A337183 If n is composite, a(n) is the number represented by the list of prime factors of n (with multiplicity, sorted in nondecreasing order) in base the sum of that list. If n is prime, a(n) = n.

2, 3, 10, 5, 13, 7, 86, 21, 19, 11, 115, 13, 25, 29, 1170, 17, 155, 19, 185, 37, 37, 23, 1641, 55, 43, 273, 271, 29, 235, 31, 22222, 53, 55, 67, 2233, 37, 61, 61, 2931, 41, 331, 43, 491, 401, 73, 47, 32211, 105, 353, 77, 625, 53, 3061, 91, 4765, 85, 91, 59, 3785, 61, 97, 553, 542906, 103, 571, 67

Offset: 2

Author

J. M. Bergot and Robert Israel, Jan 29 2021

Keywords

• nonn
• base
• look

Comments

If p is prime, a(p^k) = p*((k*p)^k-1)/(k*p-1).

If n = p*q with primes p <= q, a(n) = p^2 + p*q + q.

Examples

			For n = 12, the list of prime factors is [2,2,3], the base is 2+2+3=7, and a(12) = 223_7 = 115.
		

Links

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

Programs

Maple

f:= proc(n) local F,b,i;
  F:= sort(map(t -> t[1]$t[2],ifactors(n)[2]),`>`);
  b:= convert(F,`+`);
  (add(F[i]*b^(i-1),i=1..nops(F)));
end proc:
map(f, [$2..100]);

Mathematica

Array[If[PrimeQ@ #, #, FromDigits[#, Total[#]] &@ Flatten[ConstantArray @@@ FactorInteger[#]]] &, 66, 2] (* Michael De Vlieger, Jan 29 2021 *)