A286561 -id:A286561 - cp's OEIS frontend

A329042 a(n) = Product_{d|n, d>1} A008578(1+A286561(A122111(n),d)), where A286561(x,d) gives the exponent of the highest power of d dividing x.

1, 2, 1, 1, 1, 8, 1, 1, 6, 3, 1, 2, 1, 5, 3, 1, 1, 2, 1, 48, 3, 7, 1, 2, 1, 11, 1, 10, 1, 128, 1, 1, 3, 13, 1, 2, 1, 17, 3, 6, 1, 12, 1, 21, 3, 19, 1, 2, 1, 2, 3, 33, 1, 1, 1, 320, 3, 23, 1, 8, 1, 29, 1, 1, 1, 20, 1, 65, 3, 8, 1, 2, 1, 31, 48, 85, 1, 28, 1, 6, 1, 37, 1, 3072, 1, 41, 3, 42, 1, 8, 1, 133, 3, 43, 1, 2, 1, 1, 1, 1, 1, 44, 1, 66, 12

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A001222, A008578, A122111, A286561, A329036, A329043.

Cf. also A329037.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A329042(n) = { my(m=1,x=A122111(n),v); fordiv(n,d,if((d>1) && ((v = valuation(x,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|n, d>1} A008578(1+A286561(A122111(n),d)).

1+A001222(a(n)) = A329036(n).

A329043 a(n) = Product_{d|A122111(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the exponent of the highest power of d dividing n.

Original entry on oeis.org

1, 2, 1, 1, 1, 8, 1, 1, 6, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 48, 2, 2, 1, 5, 1, 2, 1, 6, 1, 128, 1, 1, 2, 2, 1, 3, 1, 2, 2, 10, 1, 8, 1, 6, 2, 2, 1, 7, 1, 3, 2, 6, 1, 1, 1, 320, 2, 2, 1, 12, 1, 2, 1, 1, 1, 8, 1, 6, 2, 8, 1, 3, 1, 2, 48, 6, 1, 8, 1, 21, 1, 2, 1, 3072, 1, 2, 2, 20, 1, 8, 1, 6, 2, 2, 1, 11, 1, 1, 1, 1, 1, 8, 1, 20, 8

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A001222, A008578, A122111, A286561, A329036, A329042.

Cf. also A327167.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A329043(n) = { my(m=1,v); fordiv(A122111(n),d,if((d>1) && ((v = valuation(n,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|A122111(n), d>1} A008578(1+A286561(n,d)).

1+A001222(a(n)) = A329036(n).

A286562 Transpose of square array A286561.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 20 2017

See A286561 and A286563.

			The top left 16 X 16 corner of the array:
  n \ k
     \ 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
     .-----------------------------------------------
   1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
   2 | 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4
   3 | 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0
   4 | 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2
   5 | 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0
   6 | 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
   7 | 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0
   8 | 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1
   9 | 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
  10 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0
  11 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0
  12 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
  13 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0
  14 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0
  15 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0
  16 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
The array is read by descending antidiagonals.

Antti Karttunen, Table of n, a(n) for n = 1..7875; the first 125 antidiagonals of array

Cf. A286561, A286563.

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)][::-1]) # Indranil Ghosh, May 20 2017

(define (A286562 n) (A286561bi (A004736 n) (A002260 n))) ;; For A286561bi see A286561.

A106177 Functional composition table for "n o m" = "n composed with m", where n and m are the "primal codes" of finite partial functions on the positive integers and 1 is the code for the empty function.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 5, 2, 9, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 7, 1, 25, 1, 3, 1, 1, 1, 1, 1, 36, 1, 2, 1, 8, 1, 1, 1, 1, 49, 1, 5, 1, 27, 1, 1, 1, 10, 3, 1, 1, 6, 1, 1, 1, 2, 1, 1, 11, 1, 1, 2, 7, 1, 125, 4, 3, 1, 1, 1, 3, 1, 100, 1, 1, 1, 216, 1, 1, 1, 4, 1, 1, 13

