A039696 -id:A039696 - cp's OEIS frontend

A303278 If n = Product_j p_j^k_j where the p_j are distinct primes then a(n) = (Product_j k_j)^(Product_j p_j).

1, 1, 1, 4, 1, 1, 1, 9, 8, 1, 1, 64, 1, 1, 1, 16, 1, 64, 1, 1024, 1, 1, 1, 729, 32, 1, 27, 16384, 1, 1, 1, 25, 1, 1, 1, 4096, 1, 1, 1, 59049, 1, 1, 1, 4194304, 32768, 1, 1, 4096, 128, 1024, 1, 67108864, 1, 729, 1, 4782969, 1, 1, 1, 1073741824, 1, 1, 2097152, 36, 1, 1, 1, 17179869184, 1, 1, 1, 46656, 1, 1, 32768

Offset: 1

Ilya Gutkovskiy, Apr 20 2018

nonn

This is different from A008477, which is Product_j k_j^p_j. - N. J. A. Sloane, May 01 2021

			a(36) = a(2^2 * 3^2) = (2*2)^(2*3) = 4^6 = 4096.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A000026, A000142, A005117 (indices of ones), A005361, A007947, A008477, A034386, A039696, A135291, A285769, A303277.

Table[Times@@Transpose[FactorInteger[n]][[2]]^Last[Select[Divisors[n], SquareFreeQ]], {n, 75}]

a(n) = my(f=factor(n)); factorback(f[, 2])^factorback(f[, 1]); \\ Michel Marcus, Apr 21 2018

a(n) = tau(n/rad(n))^rad(n) = A005361(n)^A007947(n).
a(p^k) = k^p where p is a prime.
a(A000142(k)) = A135291(k)^A034386(k).

Definition clarified by N. J. A. Sloane, May 01 2021

A039777 Integers m such that phi(m) is equal to the sum of (the product of prime factors) and (the product of exponents) of m-1.

Original entry on oeis.org

2, 5, 21, 45, 285, 765, 27645, 196605, 41067645, 72787965, 250871805, 4295098365, 12884901885, 23307153405, 172130669565, 1766029428523005, 20978888016396285

Offset: 1

Olivier Gérard

No other terms below 10^24. Some large terms: 1039619980803100740810795122685, 32576974833437288924302842789885. - Max Alekseyev, Jul 28 2024
All listed terms represent solutions to phi(m) = (m+3)/2 such that (m-1)/2 is an even squarefree number. Cf. A350777. - Max Alekseyev, Jul 21 2024
a(1)=2 is the only even term below 10^100000. - Max Alekseyev, Jul 22 2024

			21 is a term since 21-1 = 2^2*5^1 and (2*5)+(2*1) = 12 = phi(21).

Cf. A000010, A039696, A350777.

More terms from Jud McCranie
Corrected example and a(11)-a(14) from Donovan Johnson, Nov 14 2010
a(15)-a(17) from Max Alekseyev, Jul 21 2024

cp's OEIS Frontend

A303278 If n = Product_j p_j^k_j where the p_j are distinct primes then a(n) = (Product_j k_j)^(Product_j p_j).

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