A309891 -id:A309891 - cp's OEIS frontend

A309947 a(n) = A309891(A025487(n)): For each prime signature, gives the sum of the number of trailing zeros for all bases b >= 2 for some number m with that prime signature. The prime signatures are chosen in order in which they are first seen in the positive integers.

0, 1, 3, 3, 5, 6, 8, 9, 7, 10, 11, 13, 12, 14, 15, 16, 17, 16, 21, 20, 21, 15, 21, 23, 20, 25, 27, 24, 24, 28, 28, 23, 32, 36, 29, 33, 33, 33, 35, 27, 37, 36, 38, 38, 43, 33, 43, 42, 43, 40, 29, 48, 43, 31, 51, 44, 53, 38, 52, 47, 57, 47, 35, 57, 48, 48, 66, 55, 57

Offset: 1

David A. Corneth, Aug 24 2019

A309891(n) only depends on the prime signature of n. This sequence lists the values for different prime signatures based on A025487 which lists the least positive integer for each prime signature.

			A309891(A025487(6)) = A309891(12) = 6.

Cf. A025487, A309891.

A111003 Decimal expansion of Pi^2/8.

Original entry on oeis.org

1, 2, 3, 3, 7, 0, 0, 5, 5, 0, 1, 3, 6, 1, 6, 9, 8, 2, 7, 3, 5, 4, 3, 1, 1, 3, 7, 4, 9, 8, 4, 5, 1, 8, 8, 9, 1, 9, 1, 4, 2, 1, 2, 4, 2, 5, 9, 0, 5, 0, 9, 8, 8, 2, 8, 3, 0, 1, 6, 6, 8, 6, 7, 2, 0, 2, 7, 5, 0, 5, 6, 0, 2, 8, 0, 2, 4, 0, 0, 6, 5, 5, 3, 7, 5, 2, 2, 1, 6, 7, 5, 4, 6, 4, 8, 1, 9, 0, 2, 8, 9, 7, 8, 0, 0

Offset: 1

Sam Alexander, Oct 01 2005

According to Beckmann, Euler discovered the formula for this number as a sum of squares of reciprocals of odd numbers, along with similar formulas for Pi^2/6 and Pi^2/12. - Alonso del Arte, Apr 01 2013

Equals the asymptotic mean of the abundancy index of the odd numbers. - Amiram Eldar, May 12 2023

			1.23370055013616982735431137498451889191421242590509882830166867202...
1 + 1/9 + 1/25 + 1/49 + 1/81 + 1/121 + 1/169 + 1/225 + ... - _Bruno Berselli_, Mar 06 2017

F. Aubonnet, D. Guinin and B. Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.
Petr Beckmann, A History of Pi, 5th Ed. Boulder, Colorado: The Golem Press (1982): p. 153.
George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 122.
Calvin C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 98.
L. B. W. Jolley, Summation of Series, Dover (1961).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 54.

Eric Weisstein's World of Mathematics, Dirichlet Lambda Function. See (5).
Index entries for transcendental numbers

Cf. A013661 (Pi^2/6), A002388, A102753, A091476, A309891.

RealDigits[Pi^2/8, 10, 105][[1]] (* Robert G. Wilson v *)

Pi^2/8 \\ Charles R Greathouse IV, Dec 04 2016

sumpos(n=1,(2*n-1)^-2) \\ Charles R Greathouse IV, Mar 02 2018

Equals 1 + 1/(2*3) + (1/3)*(1*2)/(3*5) + (1/4)*(1*2*3)/(3*5*7) + ... [Jolley eq 276]
Equals Sum_{k >= 1} 1/(2*k - 1)^2 [Clawson and Wells]. - Alonso del Arte, Aug 15 2012
Equals 2*(Integral_{t=0..1} sqrt(1 - t^2) dt)^2. - Alonso del Arte, Mar 29 2013
Equals Sum_{k >= 1} 2^k/(k^2*binomial(2*k, k)). - Jean-François Alcover, Apr 29 2013
Equals Integral_{x=0..1} log((1+x^2)/(1-x^2))/x dx. - Bruno Berselli, May 13 2013
Equals limit_{p->0} Integral_{x=0..Pi/2} x*tan(x)^p dx. [Jean-François Alcover, May 17 2013, after Boros & Moll p. 230]
Equals A002388/8 = A102753/4 = A091476/2. - R. J. Mathar, Oct 13 2015
Equals Integral_{x>=0} x*K_0(x)*K_1(x)dx where K are modified Bessel functions [Gradsteyn-Ryzhik 6.576.4]. - R. J. Mathar, Oct 22 2015
Equals (3/4)*zeta(2) = (3/4)*A013661. - Wolfdieter Lang, Sep 02 2019
From Amiram Eldar, Jul 17 2020: (Start)
Equals -Integral_{x=0..1} log(x)/(1 - x^2) dx = Integral_{x>=1} log(x)/(x^2-1) dx.
Equals -Integral_{x=0..oo} log(x)/(1 - x^4) dx.
Equals Integral_{x=0..oo} arctan(x)/(1 + x^2) dx. (End)
Equals Integral_{x=0..1} log(1+x+x^2+x^3)/x dx (Aubonnet). - Bernard Schott, Feb 04 2022
Equals Sum_{n>=1} A309891(n)/n^2. - Friedjof Tellkamp, Jan 25 2025
Equals lambda(2), where lambda is the Dirichlet lambda function. - Michel Marcus, Aug 15 2025

More terms from Robert G. Wilson v, Oct 04 2005

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

A078632 Number of geometric subsequences of [1,...,n] with integral successive-term ratio and length > 1.

