A061509 -id:A061509 - cp's OEIS frontend

A054842 If n = a + 10 * b + 100 * c + 1000 * d + ... then a(n) = (2^a) * (3^b) * (5^c) * (7^d) * ...

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 27, 54, 108, 216, 432, 864, 1728, 3456, 6912, 13824, 81, 162, 324, 648, 1296, 2592, 5184, 10368, 20736, 41472, 243, 486, 972, 1944

Offset: 0

Henry Bottomley, Apr 11 2000

a((10^k-1)/9) = Primorial(k)= A061509((10^k-1)/9). This is a rearrangement of whole numbers. a(m) = a(n) iff m = n. (Unlike A061509, in which a(n) = a(n*10^k).) - Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 14 2003

Part of the previous comment is incorrect: as a set, this sequence consists of numbers n such that the largest exponent appearing in the prime factorization of n is 9. So this cannot be a rearrangement (or permutation) of the natural numbers. - Tom Edgar, Oct 20 2015

			a(15)=96 because 3^1 * 2^5 = 3*32 = 96.

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999

Cf. A054841, A069877, A085840.

Cf. analogous sequences for other bases: A019565 (base 2), A101278 (base 3), A101942 (base 4), A101943 (base 5), A276076 (factorial base), A276086 (primorial base).

a054842 = f a000040_list 1 where
   f _      y 0 = y
   f (p:ps) y x = f ps (y * p ^ d) x'  where (x', d) = divMod x 10
-- Reinhard Zumkeller, Aug 03 2015

A054842[n_] := Times @@ (Prime[Range[Length[#], 1, -1]]^#) & [IntegerDigits[n]];
Array[A054842, 100, 0] (* Paolo Xausa, Nov 25 2024 *)

a(n)= my(d=Vecrev(digits(n))); factorback(primes(#d), d); \\ Ruud H.G. van Tol, Nov 28 2024

a(n) = f(n, 1, 1) with f(x, y, z) = if x > 0 then f(floor(x/10), y*prime(z)^(x mod 10), z+1) else y. - Reinhard Zumkeller, Mar 13 2010

A069877 Smallest number with a prime signature whose indices are the decimal digits of n.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 2, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 4, 12, 36, 72, 144, 288, 576, 1152, 2304, 4608, 8, 24, 72, 216, 432, 864, 1728, 3456, 6912, 13824, 16, 48, 144, 432, 1296, 2592, 5184, 10368, 20736, 41472, 32, 96, 288, 864, 2592, 7776, 15552, 31104, 62208, 124416, 64, 192, 576, 1728, 5184, 15552, 46656, 93312, 186624, 373248, 128

Offset: 0

Amarnath Murthy, Apr 25 2002

From Antti Karttunen, Nov 17 2016: (Start)
This is a filter-sequence for decimal base: a(n) = the least number with the same prime signature as A054842(n).
This sequence can be used for filtering certain base-10 related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A054842(n)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").
Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.
Any such sequence should match where the result is computed from the nonzero decimal digits of n, but does not depend on their order. These include for example, A007953 (digital sum and any of its variants), A010888 (digital root of n) and A051801 (product of the nonzero digits of n). As of Nov 11 2016, there were a couple of hundred such sequences that seemed to match with this one. These are given at the "List of sequences whose equivalence classes ..." link.
(End)

			a(12) = 2^2 * 3^1 = 12. a(231) = 2^3 * 3^2 * 5^1 = 360.

Cf. A046523, A054842, A061509.
Cf. A278222, A278226, A278236 for similar filter sequences constructed for other bases.
Sequences that partition N into same or coarser equivalence classes: too numerous to list all here, but at least A007953, A010888, A051801 are included. See the separate list given in links.

a(n) = A046523(A054842(n)). - Antti Karttunen, Nov 16 2016

More terms from Larry Reeves (larryr(AT)acm.org), Jul 05 2002

a(0)=1 prepended and more terms added by Antti Karttunen, Nov 16 2016

A189398 a(n) = 2^d(1) * 3^d(2) * ... * prime(k)^d(k), where d(1)d(2)...d(k) is the decimal representation of n.

