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A345992
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Let m = A344005(n) = smallest m such that n divides m*(m+1); a(n) = gcd(n,m).
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26
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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
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OFFSET
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1,6
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COMMENTS
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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
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)
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LINKS
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FORMULA
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MAPLE
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ans:=[];
for n from 1 to 40 do t1:=Findm(n)[1]; ans:=[op(ans), igcd(n, t1)]; od:
ans;
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MATHEMATICA
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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 *)
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PROG
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(PARI) f(n) = my(m=1); while ((m*(m+1)) % n, m++); m; \\ A344005
(Python 3.8+)
from math import gcd, prod
from itertools import combinations
from sympy import factorint
from sympy.ntheory.modular import crt
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
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CROSSREFS
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Cf. A011772, A137827, A182665, A344000, A344005, A345993, A345994, A345995, A354930, A354931 (the least occurrence of each n=1..), A354984.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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