A061537 -id:A061537 - cp's OEIS frontend

A306013 Let P(m) be the product of unitary divisors of m; then a(n) is the position of P(n) when all the numbers P(m) are arranged in increasing order.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 144, 196, 225, 324, 400, 441, 484, 576, 676, 784, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1936, 2025, 2116

Offset: 1

Clark Kimberling, Jun 24 2018

nonn

P(m) = A061537(m).

Cf. A077610, A229994, A229996, A305995, A306010, A306011, A306012.

z = 100; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &];
k[n_] := Length[r[n]];
Table[r[n], {n, 1, z}]
a[n_] := Apply[Times, r[n]]
u = Table[a[n], {n, 1, z}]
Sort[u]

A384716 The totient of the product of unitary divisors of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 12, 6, 4, 6, 40, 10, 48, 12, 84, 120, 8, 16, 108, 18, 160, 252, 220, 22, 192, 20, 312, 18, 336, 28, 216000, 30, 16, 660, 544, 840, 432, 36, 684, 936, 640, 40, 889056, 42, 880, 1080, 1012, 46, 768, 42, 1000, 1632, 1248, 52, 972, 2200, 1344, 2052

Offset: 1

Darío Clavijo, Jun 11 2025

nonn

a(n) is the totient of the product over all unitary divisors d of n; i.e., those divisors satisfying gcd(d, n/d) = 1.
Growth rate of a(n) is ~ n^O(log log n).
Also, a(n) is lower bounded by A000010(n).

Cf. A000010, A000225, A006093, A061537, A384763.

f[p_, e_, m_] := (p-1)*p^(e*m-1); a[n_] := Module[{fct = FactorInteger[n]}, Times @@ (f[#1, #2, 2^(Length[fct]-1)] & @@@ fct)]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jun 11 2025 *)

from sympy import factorint
def a(n):
    if n == 1: return 1
    factors = factorint(n)
    phi, w = 1, len(factors)
    for p, e in factors.items():
        phi *= (p - 1) * p**(e - 1)
    return n**((1 << (w-1)) - 1) * phi
print([a(n) for n in range(1, 58)])

a(n) = phi(Product_{d|n} d if gcd(d, n/d) = 1).
a(n) = n^(2^(omega(n)-1)-1) * phi(n).
a(n) = A000010(A061537(n)).
a(p) = p-1 for p prime.

cp's OEIS Frontend

A306013 Let P(m) be the product of unitary divisors of m; then a(n) is the position of P(n) when all the numbers P(m) are arranged in increasing order.

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