Offset: 1

Jon Awbrey, May 23 2005

The right diagonal labeled by the prime power of the form j:k = (prime(j))^k contains the j^th power primes in the factorization raised to the k^th power. For example, the right diagonal labeled by the number 2 = 1:1 = (prime(1))^1 contains the power-free parts of each positive integer, specifically A055231 and the right diagonal labeled by the number 4 = 1:2 = (prime(1))^2 contains the squares of the squarefree parts of positive integers.
In general, then the right diagonal labeled by m = (j_i : k_i)_i = Product_i prime(j_i)^(k_i) contains the product over i of the (j_i)th power primes in the factorization raised to the (k_i)th powers.
For example, the operator 5 = 3:1 extracts the 3rd power primes in the factorization of each n and raises them to the first power, thus sending 8 = 1:3 to 2 = 1:1, 27 = 2:3 to 3 = 2:1 and so on.

			` ` ` ` ` ` ` ` ` ` `n o m
` ` ` ` ` ` ` ` ` ` ` \ /
` ` ` ` ` ` ` ` ` ` `1 . 1
` ` ` ` ` ` ` ` ` ` \ / \ /
` ` ` ` ` ` ` ` ` `2 . 1 . 2
` ` ` ` ` ` ` ` ` \ / \ / \ /
` ` ` ` ` ` ` ` `3 . 1 . 1 . 3
` ` ` ` ` ` ` ` \ / \ / \ / \ /
` ` ` ` ` ` ` `4 . 1 . 2 . 1 . 4
` ` ` ` ` ` ` \ / \ / \ / \ / \ /
` ` ` ` ` ` `5 . 1 . 3 . 1 . 1 . 5
` ` ` ` ` ` \ / \ / \ / \ / \ / \ /
` ` ` ` ` `6 . 1 . 1 . 1 . 4 . 1 . 6
` ` ` ` ` \ / \ / \ / \ / \ / \ / \ /
` ` ` ` `7 . 1 . 5 . 2 . 9 . 1 . 1 . 7
` ` ` ` \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` `8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 8
` ` ` \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` `9 . 1 . 7 . 1 . 25. 1 . 3 . 1 . 1 . 9
` ` \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` 10 . 1 . 1 . 1 . 36. 1 . 2 . 1 . 8 . 1 . 10
Primal codes of finite partial functions on positive integers:
1 = { }
2 = 1:1
3 = 2:1
4 = 1:2
5 = 3:1
6 = 1:1 2:1
7 = 4:1
8 = 1:3
9 = 2:2
10 = 1:1 3:1
11 = 5:1
12 = 1:2 2:1
13 = 6:1
14 = 1:1 4:1
15 = 2:1 3:1
16 = 1:4
17 = 7:1
18 = 1:1 2:2
19 = 8:1
20 = 1:2 3:1
From _Antti Karttunen_, Nov 16 2019: (Start)
When the sequence is viewed as a square array read by falling antidiagonals, the top left 15 X 15 corner looks like this:
k=  | 1  2   3  4    5    6    7  8  9    10    11  12    13    14    15
----+--------------------------------------------------------------------
n= 1| 1, 1,  1, 1,   1,   1,   1, 1, 1,    1,    1,  1,    1,    1,    1,
   2| 1, 2,  3, 1,   5,   6,   7, 1, 1,   10,   11,  3,   13,   14,   15,
   3| 1, 1,  1, 2,   1,   1,   1, 1, 3,    1,    1,  2,    1,    1,    1,
   4| 1, 4,  9, 1,  25,  36,  49, 1, 1,  100,  121,  9,  169,  196,  225,
   5| 1, 1,  1, 1,   1,   1,   1, 2, 1,    1,    1,  1,    1,    1,    1,
   6| 1, 2,  3, 2,   5,   6,   7, 1, 3,   10,   11,  6,   13,   14,   15,
   7| 1, 1,  1, 1,   1,   1,   1, 1, 1,    1,    1,  1,    1,    1,    1,
   8| 1, 8, 27, 1, 125, 216, 343, 1, 1, 1000, 1331, 27, 2197, 2744, 3375,
   9| 1, 1,  1, 4,   1,   1,   1, 1, 9,    1,    1,  4,    1,    1,    1,
  10| 1, 2,  3, 1,   5,   6,   7, 2, 1,   10,   11,  3,   13,   14,   15,
  11| 1, 1,  1, 1,   1,   1,   1, 1, 1,    1,    1,  1,    1,    1,    1,
  12| 1, 4,  9, 2,  25,  36,  49, 1, 3,  100,  121, 18,  169,  196,  225,
  13| 1, 1,  1, 1,   1,   1,   1, 1, 1,    1,    1,  1,    1,    1,    1,
  14| 1, 2,  3, 1,   5,   6,   7, 1, 1,   10,   11,  3,   13,   14,   15,
  15| 1, 1,  1, 2,   1,   1,   1, 2, 3,    1,    1,  2,    1,    1,    1,
(End)

