A308317 Multiplicative with a(prime(k)^e) = A005117(e+1)^(2^(k-1)).

1, 2, 4, 3, 16, 8, 256, 5, 9, 32, 65536, 12, 4294967296, 512, 64, 6, 18446744073709551616, 18, 340282366920938463463374607431768211456, 48, 1024, 131072, 115792089237316195423570985008687907853269984665640564039457584007913129639936, 20, 81, 8589934592, 25

Offset: 1

Author

Rémy Sigrist, May 19 2019

Keywords

• nonn
• mult

Comments

This sequence is a permutation of the natural numbers (with inverse A308328).

The property of being a bijection is easily deduced from the Fermi-Dirac representation of a number.

The first known fixed points are: 1, 2, 9, 12, 18, 3584, 32256.

Examples

			a(3) = a(prime(2)) = A001146(1) = 2^(2^1) = 4.
a(2^5) = A005117(6) = 7.
a(96) = a(2^5 * 3) = a(2^5) * a(3) = 7 * 4 = 28.
		

Links

• OEIS Wiki, "Fermi-Dirac representation" of n
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000120, A001146, A001221, A005117, A267116, A308328 (inverse), A318363.

Programs

PARI

A005117(n) = for (k=1, oo, if (issquarefree(k), if (n--==0, return (k))))
a(n) = my (f=factor(n)); prod (i=1, #f~, A005117(1+f[i,2])^(2^(primepi(f[i,1])-1)))

Formula

a(2^e) = A005117(e+1) for any e >= 0.

a(prime(k)) = A001146(k-1) for any k > 0.

A000120(A267116(a(n))) = A001221(n).