A000026 Mosaic numbers or multiplicative projection of n: if n = Product (p_j^k_j) then a(n) = Product (p_j * k_j).

1, 2, 3, 4, 5, 6, 7, 6, 6, 10, 11, 12, 13, 14, 15, 8, 17, 12, 19, 20, 21, 22, 23, 18, 10, 26, 9, 28, 29, 30, 31, 10, 33, 34, 35, 24, 37, 38, 39, 30, 41, 42, 43, 44, 30, 46, 47, 24, 14, 20, 51, 52, 53, 18, 55, 42, 57, 58, 59, 60, 61, 62, 42, 12, 65, 66, 67, 68, 69, 70, 71, 36

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• mult

Comments

a(n) = n if n is squarefree.

a(2n) = 2n if and only if n is squarefree. - Peter Munn, Feb 05 2017

Examples

			24 = 2^3*3^1, a(24) = 2*3*3*1 = 18.
		

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• R. A. Gillman, The Average Size of a Certain Arithmetic Function, A6660 solution, Amer. Math. Monthly, 100 (1993), pp. 296-298.
• B. Gordon and M. M. Robertson, Two theorems on mosaics, Canad. J. Math., 17 (1965), 1010-1014.
• A. A. Mullin, Some related number-theoretic functions, Research Problem 4, Bull. Amer. Math. Soc., 69 (1963), 446-447.
• Daniel Tsai, A recurring pattern in natural numbers of a certain property, Integers (2021) Vol. 21, Article #A32.

Crossrefs

Cf. A005117, A005361, A007947, A008474, A013661, A193551.

Programs

Haskell

a000026 n = f a000040_list n 1 (0^(n-1)) 1 where
   f _  1 q e y  = y * e * q
   f ps'@(p:ps) x q e y
     | m == 0    = f ps' x' p (e+1) y
     | e > 0     = f ps x q 0 (y * e * q)
     | x < p * p = f ps' 1 x 1 y
     | otherwise = f ps x 1 0 y
     where (x', m) = divMod x p
a000026_list = map a000026 [1..]
-- Reinhard Zumkeller, Aug 27 2011

Maple

A000026 := proc(n) local e,j; e := ifactors(n)[2]:
mul(e[j][1]*e[j][2], j=1..nops(e)) end:
seq(A000026(n), n=1..80); # Peter Luschny, Jan 17 2011

Mathematica

Array[ Times@@Flatten[ FactorInteger[ # ] ]&, 100 ]

PARI

a(n)=local(f); if(n<1,0,f=factor(n); prod(k=1,matsize(f)[1],f[k,1]*f[k,2]))

PARI

a(n)=my(f=factor(n)); factorback(f[,1])*factorback(f[,2]) \\ Charles R Greathouse IV, Apr 04 2016

Python

from math import prod
from sympy import factorint
def a(n): f = factorint(n); return prod(p*f[p] for p in f)
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, May 27 2021

Formula

n = Product (p_j^k_j) -> a(n) = Product (p_j * k_j).

Multiplicative with a(p^e) = p*e. - David W. Wilson, Aug 01 2001

a(n) = A005361(n) * A007947(n). - Enrique Pérez Herrero, Jun 24 2010

a(A193551(n)) = n and a(m) != n for m < A193551(n). - Reinhard Zumkeller, Aug 27 2011

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)^2/2) * Product_{p prime} (1 - 3/p^2 + 2/p^3 + 1/p^4 - 1/p^5) = 0.4175724194... . - Amiram Eldar, Oct 25 2022

Extensions

Example, program, definition, comments and more terms added by Olivier Gérard (02/99).