A061510 -id:A061510 - cp's OEIS frontend

A061509 Write n in decimal, omit 0's, replace the k-th digit d[k] with the k-th prime, raised to d[k]-th power and multiply.

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 8, 24, 72, 216, 648, 1944, 5832, 17496, 52488, 157464, 16, 48, 144, 432, 1296, 3888, 11664, 34992, 104976, 314928

Offset: 0

Amarnath Murthy, May 06 2001

Not the same as A189398: see formula.

			a(4) = 2^4 = 16, a(123) = (2^1)*(3^2)*(5^3) = 2250.
For n = 0, the list of nonzero digits is empty, and the empty product equals 1.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (a(0) = 1 inserted by _M. F. Hasler_, Oct 12 2018)

Cf. A061510, A000040.

a061509 n = product $ zipWith (^)
  a000040_list (map digitToInt $ filter (/= '0') $ show n)
-- Reinhard Zumkeller, May 03 2011

A061509[n_] := If[n == 0, 1, Times @@ (Prime[Range[Length[#]]]^#) & [DeleteCases[IntegerDigits[n], 0]]];
Array[A061509, 100, 0] (* Paolo Xausa, Nov 26 2024 *)

A061509(n)=prod(k=1,#n=select(t->t,digits(n)),prime(k)^n[k]) \\ M. F. Hasler, Aug 16 2014

a(n) = a(n*10^k). a((10^k-1)/9) = primorial(k) = A002110(k).
a(n) = A189398(n) for n <= 100; a(101)=2^1*3^1 = 6 <> A189398(101) = 2^1*3^0*5^1 = 10; a(A052382(n)) = A189398(A052382(n)); a(n) = A000079(A000030(n)) if n has only one nonzero digit; A001221(a(n)) = A055640(n); A001222(a(n)) = A007953(n). - Reinhard Zumkeller, May 03 2011
If n=d[1]d[2]...d[m] in decimal (0M. F. Hasler, Aug 16 2014
A007814(a(n)) = A000030(n). - M. F. Hasler, Aug 18 2014

Corrected and extended by Matthew Conroy, May 13 2001
Offset corrected by Reinhard Zumkeller, May 03 2011
Definition corrected by M. F. Hasler, Aug 16 2014
Extended to a(0) = 1 by M. F. Hasler, Oct 12 2018

A334828 Numbers that divide the multiplication of its digits raised to their own powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 25, 36, 64, 96, 125, 128, 135, 162, 175, 216, 250, 256, 375, 378, 384, 432, 486, 567, 576, 625, 648, 672, 675, 729, 735, 756, 768, 784, 864, 875, 972, 1024, 1176, 1250, 1296, 1372, 1715, 1764, 1944, 2048, 2304, 2500, 2744, 2916, 3087, 3125, 3375, 3456, 3645, 3675, 4096

Offset: 1

Scott R. Shannon, May 13 2020

As in A045503 we take 0^0 = 1.

Numbers m that divide A061510(m).

			5 is a term as 5^5 = 3125 which is divisible by 5.
16 is a term as 1^1*6^6 = 46656 which is divisible by 16.
375 is a term as 3^3*7^7*5^5 = 69486440625 which is divisible by 375.
1176 is a term as 1^1*1^1*7^7*6^6 = 38423222208 which is divisible by 1176.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000312, A045503, A046253, A061510, A243507, A108302.

pow[n_] := If[n == 0, 1, n^n]; Select[Range[2^12], Divisible[Times @@ (pow /@ IntegerDigits[#]), #] &] (* Amiram Eldar, May 13 2020 *)

isok(m) = my(d=digits(m)); (prod(k=1, #d, d[k]^d[k]) % m) == 0; \\ Michel Marcus, May 14 2020

A378246 Integers that are equal to the product of their nonzero digits raised to their own power.

Original entry on oeis.org

1, 1024, 12500

Offset: 1

Jason Hammerman, Nov 20 2024

If one accepts 0^0 = 1, the "nonzero" part of the description is unnecessary.
From Michael S. Branicky, Nov 25 2024: (Start)
Terms must be 7-smooth (A002473).
a(4) > 10^200 if it exists. (End)

			1024  = 1^1 * 2^2 * 4^4.
12500 = 1^1 * 2^2 * 5^5.

Jason Hammerman, Python program
Wikipedia, Zero to the power of zero

Fixed points of A061510.
Inspired by A046253.
Cf. A002473.

F[a_]:=If[a==0,1,a^a];Select[Range[10^5],#==Times@@F/@IntegerDigits[#]&] (* James C. McMahon, Dec 14 2024 *)

isok(k) = my(d=digits(k)); k == vecprod(vector(#d, i, d[i]^d[i])); \\ Michel Marcus, Nov 22 2024

Python
```
# See Python program link.
```

from math import prod
def ok(n): return n == prod(d**d for d in map(int, str(n)) if d > 1)
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Nov 24 2024

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A061509 Write n in decimal, omit 0's, replace the k-th digit d[k] with the k-th prime, raised to d[k]-th power and multiply.

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A334828 Numbers that divide the multiplication of its digits raised to their own powers.

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A378246 Integers that are equal to the product of their nonzero digits raised to their own power.

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Mathematica

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