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A067029
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Exponent of least prime factor in prime factorization of n, a(1)=0.
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143
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0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 1
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OFFSET
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1,4
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COMMENTS
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Even bisection is A001511: a(2n) = A007814(n) + 1. - Ralf Stephan, Jan 31 2004
Number of occurrences of the smallest part in the partition with Heinz number n. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Example: a(24)=3 because the partition with Heinz number 24 = 3*2*2*2 is [2,1,1,1]. - Emeric Deutsch, Oct 02 2015
Together with A028234 is useful for defining sequences that are multiplicative with a(p^e) = f(e), as recurrences of the form: a(1) = 1 and for n > 1, a(n) = f(A067029(n)) * a(A028234(n)). - Antti Karttunen, May 29 2017
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
Project Euler Prime factor and exponent, (2022).
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FORMULA
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a(n) = A124010(n,1). - Reinhard Zumkeller, Aug 27 2011
A028233(n) = A020639(n)^a(n). - Reinhard Zumkeller, May 13 2006
a(A247180(n)) = 1. - Reinhard Zumkeller, Nov 23 2014
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (Product_{i=1..k-1} (1 - 1/prime(i)))/(prime(k)-1) = 1/(prime(1)-1) + (1-1/prime(1))*(1/(prime(2)-1) + (1-1/prime(2))*(1/(prime(3)-1) + (1-1/prime(3))*( ... ))) = 1.6125177915... - Amiram Eldar, Oct 26 2021
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EXAMPLE
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a(18) = a(2^1 * 3^2) = 1.
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MAPLE
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A067029 := proc(n)
local f, lp, a;
a := 0 ;
lp := n+1 ;
for f in ifactors(n)[2] do
p := op(1, f) ;
if p < lp then
a := op(2, f) ;
lp := p;
fi;
end do:
a ;
end proc: # R. J. Mathar, Jul 08 2015
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MATHEMATICA
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Join[{0}, Table[FactorInteger[n][[1, 2]], {n, 2, 100}]] (* Harvey P. Dale, Oct 14 2011 *)
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PROG
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(Haskell)
a067029 = head . a124010_row
-- Reinhard Zumkeller, Jul 05 2013, Jun 04 2012
(Python)
from sympy import factorint
def a(n):
f=factorint(n)
return 0 if n==1 else f[min(f)] # Indranil Ghosh, May 15 2017
(PARI) a(n) = if (n==1, 0, factor(n)[1, 2]); \\ Michel Marcus, May 15 2017
(Scheme)
;; Naive implementation of A020639 is given under that entry. All of these functions could be also defined with definec to make them faster on the later calls. See http://oeis.org/wiki/Memoization#Scheme
(define (A067029 n) (if (< n 2) 0 (let ((mp (A020639 n))) (let loop ((e 0) (n (/ n mp))) (cond ((integer? n) (loop (+ e 1) (/ n mp))) (else e)))))) ;; Antti Karttunen, May 29 2017
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CROSSREFS
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Cf. A051903, A020639, A028233, A034684, A071178, first column of A124010, A247180.
Sequence in context: A360721 A095691 A182426 * A087179 A290109 A302045
Adjacent sequences: A067026 A067027 A067028 * A067030 A067031 A067032
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KEYWORD
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nonn,nice
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AUTHOR
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Reinhard Zumkeller, Feb 17 2002
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STATUS
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approved
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