (see A187399) it has been replaced by an explicit definition. - Editors of OEIS, Dec 16 2019
%C The following is the original definition. Assume n and s are positive integers. We say a prime p is 'reachable' from n if there exists an s such that 8*(p - (s + 1)*n) + 1 is a perfect square, and that a prime p is 'hidden' if it is not reachable from any n.
%C Equivalently, a prime p is reachable if there exists m >= n such that p = m(m+1)/2 - n(n-3)/2.
%C A description of the sequence as an arithmetic training game for children was given on the Sequence Fans Mailing List. A representation as a sieve is given in the Maple script.
%C The game is to start at n and (cumulatively) add n, n+1, n+2, ..., m until a prime is reached, which appears to happen for all n, usually with m close to n, except for n = 3.
%C Conjecture: The sequence is infinite.
%C For comparison the number of primes < 10^n:
%C n : 1 2 3 4 5 6 7 8
%C Ramanujan p. : 1, 10, 72, 559, 4459, 36960, 316066, 2760321, ...
%C Hidden primes : 2, 10, 49, 271, 1768, 34181, 601549,
%C Lesser twin p. : 2, 8, 35, 205, 1224, 8169, 58980, 440312, ...
%C All terms except a(1) = 3 are congruent to 5 (mod 6), i.e., in A007528. Indeed, any prime p = 6k + 1 is reached from n = 2k in 2 steps. - _M. F. Hasler_, Dec 16 2019
%C Only one prime is eliminated (for n != 3) by each (variable sized) "grid" G(n) = (2n, 3n + 1, 4n + 3, 5n + 6, ..., (m+2)n + T(m), ...), since the scan stops as soon as the first prime is found. If used as a sieve in the usual sense, the grid G(n) should also eliminate all subsequent primes of the form (m+2)n + T(m). If this were done, only Fermat primes A019434 = {3, 5, 17, 257, 65537, ?} would remain. - _M. F. Hasler_, Dec 17 2019
%H Peter Luschny, <a href="http://list.seqfan.eu/oldermail/seqfan/2019-December/">Hopping for primes</a>, SeqFan list, Dec 13 2019.
%p aList := proc(lim) local n, p, k, L:
%p L := select(isprime, {$1..lim}):
%p for n from 1 to iquo(lim, 2) do
%p p := n:
%p for k from n to 10000 do
%p p := p + k:
%p if isprime(p)
%p then L := L minus {p}: break fi;
%p if p > lim then break fi;
%p od:
%p od: sort(L) end:
%p aList(1111);
%o (SageMath)
%o def aSieve(lim):
%o S = Set(prime_range(lim))
%o for n in (1..lim//2):
%o p = n
%o for k in (n..10000):
%o p += k
%o if p > lim: break
%o if is_prime(p):
%o S = S.difference({p})
%o break
%o return sorted(S)
%o aSieve(1111)
%o (PARI) A329946=setminus(primes(199), Set(apply((n,p=n)->while(!isprime(p+=n), n++); p, [1..1199][^3]))) \\ _M. F. Hasler_, Dec 16 2019
%Y Cf. A000040, A000217 (triangular numbers), A000096 (n(n+3)/2), A187399, A330501 (least prime m(m+1)/2 - n(n-3)/2, m >= n), A330502 (corresponding m).
%K nonn
%O 1,1
%A _Peter Luschny_, Dec 16 2019
%E Edited by _M. F. Hasler_ and _N. J. A. Sloane_, Dec 16 2019
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