A085073 Smallest k such that n+k and n*k have the same prime signature, or 0 if no such number exists.

2, 1, 7, 41, 15, 134, 3, 127, 11, 2, 3, 548, 2, 1, 3, 389, 5, 582, 2, 316, 1, 38, 3, 2216, 3, 2, 13, 212, 5, 2742, 2, 1669, 1, 1, 31, 2764, 2, 1, 13, 1094, 4, 2298, 3, 1, 123, 14, 11, 8912, 3, 202, 17, 2, 2, 1146, 23, 904, 1, 26, 3, 11028, 13, 22, 57, 3581, 37, 1194, 2, 172, 15

Offset: 1

Author

Amarnath Murthy, Jul 01 2003

Keywords

• nonn

Examples

			a(6) = 379 as 6*379 = 2*3*379 and 6+379 = 385 = 5*7*11 both have prime signature p*q*r.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..3000

Crossrefs

Cf. A052213 (a(n)=1), A085072.

Programs

Maple

s:= proc(n) s(n):= sort(map(i-> i[2], ifactors(n)[2])) end:
a:= proc(n) option remember; local k; for k
       while s(n*k)<>s(n+k) do od; k
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 06 2019

Mathematica

kmax = 10^6;
s[n_] := FactorInteger[n][[All, 2]] // Sort;
a[n_] := Module[{k}, If[n == 1, Return[2]]; For[k = 1, k <= kmax, k++, If[s[n k] == s[n+k], Return[k]]]; 0];
Array[a, 70] (* Jean-François Alcover, Nov 17 2020 *)

PARI

sgntr(n) = vecsort(factor(n)[, 2]~);
a(n) = {my(k=1); while (sgntr(n+k) != sgntr(n*k), k++); k; } \\ Michel Marcus, Nov 17 2020

Extensions

Corrected by Jason Earls, Jul 10 2003

More terms from David Wasserman, Jan 12 2005