A334878 For any n > 0 with prime factorization Product_{k > 0} prime(k)^e_k (where prime(k) denotes the k-th prime number), let b_k = 1 + max_{k > 0} e_k; a(n) = Sum_{k > 0} e_k * b_k^(k-1).

0, 1, 2, 2, 4, 3, 8, 3, 6, 5, 16, 5, 32, 9, 6, 4, 64, 7, 128, 11, 10, 17, 256, 7, 18, 33, 12, 29, 512, 7, 1024, 5, 18, 65, 12, 8, 2048, 129, 34, 19, 4096, 11, 8192, 83, 15, 257, 16384, 9, 54, 19, 66, 245, 32768, 13, 20, 67, 130, 513, 65536, 14, 131072, 1025

Offset: 1

Author

Rémy Sigrist, May 14 2020

Keywords

• nonn
• base

Comments

In other words, a(n) encodes the prime factorization of n in base 1 + A051903(n).

Every nonnegative integer appears finitely many times in this sequence.

Examples

			For n = 84:
- 84 = 7 * 3 * 2^2 = prime(4) * prime(2) * prime(1)^2,
- b_84 = 1 + 2 = 3,
- so a(84) = 1*3^(4-1) + 1*3^(2-1) + 2*3^(1-1) = 32.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A051903, A087207.

Programs

PARI

a(n) = { if (n==1, 0, my (f=factor(n), b=1+vecmax(f[,2]~)); sum(k=1, #f~, f[k,2]*b^(primepi(f[k,1])-1))) }

Formula

a(2^e) = e for any e >= 0.

a(prime(k)) = 2^(k-1) for any k > 0.

a(prime(k)^e) = e*(e+1)^(k-1) for any k > 0 and e >= 0.

a(n) = A087207(n) for any squarefree number n.