Antti Karttunen, Table of n, a(n) for n = 1..25425; the first 225 antidiagonals of the array
J. Awbrey, Riffs and Rotes

Cf. A000040, A061396, A062504, A062537, A062860, A106178, A181819, A286561.
Rows 1 - 3: A000012, A055231, A329376.
Columns 1 - 2: A000012, A006519.

up_to = 105;
A106177sq(n,k) = { my(f = factor(k)); prod(i=1,#f~,f[i, 1]^valuation(n, prime(f[i, 2]))); };
A106177list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A106177sq(col,(a-(col-1))))); (v); };
v106177 = A106177list(up_to);
A106177(n) = v106177[n]; \\ Antti Karttunen, Nov 16 2019

If k = Product p_i^e_i, A(n,k) = p_i^A286561(n, A000040(e_i)), where A286561(x,y) gives the y-valuation of x. - Antti Karttunen, Nov 16 2019

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.

A168512 Sum of divisors of n weighted by divisor multiplicity in n.

Original entry on oeis.org

1, 3, 4, 9, 6, 12, 8, 19, 16, 18, 12, 30, 14, 24, 24, 41, 18, 42, 20, 44, 32, 36, 24, 64, 36, 42, 46, 58, 30, 72, 32, 75, 48, 54, 48, 102, 38, 60, 56, 94, 42, 96, 44, 86, 81, 72, 48, 134, 64, 98, 72, 100, 54, 126, 72, 124, 80, 90, 60, 170, 62, 96, 107, 153, 84, 144, 68, 128, 96

Offset: 1

Joseph L. Pe, Nov 28 2009

nonn

If d > 1 divides n, the multiplicity of d in n is the largest integer i such that d^i divides n; e.g. the multiplicity of 4 in 16 is 2. If d = 1 (degenerate case), then the multiplicity of d is defined as 1.

			The divisors of 16 are 1, 2, 4, 8, 16, which are of multiplicity 1, 4, 2, 1, 1, respectively, in 16. So a(16) = 1*1 + 4*2 + 2*4 + 1*8 + 1*16 = 41.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A169594, A286561, A286563.

Table[1 + Total[Function[i, i*Select[Range[Log[i, n]], Divisible[n, i^#] &][[-1]]] /@ Rest@Divisors@n], {n, 69}] (* Ivan Neretin, Jul 26 2015 *)
Table[1 + DivisorSum[n, # IntegerExponent[n, #] &, # > 1 &], {n, 69}] (* 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)); };
A168512(n) = sumdiv(n,d,A286561(n,d)*d); \\ Antti Karttunen, May 20 2017

a(n) = Sum_{d|n} A286561(n,d)*d. - Antti Karttunen, May 20 2017

Extended by Ray Chandler, Dec 08 2009

A185633 For odd n, a(n) = 2; for even n, a(n) = denominator of Bernoulli(n)/n; The number 2 alternating with the elements of A006953.

