A286564 -id:A286564 - cp's OEIS frontend

A286561 Square array A(n,k): A(n,1) = 1, and for k > 1, A(n,k) = the highest exponent e such that k^e divides n, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc.

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1

Offset: 1

Antti Karttunen, May 20 2017

			The top left 18 X 18 corner of the array:
  n \k 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18
     .-----------------------------------------------------
   1 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   2 | 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   3 | 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   4 | 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   5 | 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   6 | 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   7 | 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   8 | 1, 3, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   9 | 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0
  10 | 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0
  11 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
  12 | 1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0
  13 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0
  14 | 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
  15 | 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0
  16 | 1, 4, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0
  17 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0
  18 | 1, 1, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1
  ---------------------------------------------------------
A(18,2) = 1, because 2^1 divides 18, but 2^2 does not. A(18,3) = 2, because 3^2 divides 18 (but 3^3 does not). A(18,4) = 0, because 4^0 (= 1) divides 18, but 4^1 does not. A(18,18) = 1, because 18^1 divides 18, but 18^2 does not.
A(2,18) = 0, because 18^0 divides 2, but 18^1 does not.

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array

Cf. A286562 (transpose), A286563 (lower triangular region), A286564 (lower triangular region reversed).
Cf. A169594 (row sums), also A168512, A178638, A186643.
Cf. also array A286156.

Table[Function[m, If[k == 1, 1, IntegerExponent[m, k]]][n - k + 1], {n, 15}, {k, n}] // TableForm (* Michael De Vlieger, May 20 2017 *)

A286561(n,k) = if(1==k, 1, valuation(n, k)); \\ Antti Karttunen, May 27 2017

def a(n, k):
    i=1
    if k==1: return 1
    while n%(k**i)==0:
        i+=1
    return i-1
for n in range(1, 21): print([a(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, May 20 2017

(define (A286561 n) (A286561bi (A002260 n) (A004736 n)))
(define (A286561bi row col) (if (= 1 col) 1 (let loop ((i 1)) (if (not (zero? (modulo row (expt col i)))) (- i 1) (loop (+ 1 i))))))

A169594 Number of divisors of n, counting divisor multiplicity in n.

Original entry on oeis.org

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

Offset: 1

Joseph L. Pe, Dec 02 2009

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) depends only on the prime signature of n. - David A. Corneth, Mar 28 2021

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

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

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.
Cf. A003238, A003242, A069916, A122651, A309891, A318991, A318992, A325545.
First differences of A078651.

a := n -> ifelse(n < 2, 1, 1 + add(padic:-ordp(n, k), k = 2..n)):
seq(a(n), n = 1..98);  # Peter Luschny, Apr 10 2025

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 *)

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

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

A169594(n) = {my(s=0, k=2); while(k<=n, s+=valuation(n, k); k=k+1); s + 1} \\ Zhuorui He, Aug 28 2025

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

(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

From Friedjof Tellkamp, Feb 29 2024: (Start)
a(n) = A309891(n) + 1.
G.f.: x/(1-x) + Sum_{k>=2, j>=1} x^(k^j)/(1-x^(k^j)).
Dirichlet g.f.: zeta(s) * (1 + Sum_{k>=1} (zeta(k*s) - 1)).
Sum_{n>=1} a(n)/n^2 = (7/24) * Pi^2. (End)

Extended by Ray Chandler, Dec 08 2009

A286563 Triangular table T(n,k) read by rows: T(n,1) = 1, and for 1 < k <= n, T(n,k) = the highest exponent e such that k^e divides n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 3, 0, 1, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, May 20 2017

T(n,k) > 0 for k in row n of A027750. - Michael De Vlieger, May 20 2017

Compare rows to those of triangle A279907, smallest exponent e of n divisible by k. The values of k > -1 in row n of A279907 pertain to k in row n of A162306 rather than k in row n of A027750. - Michael De Vlieger, May 21 2017

			The first fifteen rows of this triangular table:
  1,
  1, 1,
  1, 0, 1,
  1, 2, 0, 1,
  1, 0, 0, 0, 1,
  1, 1, 1, 0, 0, 1,
  1, 0, 0, 0, 0, 0, 1,
  1, 3, 0, 1, 0, 0, 0, 1,
  1, 0, 2, 0, 0, 0, 0, 0, 1,
  1, 1, 0, 0, 1, 0, 0, 0, 0, 1,
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1,
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1,
  1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 rows of the triangle

Lower triangular region of A286561.
Cf. A286564 (same triangle reversed).
Cf. A169594 (row sums).
Cf. also arrays A051731, A286158, A027750, A279907, A280269.

T := (n, k) -> ifelse(k = 1, 1, padic:-ordp(n, k)):
for n from 1 to 12 do seq(T(n, k), k = 1..n) od;  # Peter Luschny, Apr 07 2025

Table[If[k == 1, 1, IntegerExponent[n, k]], {n, 15}, {k, n}] // Flatten (* Michael De Vlieger, May 20 2017 *)

def T(n, k):
    i=1
    if k==1: return 1
    while n%(k**i)==0:
        i+=1
    return i-1
for n in range(1, 21): print([T(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 20 2017

(define (A286563 n) (A286561bi (A002024 n) (A002260 n))) ;; For A286561bi see A286561.

T(n,k) = A286561(n,k) listed row by row for n >= 1, k = 1 .. n.

cp's OEIS Frontend

A286561 Square array A(n,k): A(n,1) = 1, and for k > 1, A(n,k) = the highest exponent e such that k^e divides n, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc.

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A169594 Number of divisors of n, counting divisor multiplicity in n.

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A286563 Triangular table T(n,k) read by rows: T(n,1) = 1, and for 1 < k <= n, T(n,k) = the highest exponent e such that k^e divides n.

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