Original entry on oeis.org

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, 32

Offset: 1

Reinhard Zumkeller, May 03 2011

Not the same as A061509: a(n) = A061509(n) for n <= 100; a(101)=2^1*3^0*5^1=10 <> A061509(101)=2^1*3^1=6;
a(A052382(n)) = A000079(A000030(a052382(n))) = A061509(A052382(n));
a(A002275(n)) = A002110(n): a(n-th rep-unit) = n-th primorial;
a(n*A011557(k)) = a(n): trailing zeros don't matter;
A001221(a(n)) = A055640(n): number of distinct prime factors of a(n) = number of nonzero digits of n;
A001222(a(n)) = A007953(n): number of all prime factors of a(n) = sum of digits of n;
a(81312000) = 2^8*3^1*5^3*7^1*11^2*13^0*17^0*19^0 = 81312000, the smallest fixed point, is called the Meertens number.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Richard S. Bird, Functional Pearl: Meertens number, Journal of Functional Programming 8 (1), Jan 1998, 83-88.
Wikipedia, Meertens number

Cf. A000040.

import Data.Char (digitToInt)
import Data.List (findIndices)
a189398 n = product $ zipWith (^) a000040_list (map digitToInt $ show n)
-- Two computations of the Meertens number: the first is brute force,
meertens = map succ $ findIndices (\x -> a189398 x == x) [1..]
-- ... and the second is more efficient, from Bird reference, page 87:
meertens' k = [n | (n,g) <- candidates (0,1), n == g] where
  candidates        = concat . map (search pps) . tail . labels ps
  ps : pps          = map (\p -> iterate (p *) 1) $ take k a000040_list
  search [] x       = [x]
  search (ps:pps) x = x : concat (map (search pps) (labels ps x))
  labels ps (n,g)   = zip (map (10*n +) [0..9]) (chop $ map (g *) ps)
  chop              = takeWhile (< 10^k)
-- Time and space required, GHC interpreted, Mac OS X, 2.66 GHz:
-- for >head meertens: (466.87 secs, 254780027728 bytes);
-- for >meertens' 8:   (  0.28 secs,     62027124 bytes).

a:= n-> `if`(n=0, 1, ithprime(length(n))^irem(n, 10, 'm') *a(m)):
seq(a(n), n=1..110);  # Alois P. Heinz, May 04 2011

a[n_] := (p = Prime[Range[Length[d = IntegerDigits[n]]]]; Times @@ (p^d)); Array[a, 50] (* Jean-François Alcover, Jan 09 2016 *)

a(n)=my(d=digits(n),p=primes(#d)); prod(i=1,#d,p[i]^d[i]) \\ Charles R Greathouse IV, Aug 19 2014

from sympy import prime
from operator import mul
from functools import reduce
def A189398(n):
    return reduce(mul, (prime(i)**int(d) for i,d in enumerate(str(n),start=1)))
# Chai Wah Wu, Aug 31 2014

# implementation using recursion
from sympy import prime
def _A189398(n):
    nlen = len(n)
    return _A189398(n[:-1])*prime(nlen)**int(n[-1]) if nlen > 1 else 2**int(n)
def A189398(n):
    return _A189398(str(n))
# Chai Wah Wu, Aug 31 2014

A061510 Write n in decimal, omit 0's, raise each digit k to k-th power and multiply.

Original entry on oeis.org

1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 4, 4, 16, 108, 1024, 12500, 186624, 3294172, 67108864, 1549681956, 27, 27, 108, 729, 6912, 84375, 1259712, 22235661, 452984832, 10460353203

Offset: 0

Amarnath Murthy, May 06 2001

The empty product is 1. If one accepts 0^0 = 1, then "omit 0's" is unnecessary. - Michael S. Branicky, Nov 26 2024

			a(4) = 4^4 = 256.
a(123) =  1^1 * 2^2 * 3^3 = 108.
a(1024) = 1^1 * 2^2 * 4^4 = 1024.

Michael S. Branicky, Table of n, a(n) for n = 0..10000
Wikipedia, Zero to the power of zero

Cf. A061509.

a:= n-> mul(i^i,i=convert(n, base, 10)):
seq(a(n), n=0..39);  # Alois P. Heinz, Nov 26 2024

A061510[n_] := If[n == 0, 1, Times @@ (#^#) & [DeleteCases[IntegerDigits[n], 0]]];
Array[A061510, 50, 0] (* Paolo Xausa, Nov 26 2024 *)

a(n) = my(d=digits(n)); vecprod(vector(#d, k, d[k]^d[k])); \\ Michel Marcus, Nov 25 2024

from math import prod
def a(n): return prod(d**d for d in map(int, str(n)) if d > 1)
print([a(n) for n in range(40)]) # Michael S. Branicky, Nov 25 2024

Corrected and extended by Matthew Conroy, May 13 2001

cp's OEIS Frontend

A085940 After correction, duplicate of A061509.

Views

Author

Keywords

A054842 If n = a + 10 * b + 100 * c + 1000 * d + ... then a(n) = (2^a) * (3^b) * (5^c) * (7^d) * ...

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A069877 Smallest number with a prime signature whose indices are the decimal digits of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A189398 a(n) = 2^d(1) * 3^d(2) * ... * prime(k)^d(k), where d(1)d(2)...d(k) is the decimal representation of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Python

A061510 Write n in decimal, omit 0's, raise each digit k to k-th power and multiply.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Extensions