A188591 -id:A188591 - cp's OEIS frontend

A188550 Maximal number of divisors d>1 of n-k such that n-d is a multiple of k, when k runs through values 2, 3, ..., floor(sqrt(n)).

1, 1, 2, 1, 2, 2, 3, 2, 2, 1, 4, 3, 2, 3, 4, 2, 3, 4, 4, 3, 2, 3, 6, 4, 4, 3, 4, 3, 4, 4, 5, 4, 4, 3, 6, 6, 3, 3, 6, 3, 4, 4, 4, 5, 4, 4, 8, 5, 6, 3, 4, 4, 4, 6, 6, 4, 4, 1, 8, 6, 4, 6, 6, 3, 5, 4, 4, 3, 4, 3, 9, 8, 6, 5, 6, 3, 4, 4, 8, 5, 6, 5, 8, 6, 4, 3, 6, 6, 6, 8, 6, 6, 4, 3, 10, 6, 8, 5, 6, 4, 6, 6, 6, 7, 4, 3, 8, 9, 4, 4, 8, 5, 6, 6, 4, 5, 4

Offset: 4

Vladimir Shevelev, Apr 04 2011

nonn

Conjecture: if the definition is changed so that k runs through values 2, 3, ..., floor((n-2)/2) then, beginning with n=6, the sequence remains without changes. - Vladimir Shevelev, Apr 10 2011
From Vladimir Shevelev, Jan 21 2013: (Start)
Other conjectures:
1) Primes 5, 7, 13 are only primes p for which a(p) = 1;
2) Primes 11 and 19 are only primes p for which a(p) = 2;
3) Let n = m^2 and m be the least value of k for which the number of divisors d > 1 of n-k, such that k|(n-d), equals a(n). Then m is prime or even power of a prime. (End)

Alois P. Heinz, Table of n, a(n) for n = 4..10004

Cf. A000005, A188579, A188586, A188591, A188592.

with(numtheory):
a:= n-> max(seq(nops(select(x-> irem(x, k)=0,
    [seq(n-d, d=divisors(n-k) minus{1})])), k=2..floor(sqrt(n)))):
seq(a(n), n=4..120);  # Alois P. Heinz, Apr 04 2011

a[n_] := Max @ Table[ Length @ Select[Table[n-d, {d, Divisors[n-k] // Rest} ], Mod[#, k] == 0&], {k, 2, Floor[Sqrt[n]]}]; Table[a[n], {n, 4, 120}] (* Jean-François Alcover, Feb 06 2016, after Alois P. Heinz *)

lim sup_{n -> infinity} a(n) = infinity. Indeed, it is easy to show that a(2^(2^n+1)) >= 2^n. Moreover, for n>5, we have a(2^(2^n+1)) > 2^n. - Vladimir Shevelev, Apr 09 2011

A188592 Places of records of A188550.

Original entry on oeis.org

4, 6, 10, 14, 26, 50, 74, 98, 122, 242, 362, 482, 722, 1442, 1682, 2522, 3362, 5042, 10082, 15122, 20162, 30242, 40322, 50402, 55442, 90722, 100802, 110882, 166322, 221762, 332642, 443522

Offset: 1

Vladimir Shevelev, Apr 04 2011

Questions:
1) Are there any terms after a(9) which are not of the form 10*k+2?
2) For which primes p do there exist terms in the sequence of the form 2*p^2? The sequence of these primes starts with 5, 7, 11, 19, 29, 41, 71, ...

Cf. A188550, A188579, A188591.

More terms from Alois P. Heinz, Apr 04 2011

cp's OEIS Frontend

A188550 Maximal number of divisors d>1 of n-k such that n-d is a multiple of k, when k runs through values 2, 3, ..., floor(sqrt(n)).

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