A246827 Smallest x such that sigma(x)/x = 2*sigma(n)/n where sigma(n) is the sum of divisors of n.

6, 120, 84, 4320, 30, 30240, 42, 293760, 252, 3360, 66, 208565280, 78, 840, 420, 760320, 102, 18506880, 114, 131040, 1890, 1320, 138, 14182439040, 150, 1560, 756, 30240, 174, 668304000, 186, 1272960, 924, 2040, 210, 2068967577600, 222, 2280, 1092, 8910720, 246

Offset

1,1

Comments

When n is odd, and if there are infinitely many Mersenne primes, then a(n) would be at most equal to n multiplied by the smallest perfect number (A000396) whose prime Mersenne component (A000668) is coprime to n.

When n is even, there is no such obvious upper bound.

Conjecture: a(n) exists for all n.

It appears that a(n) is divisible by n.

Links

Michel Marcus, Table of n, a(n) for n = 1..500

M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious pairs, p. 16, 20, IJNT, to appear.

Michel Marcus, solveBA PARI script

P. Pollack and C. Pomerance, Some problems of Erdős on the sum-of-divisors function, (2015), p. 17, 22.

P. Pollack, C. Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B 3 (2016), 1-26.

Prog

(PARI) a(n) =  {nv = 2*sigma(n)/n; lim = 1; sv = []; while (#sv == 0, lim *= 10^10; sv = vecsort(solveBA(numerator(nv), denominator(nv), lim))); return (sv[1]); }

Crossrefs

Cf. A000203, A000396, A000668, A017665, A017666.

Sequence in context: A271648 A353691 A290341 * A127726 A117063 A178911

Adjacent sequences: A246824 A246825 A246826 * A246828 A246829 A246830

Keyword

nonn,look

Author

Michel Marcus, Sep 04 2014

Status

approved