A345992 Let m = A344005(n) = smallest m such that n divides m*(m+1); a(n) = gcd(n,m).

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 5, 1, 1, 2, 1, 4, 3, 2, 1, 8, 1, 2, 1, 7, 1, 5, 1, 1, 11, 2, 7, 4, 1, 2, 3, 5, 1, 6, 1, 11, 9, 2, 1, 3, 1, 2, 17, 4, 1, 2, 5, 7, 3, 2, 1, 15, 1, 2, 9, 1, 5, 11, 1, 4, 23, 14, 1, 8, 1, 2, 3, 19, 7, 6, 1, 5, 1, 2, 1, 4, 17, 2, 29, 8, 1, 9, 13, 23, 3, 2, 19, 32, 1, 2, 11, 4, 1, 17

Offset: 1

Author

Robert Dougherty-Bliss and N. J. A. Sloane, Jul 15 2021

Keywords

• nonn

Comments

By definition, a(n)*A345993(n) = n.

a(n) is even iff n/2 is in A344000. This is true, but essentially trivial, and does not provide any insight into either sequence.

Empirical: For n >= 3, a(n) <= n/3, and a(n) = n/3 iff n is in 3*{2^odd, primes == -1 mod 6}.

If n = 2*p^k where p is an odd prime then m = A344005(n) = p^k - 1 and a(n) = 2. Conversely, it appears that if a(n) = 2 then n is twice an odd prime power. (Corrected by Antti Karttunen, Jun 14 2022)

a(n) = 1 if n is a prime power. - Chai Wah Wu, Jun 01 2022

From Antti Karttunen, Jun 14 2022: (Start)

Conversely, if a(n) = 1 [i.e., A345993(n) = n] then n is a power of prime. (This follows from N. J. A. Sloane's Jul 11 2021 theorem given in A344005).

Apparently, a(n) = 3 iff n = A354984(k) = 3*A137827(k), for some k >= 1.

(End)

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A011772, A137827, A182665, A344000, A344005, A345993, A345994, A345995, A354930, A354931 (the least occurrence of each n=1..), A354984.

Cf. also A007528, A051119, A284600.

Programs

Maple

# load Findm from A344005
ans:=[];
for n from 1 to 40 do t1:=Findm(n)[1]; ans:=[op(ans), igcd(n,t1)]; od:
ans;

Mathematica

smd[n_]:=Module[{m=1},While[Mod[m(m+1),n]!=0,m++];GCD[n,m]]; Array[smd,110] (* Harvey P. Dale, Jan 07 2022 *)

PARI

f(n) = my(m=1); while ((m*(m+1)) % n, m++); m; \\ A344005
a(n) = gcd(n,f(n)); \\ Michel Marcus, Aug 06 2021
(Python 3.8+)
from math import gcd, prod
from itertools import combinations
from sympy import factorint
from sympy.ntheory.modular import crt
def A345992(n):
    if n == 1:
        return 1
    plist = tuple(p**q for p, q in factorint(n).items())
    return 1 if len(plist) == 1 else gcd(n,int(min(min(crt((m, n//m), (0, -1))[0], crt((n//m, m), (0, -1))[0]) for m in (prod(d) for l in range(1, len(plist)//2+1) for d in combinations(plist, l))))) # Chai Wah Wu, Jun 01 2022

Formula

a(n) = gcd(n, A182665(n)) = gcd(A182665(n), A344005(n)). - Antti Karttunen, Jun 13 2022