A340592 a(n) is the concatenation of the prime factors (with multiplicity) of n mod n.

0, 0, 2, 0, 5, 0, 6, 6, 5, 0, 7, 0, 13, 5, 14, 0, 17, 0, 5, 16, 13, 0, 15, 5, 5, 9, 3, 0, 25, 0, 14, 14, 13, 22, 1, 0, 29, 1, 25, 0, 27, 0, 11, 20, 39, 0, 47, 28, 5, 11, 29, 0, 11, 16, 43, 34, 55, 0, 15, 0, 45, 22, 14, 58, 1, 0, 41, 47, 47, 0, 57, 0, 15, 55, 15, 18, 51, 0, 65, 12, 77, 0, 53, 7

Offset: 2

Author

J. M. Bergot and Robert Israel, Jan 12 2021

Keywords

• nonn
• base

Comments

a(n) = 0 if n is prime.

The first composite n for which a(n)=0 is 28749. Are there others?

There are no other composite n terms for which a(n)=0 up to 5 million. - Harvey P. Dale, Jul 17 2023

Examples

			For n = 20 = 2*2*5, a(20) = 225 mod 20 = 5.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A037276, A340594, A340595.

Programs

Maple

dcat:= proc(L) local i,x;
  x:= L[-1];
  for i from nops(L)-1 to 1 by -1 do
    x:= 10^(1+ilog10(x))*L[i]+x
  od;
  x
end proc:
f:= proc(n) local F;
  F:= sort(ifactors(n)[2],(a,b) -> a[1] < b[1]);
  dcat(map(t -> t[1]$t[2], F)) mod n;
end proc:
map(f, [$2..100]);

Mathematica

Table[Mod[FromDigits[Flatten[IntegerDigits/@Table[#[[1]],#[[2]]]&/@FactorInteger[n]]],n],{n,2,100}] (* Harvey P. Dale, Jul 17 2023 *)

Python

from sympy import factorint
def a(n):
    if n == 1: return 0
    return int("".join(str(f) for f in factorint(n, multiple=True)))%n
print([a(n) for n in range(2, 86)]) # Michael S. Branicky, Jan 18 2022

Formula

a(n) = A037276(n) mod n.