Original entry on oeis.org

0, 1, 2, 5, 6, 9, 10, 15, 18, 21, 22, 28, 29, 32, 35, 43, 44, 50, 51, 57, 60, 63, 64, 73, 76, 79, 84, 90, 91, 98, 99, 109, 112, 115, 118, 129, 130, 133, 136, 145, 146, 153, 154, 160, 166, 169, 170, 183, 186, 192, 195, 201, 202, 211, 214, 223, 226, 229, 230, 242

Offset: 1

Robert E. Sawyer (rs.1(AT)mindspring.com)

The number of geometric subsequences of [1,...,n] with integral successive-term ratio r and length k is floor(n/r^(k-1))(n > 0, r > 1, k > 0).

			a(2): [1,2]; a(3): [1,2],[1,3]; a(4): [1,2],[1,3],[1,4],[2,4],[1,2,4].

Zhuorui He, Table of n, a(n) for n = 1..10000

Cf. A078651.
Row sums of triangle A090623.
Partial sums of A309891.

g := (n, b) -> local i; add(iquo(n, b^i), i = 1..floor(log(n, b))):
a := n -> local b; add(g(n, b), b = 2..n):
seq(a(n), n = 1..60);  # Peter Luschny, Apr 03 2025

Accumulate[Table[Total[IntegerExponent[n, Rest[Divisors[n]]]], {n, 100}]] (* Paolo Xausa, Aug 27 2025 *)

A078632(n) = {my(s=0, k=2); while(k<=n, s+=(n - sumdigits(n, k))/(k-1); k=k+1); s} \\ Zhuorui He, Aug 26 2025

a(n) = Sum_{r > 1, j > 0} floor(n/r^j).

A369180 Alternating sum of the k-adic valuations (ruler functions) of n.

Original entry on oeis.org

0, 1, -1, 3, -1, 1, -1, 5, -3, 1, -1, 4, -1, 1, -3, 8, -1, 0, -1, 4, -3, 1, -1, 7, -3, 1, -5, 4, -1, 1, -1, 10, -3, 1, -3, 5, -1, 1, -3, 7, -1, 1, -1, 4, -6, 1, -1, 11, -3, 0, -3, 4, -1, -1, -3, 7, -3, 1, -1, 6, -1, 1, -6, 14, -3, 1, -1, 4, -3, 1, -1, 9, -1, 1, -6, 4, -3, 1

Offset: 1

Friedjof Tellkamp, Jan 15 2024

sign

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A007814, A007949, A222171, A235127, A309891.

a:= n-> add((-1)^i*padic[ordp](n, i), i=2..n):
seq(a(n), n=1..78);  # Alois P. Heinz, Jan 15 2024

z = 70; Sum[(-1)^k IntegerExponent[Range[z], k], {k, 2, z}]

a(n) = sum(k=2, n, (-1)^k * valuation(n,k)); \\ Michel Marcus, Jan 18 2024

a(n)=sumdiv(n,k, if(k>1, (-1)^k * valuation(n, k))) \\ Charles R Greathouse IV, Jan 23 2025

a(n) = Sum_{k=2..n} (-1)^k * valuation(n,k).
a(n) = A007814(n) - A007949(n) + A235127(n) - (...).
G.f.: Sum_{k>=2, j>=1} (-1)^k x^(k^j)/(1-x^(k^j)).
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{n=1..m} a(n) = log(2).
Dirichlet g.f.: zeta(s) * Sum_{k>=1} (1 - eta(ks)).
Sum_{n>=1} a(n)/n^2 = Pi^2/24.

A351923 Number of ordered pairs of positive integers (s,t), s,t <= n, such that (s^t) | n.

Original entry on oeis.org

1, 3, 4, 7, 6, 9, 8, 13, 12, 13, 12, 18, 14, 17, 18, 24, 18, 24, 20, 26, 24, 25, 24, 33, 28, 29, 32, 34, 30, 37, 32, 42, 36, 37, 38, 47, 38, 41, 42, 49, 42, 49, 44, 50, 51, 49, 48, 61, 52, 56, 54, 58, 54, 63, 58, 65, 60, 61, 60, 72, 62, 65, 69, 78, 68, 73, 68, 74, 72, 77, 72, 87, 74

Offset: 1

Wesley Ivan Hurt, Feb 25 2022

nonn

			a(4) = 7; The 7 pairs are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2) and (4,1) since all of 1^1, 1^2, 1^3, 1^4, 2^1, 2^2 and 4^1 divide 4.

Cf. A309891.

seq(add(padic[ordp](n,d), d in numtheory[divisors](n) minus {1}) + n, n=1..80); # Ridouane Oudra, Sep 28 2024

a(n) = Sum_{k=1..n} Sum_{i=1..n} (1 - ceiling(n/(k^i)) + floor(n/(k^i))).

a(n) = A309891(n) + n. - Ridouane Oudra, Sep 28 2024

cp's OEIS Frontend

A309947 a(n) = A309891(A025487(n)): For each prime signature, gives the sum of the number of trailing zeros for all bases b >= 2 for some number m with that prime signature. The prime signatures are chosen in order in which they are first seen in the positive integers.

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A111003 Decimal expansion of Pi^2/8.

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

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A078632 Number of geometric subsequences of [1,...,n] with integral successive-term ratio and length > 1.

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Maple

Mathematica

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A369180 Alternating sum of the k-adic valuations (ruler functions) of n.

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Maple

Mathematica

PARI

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A351923 Number of ordered pairs of positive integers (s,t), s,t <= n, such that (s^t) | n.

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