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A169594 Number of divisors of n, counting divisor multiplicity in n. 8
1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 7, 2, 4, 4, 9, 2, 7, 2, 7, 4, 4, 2, 10, 4, 4, 6, 7, 2, 8, 2, 11, 4, 4, 4, 12, 2, 4, 4, 10, 2, 8, 2, 7, 7, 4, 2, 14, 4, 7, 4, 7, 2, 10, 4, 10, 4, 4, 2, 13, 2, 4, 7, 15, 4, 8, 2, 7, 4, 8, 2, 16, 2, 4, 7, 7, 4, 8, 2, 14, 9, 4, 2, 13, 4, 4, 4, 10, 2, 13, 4, 7, 4, 4, 4, 17, 2, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The multiplicity of a divisor d > 1 in n is defined as the largest power i for which d^i divides n; and for d = 1 it is defined as 1.

a(n) is also the sum of the multiplicities of the divisors of n.

In other words, a(n) = 1 + sum of the highest exponents e_i for which each number k_i in range 2 .. n divide n, as {k_i}^{e_i} | n. For nondivisors of n this exponent e_i is 0, for n itself it is 1. - Antti Karttunen, May 20 2017

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

EXAMPLE

The divisors of 8 are 1, 2, 4, 8 of multiplicity 1, 3, 1, 1, respectively. So a(8) = 1 + 3 + 1 + 1 = 6.

MATHEMATICA

divmult[d_, n_] := Module[{output, i}, If[d == 1, output = 1, If[d == n, output = 1, i = 0; While[Mod[n, d^(i + 1)] == 0, i = i + 1]; output = i]]; output]; dmt0[n_] := Module[{divs, l}, divs = Divisors[n]; l = Length[divs]; Sum[divmult[divs[[i]], n], {i, 1, l}]]; Table[dmt0[i], {i, 1, 40}]

Table[1 + DivisorSum[n, IntegerExponent[n, #] &, # > 1 &], {n, 98}] (* Michael De Vlieger, May 20 2017 *)

PROG

(PARI)

A286561(n, k) = { my(i=1); if(1==k, 1, while(!(n%(k^i)), i = i+1); (i-1)); };

A169594(n) = sumdiv(n, d, A286561(n, d)); \\ Antti Karttunen, May 20 2017

(Scheme)

(define (A169594 n) (add (lambda (k) (A286561bi n k)) 1 n))

;; Implements sum_{i=lowlim..uplim} intfun(i)

(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

;; For A286561bi see A286561. - Antti Karttunen, May 20 2017

(Python)

def a286561(n, k):

    i=1

    if k==1: return 1

    while n%(k**i)==0:

        i+=1

    return i-1

def a(n): return sum([a286561(n, d) for d in divisors(n)]) # Indranil Ghosh, May 20 2017

CROSSREFS

Cf. A168512.

Row sums of A286561, A286563 and A286564.

Sequence in context: A193432 A129089 A255201 * A294337 A322327 A124315

Adjacent sequences:  A169591 A169592 A169593 * A169595 A169596 A169597

KEYWORD

nonn

AUTHOR

Joseph L. Pe, Dec 02 2009

EXTENSIONS

Extended by Ray Chandler, Dec 08 2009

STATUS

approved

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Last modified October 19 13:01 EDT 2019. Contains 328222 sequences. (Running on oeis4.)