A329946 - OEIS

%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%C These primes were originally called "hidden primes", but since that term is already in use (see A187399) it has been replaced by an explicit definition.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