|
%I M0467 N0171 #68 Oct 25 2022 02:41:54
%S 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,
%T 9,28,29,30,31,10,33,34,35,24,37,38,39,30,41,42,43,44,30,46,47,24,14,
%U 20,51,52,53,18,55,42,57,58,59,60,61,62,42,12,65,66,67,68,69,70,71,36
%N Mosaic numbers or multiplicative projection of n: if n = Product (p_j^k_j) then a(n) = Product (p_j * k_j).
%C a(n) = n if n is squarefree.
%C a(2n) = 2n if and only if n is squarefree. - _Peter Munn_, Feb 05 2017
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000026/b000026.txt">Table of n, a(n) for n = 1..10000</a>
%H R. A. Gillman, <a href="http://www.jstor.org/stable/2324471">The Average Size of a Certain Arithmetic Function, A6660 solution</a>, Amer. Math. Monthly, 100 (1993), pp. 296-298.
%H B. Gordon and M. M. Robertson, <a href="http://dx.doi.org/10.4153/CJM-1965-094-9">Two theorems on mosaics</a>, Canad. J. Math., 17 (1965), 1010-1014.
%H A. A. Mullin, <a href="http://dx.doi.org/10.1090/S0002-9904-1963-10961-1">Some related number-theoretic functions</a>, Research Problem 4, Bull. Amer. Math. Soc., 69 (1963), 446-447.
%H Daniel Tsai, <a href="http://math.colgate.edu/~integers/v32/v32.mail.html">A recurring pattern in natural numbers of a certain property</a>, Integers (2021) Vol. 21, Article #A32.
%F n = Product (p_j^k_j) -> a(n) = Product (p_j * k_j).
%F Multiplicative with a(p^e) = p*e. - _David W. Wilson_, Aug 01 2001
%F a(n) = A005361(n) * A007947(n). - _Enrique Pérez Herrero_, Jun 24 2010
%F a(A193551(n)) = n and a(m) != n for m < A193551(n). - _Reinhard Zumkeller_, Aug 27 2011
%F 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
%e 24 = 2^3*3^1, a(24) = 2*3*3*1 = 18.
%p A000026 := proc(n) local e,j; e := ifactors(n)[2]:
%p mul(e[j][1]*e[j][2], j=1..nops(e)) end:
%p seq(A000026(n), n=1..80); # _Peter Luschny_, Jan 17 2011
%t Array[ Times@@Flatten[ FactorInteger[ # ] ]&, 100 ]
%o (PARI) a(n)=local(f); if(n<1,0,f=factor(n); prod(k=1,matsize(f)[1],f[k,1]*f[k,2]))
%o (PARI) a(n)=my(f=factor(n)); factorback(f[,1])*factorback(f[,2]) \\ _Charles R Greathouse IV_, Apr 04 2016
%o (Haskell)
%o a000026 n = f a000040_list n 1 (0^(n-1)) 1 where
%o f _ 1 q e y = y * e * q
%o f ps'@(p:ps) x q e y
%o | m == 0 = f ps' x' p (e+1) y
%o | e > 0 = f ps x q 0 (y * e * q)
%o | x < p * p = f ps' 1 x 1 y
%o | otherwise = f ps x 1 0 y
%o where (x', m) = divMod x p
%o a000026_list = map a000026 [1..]
%o -- _Reinhard Zumkeller_, Aug 27 2011
%o (Python)
%o from math import prod
%o from sympy import factorint
%o def a(n): f = factorint(n); return prod(p*f[p] for p in f)
%o print([a(n) for n in range(1, 73)]) # _Michael S. Branicky_, May 27 2021
%Y Cf. A005117, A005361, A007947, A008474, A013661, A193551.
%K nonn,easy,nice,mult
%O 1,2
%A _N. J. A. Sloane_
%E Example, program, definition, comments and more terms added by _Olivier Gérard_ (02/99).
|
