A280163 Least number k such that sopfr(k) - sopf(k) = k/n, -1 if such a number does not exist.

4, 9, 32, 25, 12, 49, 48, 54, 20, 121, 96, 169, 28, 45, 144, 289, 162, 361, 160, 63, 44, 529, -1, 250, 52, -1, 224, 841, 60, 961, 320, 99, 68, 175, 180, 1369, 76, 117, 240, 1681, 84, 1849, 352, 270, 92, 2209, -1, 686, -1, 153, 416, 2809, -1, 275, 336, 171, 116

Offset

2,1

Comments

a(n) = n^2 for n prime.

Values equal to -1 are hypothetical (tested up to 2*10^10 by Giovanni Resta).

Links

Table of n, a(n) for n=2..58.

Example

a(6) = 12 because 12 is the least number such that sopfr(12) - sopf(12) = 7 - 5 = 2 = 12/6.

Maple

with(numtheory): P:=proc(q) local a, j, k, n;
for n from 2 to q do for j from n by n to q do
a:=ifactors(j)[2]; if add(a[k][1]*a[k][2], k=1..nops(a))-add(a[k][1], k=1..nops(a))=j/n then
lprint(n, j); break; fi; od; od; end: P(10^6);

Mathematica

Table[SelectFirst[Range[n^3], Function[k, Total@ # - Total@ Union@ # == k/n &@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, #] &@ FactorInteger@ k]] /. k_ /; MissingQ@ k -> -1, {n, 2, 58}] (* Version 10.2, or *)
Table[k = 1; While[And[Total@ # - Total@ Union@ # != k/n, k <= n^3] &@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, #] &@ FactorInteger@ k, k++]; If[k > n^3, -1, k], {n, 2, 58}] (* Michael De Vlieger, Dec 28 2016 *)

Crossrefs

Cf. A007947, A008472.

Sequence in context: A270206 A271461 A272423 * A361987 A071378 A053192

Adjacent sequences: A280160 A280161 A280162 * A280164 A280165 A280166

Keyword

sign

Author

Paolo P. Lava, Dec 27 2016

Status

approved