A367168 The largest unitary divisor of n that is a term of A138302.

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70

Offset: 1

Author

Amiram Eldar, Nov 07 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A056192 at n = 32 and from A270418 at n = 128.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A048298, A065465, A138302, A367169, A367170, A367171.

Cf. A056192, A270418.

Similar sequences: A350388, A350389, A360539.

Programs

Mathematica

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 << valuation(f[i, 2], 2), f[i, 1]^f[i, 2], 1));}

Python

from math import prod
from sympy import factorint
def A367168(n): return prod(p**e for p,e in factorint(n).items() if not(e&-e)^e) # Chai Wah Wu, Nov 10 2023

Formula

Multiplicative with a(p^e) = p^A048298(e).

a(n) <= n, with equality if and only if n is in A138302.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 0.881513... (A065465).