

A329946


Primes that are not of the form u(u+1)/2  v(v3)/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These primes were originally called "hidden primes", but since that term is already in use (see A187399) it has been replaced by an explicit definition.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(n3)/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(n3)/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