Original entry on oeis.org

2, 12, 2, 120, 2, 252, 2, 240, 2, 132, 2, 32760, 2, 12, 2, 8160, 2, 14364, 2, 6600, 2, 276, 2, 65520, 2, 12, 2, 3480, 2, 85932, 2, 16320, 2, 12, 2, 69090840, 2, 12, 2, 541200, 2, 75852, 2, 2760, 2, 564, 2, 2227680, 2, 132, 2, 6360

Offset: 1

Paul Curtz, Dec 18 2012

nonn

There is an integer sequence b(n) = A053657(n)/2^(n-1) = 1, 1, 6, 6, 360, 360, 45360, 45360, 5443200, 5443200,... which consists of the duplicated entries of A202367.

The ratios of this sequence are b(n+1)/b(n) = 1, 6, 1, 60, 1, 126 .... = a(n)/2, which is a variant of A036283.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A006953, A007395 (bisections).

Cf. A006863, A027760, A067513, A322312, A322315 (rgs-transform).

A185633 := proc(n)
    A053657(n+1)/A053657(n) ;
end proc: # R. J. Mathar, Dec 19 2012

max = 52; s = Expand[Normal[Series[(-Log[1-x]/x)^z, {x, 0, max}]]]; a[n_, k_] := Denominator[Coefficient[s, x^n*z^k]]; A053657 = Prepend[LCM @@@ Table[a[n, k], {n, max}, {k, n}], 1]; a[n_] := A053657[[n+1]]/A053657[[n]]; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Dec 20 2012 *)

A185633(n) = if(n%2,2,denominator(bernfrac(n)/(n))); \\ Antti Karttunen, Dec 03 2018

A185633(n) = { my(m=1); fordiv(n, d, if(isprime(1+d), m *= (1+d)^(1+valuation(n,1+d)))); (m); }; \\ Antti Karttunen, Dec 03 2018

a(n) = A053657(n+1)/A053657(n).
a(2*n) = 2*A036283(n).
From Antti Karttunen, Dec 03 2018: (Start)
a(n) = Product_{d|n} [(1+d)^(1+A286561(n,1+d))]^A010051(1+d) - after Peter J. Cameron's Mar 25 2002 comment in A006863.
A007947(a(n)) = A027760(n)
A001221(a(n)) = A067513(n).
A181819(a(n)) = A322312(n).
(End)

Name edited by Antti Karttunen, Dec 03 2018

A309891 a(n) is the total number of trailing zeros in the representations of n over all bases b >= 2.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 6, 1, 3, 3, 8, 1, 6, 1, 6, 3, 3, 1, 9, 3, 3, 5, 6, 1, 7, 1, 10, 3, 3, 3, 11, 1, 3, 3, 9, 1, 7, 1, 6, 6, 3, 1, 13, 3, 6, 3, 6, 1, 9, 3, 9, 3, 3, 1, 12, 1, 3, 6, 14, 3, 7, 1, 6, 3, 7, 1, 15, 1, 3, 6, 6, 3, 7, 1, 13, 8, 3, 1, 12

Offset: 1

Rémy Sigrist, Aug 21 2019

a(n) depends only on the prime signature of n.

a(n) is the sum of the k-adic valuations of n for k >= 2. - Friedjof Tellkamp, Jan 25 2025

			For n = 12: 12 has 2 trailing zeros in base 2 (1100), 1 trailing zero in bases 3, 4, 6 and 12 (110, 30, 20, 10) and no trailing zero in other bases, hence a(12) = 1*2 + 4*1 = 6.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001221, A006218, A007814, A007949, A033093, A056239, A112765, A122841, A169594, A214411, A235127, A244413, A286561, A293514, A369180.
Cf. A111003.
First differences of A078632.

