A000026 Mosaic numbers or multiplicative projection of n: if n = Product (p_j^k_j) then a(n) = Product (p_j * k_j). (Formerly M0467 N0171)

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

1,2

Comments

a(n) = n if n is squarefree.

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

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.

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

Example

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

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 ]

Prog

(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
(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
(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

Crossrefs

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

Sequence in context: A206495 A161209 A279513 * A005599 A071934 A337642

Adjacent sequences: A000023 A000024 A000025 * A000027 A000028 A000029

Keyword

nonn,easy,nice,mult

Author

N. J. A. Sloane

Extensions

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

Status

approved