OFFSET

1,1

COMMENTS

These primes were originally called "hidden primes", but since that term is already in use (see A187399) it has been replaced by an explicit definition.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

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.

Equivalently, a prime p is reachable if there exists m >= n such that p = m(m+1)/2 - n(n-3)/2.

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.

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.

Conjecture: The sequence is infinite.

For comparison the number of primes < 10^n:

n : 1 2 3 4 5 6 7 8

Ramanujan p. : 1, 10, 72, 559, 4459, 36960, 316066, 2760321, ...

Hidden primes : 2, 10, 49, 271, 1768, 34181, 601549,

Lesser twin p. : 2, 8, 35, 205, 1224, 8169, 58980, 440312, ...

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

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

LINKS

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

MAPLE

aList := proc(lim) local n, p, k, L:

L := select(isprime, {$1..lim}):

for n from 1 to iquo(lim, 2) do

p := n:

for k from n to 10000 do

p := p + k:

if isprime(p)

then L := L minus {p}: break fi;

if p > lim then break fi;

od:

od: sort(L) end:

aList(1111);

PROG

(SageMath)

def aSieve(lim):

S = Set(prime_range(lim))

for n in (1..lim//2):

p = n

for k in (n..10000):

p += k

if p > lim: break

if is_prime(p):

S = S.difference({p})

break

return sorted(S)

aSieve(1111)

(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

CROSSREFS

KEYWORD

nonn

AUTHOR

Peter Luschny, Dec 16 2019

EXTENSIONS

Edited by M. F. Hasler and N. J. A. Sloane, Dec 16 2019

STATUS

approved