Table[DivisorSum[n, IntegerExponent[n, #] &, # > 1 &], {n, 84}] (* Jon Maiga, Aug 25 2019 *)

a(n) = sumdiv(n, d, if (d>1, valuation(n,d), 0))

a(n) = {if(n == 1, return(0)); my(f = factor(n)[, 2], res = 0, t = 2, of = f, nf = f >> 1, nd(v) = prod(i = 1, #v, v[i] + 1)); while(Set(of) != [0], res += (nd(of) - nd(nf)) * (t-1); of = nf; t++; nf = f \ t); res} \\ David A. Corneth, Aug 22 2019

a(n) = Sum_{d|n, d>1} A286561(n,d), where A286561 gives the d-valuation of n.
a(p) = 1 for any prime number p.
a(p^k) = A006218(k) for any k >= 0 and any prime number p.
a(n) = 2^A001221(n) - 1 for any squarefree number n.
a(n) = 3 for any semiprime number n.
a(m*n) >= a(m) + a(n).
a(n) >= A007814(n) + A007949(n) + A235127(n) + A112765(n) + A122841(n) + A214411(n) + A244413(n).
a(n) = A056239(A293514(n)). - Antti Karttunen, Aug 22 2019
a(n) <= A033093(n). - Michel Marcus, Aug 22 2019
a(n) = A169594(n) - 1. - Jon Maiga, Aug 25 2019
From Friedjof Tellkamp, Feb 27 2024: (Start)
G.f.: Sum_{k>=2, j>=1} x^(k^j)/(1-x^(k^j)).
Dirichlet g.f.: zeta(s) * Sum_{k>=1} (zeta(k*s) - 1).
Sum_{n>=1} a(n)/n^2 = Pi^2/8 (A111003). (End)

A351543 Even numbers k such that there is an odd prime p that divides sigma(k), but valuation(k, p) differs from valuation(sigma(k), p), and p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

4, 8, 12, 16, 18, 26, 32, 36, 38, 44, 48, 50, 52, 56, 58, 64, 68, 72, 74, 76, 78, 80, 82, 86, 88, 90, 92, 96, 98, 100, 104, 108, 112, 116, 118, 122, 124, 126, 128, 132, 134, 136, 144, 146, 148, 150, 152, 156, 158, 162, 164, 166, 172, 176, 178, 180, 184, 188, 192, 194, 196, 200, 202, 204, 206, 208, 212, 218, 222, 226

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Even numbers k such that sigma(k) has an odd prime factor prime(i), but prime(i-1) is not a factor of k, and A286561(k, prime(i)) <> A286561(sigma(k), prime(i)). This differs from the definition of A351542 in that prime(i) is not here required to be a factor of k itself. The condition implies also that if there is any such odd prime factor prime(i) of sigma(k), it must be >= 5.
Even numbers k for which A351555(k) > 0.
Question: Is A351538 subsequence of this sequence?

			12 = 2^2 * 3 is present as sigma(12) = 28 = 2^2 * 7, whose prime factorization contains an odd prime 7 such that neither it nor the immediately previous prime, which is 5, divide 12 itself.
196 = 2^2 * 7^2 is present as sigma(196) = 399 = 3^1 * 7^1 * 19^1, which thus has a shared prime factor 7 with 196, but occurring with smaller exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 196.
364 = 2^2 * 7^1 * 13^1 is present as sigma(364) = 784 = 2^4 * 7^2, which thus has a shared prime factor 7 with 364, but occurring with larger exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 364.

Cf. A000203, A003961, A286561, A351555.
Subsequences: A351541, A351542, and also conjecturally A351538.
Cf. A351553 (complement among even numbers).
No common terms with A349745.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
isA351543(n) = (!(n%2) && A351555(n)>0);

cp's OEIS Frontend

A329042 a(n) = Product_{d|n, d>1} A008578(1+A286561(A122111(n),d)), where A286561(x,d) gives the exponent of the highest power of d dividing x.

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A329043 a(n) = Product_{d|A122111(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the exponent of the highest power of d dividing n.

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A286562 Transpose of square array A286561.

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A106177 Functional composition table for "n o m" = "n composed with m", where n and m are the "primal codes" of finite partial functions on the positive integers and 1 is the code for the empty function.

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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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A168512 Sum of divisors of n weighted by divisor multiplicity in n.

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