A051904 Minimal exponent in prime factorization of n.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset

1,4

Comments

The asymptotic mean of this sequence is 1 (Niven, 1969). - Amiram Eldar, Jul 10 2020

Let k = A007947(n), then for n > 1 k^a(n) is the greatest power of k which divides n; see example. - David James Sycamore, Sep 07 2023

Links

Daniel Forgues, Table of n, a(n) for n = 1..100000

Cao Hui-Zhong, The Asymptotic Formulas Related to Exponents in Factoring Integers, Math. Balkanica, Vol. 5 (1991), Fasc. 2.

Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.

Eric Weisstein's World of Mathematics, Niven's Constant

Index entries for sequences computed from exponents in factorization of n

Formula

a(n) = min_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011

a(1) = 0, for n > 1, if A001221(n) = 1 (when n is in A000961), a(n) = A001222(n), otherwise a(n) = min(A067029(n), a(A028234(n))). - Antti Karttunen, Jul 12 2017

Example

For n = 72 = 2^3*3^2, a(72) = min(exponents) = min(3,2) = 2.
For n = 72, using alternative definition: rad(72) = 6; and 6^2 = 36 divides 72 but no higher power of 6 divides 72, so a(72) = 2.
For n = 432, rad(432) = 6 and 6^3 = 216 divides 432 but no higher power of 6 divides 432, therefore a(432) = 3. - David James Sycamore, Sep 08 2023

Maple

a := proc (n) if n = 1 then 0 else min(seq(op(2, op(j, op(2, ifactors(n)))), j = 1 .. nops(op(2, ifactors(n))))) end if end proc: seq(a(n), n = 1 .. 100); # Emeric Deutsch, May 20 2015

Mathematica

Table[If[n == 1, 0, Min @@ Last /@ FactorInteger[n]], {n, 100}] (* Ray Chandler, Jan 24 2006 *)

Prog

(Haskell)
a051904 1 = 0
a051904 n = minimum $ a124010_row n  -- Reinhard Zumkeller, Jul 15 2012
(PARI) a(n)=vecmin(factor(n)[, 2]) \\ Charles R Greathouse IV, Nov 19 2012
(Scheme) (define (A051904 n) (cond ((= 1 n) 0) ((= 1 (A001221 n)) (A001222 n)) (else (min (A067029 n) (A051904 (A028234 n)))))) ;; Antti Karttunen, Jul 12 2017
(Python)
from sympy import factorint
def a(n):
    f = factorint(n)
    l = [f[p] for p in f]
    return 0 if n == 1 else min(l)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 13 2017

Crossrefs

Cf. A000961, A001221, A005361, A008479, A051903, A052409, A076558, A124010.

Cf. A007947.

Sequence in context: A158378 A052409 A327503 * A070012 A071178 A366895

Adjacent sequences: A051901 A051902 A051903 * A051905 A051906 A051907

Keyword

nonn,easy

Author

Labos Elemer, Dec 16 1999

Status

approved