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A067029 Exponent of least prime factor in prime factorization of n, a(1)=0. 141
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Project Euler Prime factor and exponent, (2022).

FORMULA

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

Join[{0}, Table[FactorInteger[n][[1, 2]], {n, 2, 100}]] (* Harvey P. Dale, Oct 14 2011 *)

PROG

(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

CROSSREFS

Cf. A051903, A020639, A028233, A034684, A071178, first column of A124010, A247180.

Sequence in context: A072909 A095691 A182426 * A087179 A290109 A302045

Adjacent sequences:  A067026 A067027 A067028 * A067030 A067031 A067032

KEYWORD

nonn,nice

AUTHOR

Reinhard Zumkeller, Feb 17 2002

STATUS

approved

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Last modified July 2 14:02 EDT 2022. Contains 355007 sequences. (Running on oeis4.)