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A329946 Primes that are not of the form u(u+1)/2 - v(v-3)/2 for any u >= v >= 1. 4
3, 5, 11, 17, 29, 41, 53, 59, 83, 89, 101, 107, 113, 131, 137, 149, 173, 197, 233, 257, 269, 293, 317, 353, 389, 419, 443, 449, 461, 467, 509, 557, 563, 569, 587, 593, 617, 653, 677, 761, 773, 797, 809, 827, 857, 929, 941, 947, 977, 1013, 1049, 1097, 1109 (list; graph; refs; listen; history; text; internal format)
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These primes were originally called "hidden primes", but since that term is already in use (see A187399) it has been replaced by an explicit definition.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

Table of n, a(n) for n=1..53.

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

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).

Sequence in context: A144105 A141262 A069233 * A063700 A078859 A054799

Adjacent sequences:  A329943 A329944 A329945 * A329947 A329948 A329949

KEYWORD

nonn

AUTHOR

Peter Luschny, Dec 16 2019

EXTENSIONS

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

STATUS

approved

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Last modified October 15 20:32 EDT 2021. Contains 348034 sequences. (Running on oeis4.)