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A258409 Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n. 6
1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 2, 1, 22, 1, 4, 1, 2, 1, 28, 1, 30, 1, 2, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 2, 1, 46, 1, 6, 1, 2, 1, 52, 1, 2, 1, 2, 1, 58, 1, 60, 1, 2, 1, 4, 1, 66, 1, 2, 1, 70, 1, 72, 1, 2, 1, 2, 1, 78, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

a(n) = 1 for even n; a(p) = p-1 for prime p.

a(n) is even for odd n (since all divisors of n are odd).

It appears that a(n) = A052409(A005179(n)), i.e., it is the largest integer power of the smallest number with exactly n divisors. - Michel Marcus, Nov 10 2015

Conjecture: GCD of all (p-1) for prime p|n. - Thomas Ordowski, Sep 14 2016

Conjecture is true, because the set of numbers == 1 (mod g) is closed under multiplication. - Robert Israel, Sep 14 2016

LINKS

Ivan Neretin, Table of n, a(n) for n = 2..10000

EXAMPLE

65 has divisors 1, 5, 13, and 65, hence a(65) = gcd(1-1,5-1,13-1,65-1) = gcd(0,4,12,64) = 4.

MAPLE

f:= n -> igcd(op(map(`-`, numtheory:-factorset(n), -1))):

map(f, [$2..100]); # Robert Israel, Sep 14 2016

MATHEMATICA

Table[GCD @@ (Divisors[n] - 1), {n, 2, 100}]

PROG

(PARI) a(n) = my(g=0); fordiv(n, d, g = gcd(g, d-1)); g; \\ Michel Marcus, May 29 2015

(PARI) a(n) = gcd(apply(x->x-1, divisors(n))); \\ Michel Marcus, Nov 10 2015

(PARI) a(n)=if(n%2==0, return(1)); if(n%3==0, return(2)); if(n%5==0 && n%4 != 1, return(2)); gcd(apply(p->p-1, factor(n)[, 1])) \\ Charles R Greathouse IV, Sep 19 2016

(Haskell)

a258409 n = foldl1 gcd $ map (subtract 1) $ tail $ a027750_row' n

-- Reinhard Zumkeller, Jun 25 2015

CROSSREFS

Cf. A049559, A057237, A060680, A063994, A187730, A027750.

Cf. A084190 (similar but with LCM).

Sequence in context: A247339 A281071 A256908 * A060680 A057237 A187730

Adjacent sequences:  A258406 A258407 A258408 * A258410 A258411 A258412

KEYWORD

nonn

AUTHOR

Ivan Neretin, May 29 2015

STATUS

approved

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Last modified May 24 04:25 EDT 2019. Contains 323528 sequences. (Running on oeis4.)