A194099 Numbers m >= 2, such that, if a prime p divides m^2 - 1, then every prime q < p divides m^2 - 1 as well.

3, 5, 7, 11, 17, 19, 29, 31, 41, 49, 71, 161, 251, 449, 769, 881, 1079, 1429, 3431, 4159, 4801, 6049, 8749, 19601, 24751, 246401, 388961, 1267111

Offset

1,1

Comments

No more terms <= 10^8.

No more terms <= 2 * 10^38. - Charles R Greathouse IV, Aug 22 2011

All terms are odd. - Kausthub Gudipati, Aug 22 2011

Links

Table of n, a(n) for n=1..28.

Florian Luca and Filip Najman, On the largest prime factor of x^2 - 1, Math. Comp. 80 (2011), 429-435.

Formula

A055932 INTERSECT A005563. - R. J. Mathar, Aug 16 2011

a(n) = 2 * A141399(n) + 1. - Thomas Ordowski, Mar 29 2026

Example

881^2-1 = 776160 = 2^5 * 3^2 * 5 * 7^2 * 11 (all primes <= 11 appear), so 881 is a term.

Mathematica

Select[Range[1, 10^4], First@ # == 1 && If[Length@ # > 1, Union@ Differences@ # == {1}, True] &@ PrimePi@ Map[First, FactorInteger[#^2 - 1]] &] (* Michael De Vlieger, Jul 02 2016 *)

Prog

(PARI) isok(n) = my(f = factor(n^2-1)); #f~ == primepi(f[#f~, 1]); \\ Michel Marcus, Jul 02 2016
(Python)
from sympy import prime, primefactors
def is_ok(m):
    pf = primefactors(m*m - 1)
    return prime(len(pf)) == pf[-1]
print([ m for m in range(3, 1270000, 2) if is_ok(m)])
# Ken Clements, Jan 01 2026
(Python)
from itertools import islice
from heapq import heappop, heappush
from sympy import integer_nthroot, nextprime
def A194099_gen(): # generator of terms
    h, hset = [(1, (1, ))], {1}
    while True:
        m, ps = heappop(h)
        a, b = integer_nthroot(m+1, 2)
        if b:
            yield a
        for p in ps:
            mp = m*p
            if mp not in hset:
                heappush(h, (mp, ps))
                hset.add(mp)
        q = nextprime(max(ps, default=1))
        mp = m*q
        if mp not in hset:
            heappush(h, (mp, (ps+(q, ))))
            hset.add(mp)
A194099_list = list(islice(A194099_gen(), 28)) # Chai Wah Wu, Apr 04 2026

Crossrefs

Cf. A005563, A055932, A141399, A391602.

Sequence in context: A290283 A163420 A155489 * A045396 A155779 A337815

Adjacent sequences: A194096 A194097 A194098 * A194100 A194101 A194102

Keyword

nonn,more

Author

Vladimir Shevelev, Aug 15 2011

Extensions

Dubious claim of finiteness removed by Max Alekseyev, Mar 29 2026

Status

approved