login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A169594 Number of divisors of n, counting divisor multiplicity in n. 18
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
From Gus Wiseman, Mar 25 2021: (Start)
Also the number of strict chains of divisors ending with n and having constant (equal) first quotients. The case starting with 1 is A089723. For example, the a(1) = 1 through a(12) = 7 chains are:
1 2 3 4 5 6 7 8 9 10 11 12
1|2 1|3 1|4 1|5 1|6 1|7 1|8 1|9 1|10 1|11 1|12
2|4 2|6 2|8 3|9 2|10 2|12
1|2|4 3|6 4|8 1|3|9 5|10 3|12
2|4|8 4|12
1|2|4|8 6|12
3|6|12
(End)
a(n) only depends on the prime signature of n. - David A. Corneth, Mar 28 2021
LINKS
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
(PARI) a(n) = { if(n == 1, return(1)); my(f = factor(n), u = vecmax(f[, 2]), cf = f, res = numdiv(f) - u + 1); for(i = 2, u, cf[, 2] = f[, 2]\i; res+=numdiv(factorback(cf)) ); res } \\ David A. Corneth, Mar 29 2021
(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.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A057567 counts chains of divisors with weakly increasing first quotients.
A067824 counts strict chains of divisors ending with n.
A253249 counts strict chains of divisors.
A334997 counts chains of divisors of n by length.
A342086 counts chains of divisors with strictly increasing first quotients.
A342496 counts partitions with equal first quotients (strict: A342515, ranking: A342522, ordered: A342495).
A342530 counts chains of divisors with distinct first quotients.
Sequence in context: A193432 A129089 A255201 * A294337 A322327 A124315
KEYWORD
nonn,easy
AUTHOR
Joseph L. Pe, Dec 02 2009
EXTENSIONS
Extended by Ray Chandler, Dec 08 2009
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)