A152649 -id:A152649 - cp's OEIS frontend

A238167 Decimal expansion of sum_(n>=1) H(n,3)/n^5 where H(n,3) = A007408(n)/A007409(n) is the n-th harmonic number of order 3.

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.046924401724676082345723014222792329619598402264...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 16.

Cf. A007408, A007409, A152648, A152649, A152651, A238166, A238168, A238169.

RealDigits[5*Zeta[2]*Zeta[5] +2*Zeta[3]*Zeta[4] -10*Zeta[7],10,100][[1]]

5*zeta(2)*zeta(5) + 2*zeta(3)*zeta(4) - 10*zeta(7) \\ G. C. Greubel, Dec 30 2017

Equals 5*zeta(2)*zeta(5) + 2*zeta(3)*zeta(4) - 10*zeta(7).

A238169 Decimal expansion of sum_(n>=1) H(n)^3/n^4 where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.38146831050385237300478512040662269993...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 16.

Cf. A152648, A152649, A152651, A238166, A238167, A238168.

RealDigits[(231/16)*Zeta[7] - (51/4)*Zeta[3]*Zeta[4] + 2*Zeta[2]*Zeta[5], 10, 100][[1]] (* G. C. Greubel, Dec 30 2017 *)

(231/16)*zeta(7) - (51/4)*zeta(3)*zeta(4) + 2*zeta(2)*zeta(5) \\ G. C. Greubel, Dec 30 2017

Equals (231/16)*Zeta(7) - (51/4)*Zeta(3)*Zeta(4) + 2*Zeta(2)*Zeta(5).

A304407 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*k_j).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 3, 4, 4, 10, 4, 12, 6, 8, 4, 16, 4, 18, 8, 12, 10, 22, 6, 8, 12, 6, 12, 28, 8, 30, 5, 20, 16, 24, 8, 36, 18, 24, 12, 40, 12, 42, 20, 16, 22, 46, 8, 12, 8, 32, 24, 52, 6, 40, 18, 36, 28, 58, 16, 60, 30, 24, 6, 48, 20, 66, 32, 44, 24, 70, 12, 72, 36, 16

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(60) = a(2^2*3*5) = (2 - 1)*2 * (3 - 1)*1 * (5 - 1)*1 = 16.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000010, A000026, A000040, A002110, A003958, A005117, A005361, A005867, A006093, A007947, A023900, A048250, A059975, A068997, A081294 (numbers n such that a(n) is odd), A087656, A090624, A152649, A173557, A304117, A304408, A304409, A304411, A304412.

Cf. A001221, A076479.

seq(mul((p-1)*padic[ordp](n, p), p in numtheory[factorset](n)), n=1..100); # Ridouane Oudra, Jun 06 2025

a[n_] := Times @@ ((#[[1]] - 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 75}]
Table[EulerPhi[Last[Select[Divisors[n], SquareFreeQ]]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 75}]

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p-1)*e)} \\ Andrew Howroyd, Jul 24 2018

a(n) = A005361(n)*abs(A023900(n)) = A005361(n)*A173557(n) = A005361(n)*A000010(A007947(n)).
a(p^k) = (p - 1)*k where p is a prime and k > 0.
a(n) = phi(n) if n is a squarefree (A005117), where phi() = A000010.
a(A002110(k)) = A005867(k).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - 4/p^2 + 3/p^3 + 1/p^4 - 1/p^5) = 0.2644703894... . - Amiram Eldar, Nov 30 2022
a(n) = (-1)^A001221(n) * (Sum_{d1|n} Sum_{d2|n} mu(d1)*gcd(d1,d2)). - Ridouane Oudra, Jun 06 2025

A158274 Numerators of antiharmonic means of divisors of n.

Original entry on oeis.org

1, 5, 5, 3, 13, 25, 25, 17, 7, 65, 61, 15, 85, 125, 65, 11, 145, 35, 181, 13, 125, 305, 265, 85, 21, 425, 41, 75, 421, 325, 481, 65, 305, 725, 325, 21, 685, 181, 425, 221, 841, 625, 925, 61, 91, 1325, 1105, 55, 43, 35

Offset: 1

Jaroslav Krizek, Mar 15 2009

Numbers k such that sigma_2(k)/sigma_1(k) = A001157(k)/A000203(k) are integers are in A020487.

			Antiharmonic means of divisors of n>=1: 1, 5/3, 5/2, 3, 13/2, 25/6, ...

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

Cf. A000203, A001157, A020487, A065473, A152649, A158275 (denominators)

Table[Numerator[DivisorSigma[2, n]/DivisorSigma[1, n]], {n, 50}] (* Ivan Neretin, May 22 2015 *)

a(n) = numerator(sigma(n,2)/sigma(n)); \\ Amiram Eldar, Nov 21 2022

Antiharmonic mean of divisors of number n = Product (p_i^e_i) is sigma_2(n)/sigma_1(n) = A001157(n)/A000203(n) = Product ((p_i^(e_i+1)+1)/(p_i+1)).

Sum_{k=1..n} a(k)/A158275(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - (3*p-2)/(p^3)) = A152649 * A065473 = 0.387941... . - Amiram Eldar, Nov 21 2022

A145398 a(n) = Sum_{d|n} sigma(d) - Sum_{2c|n} sigma(c) + 4*Sum_{4b|n} sigma(b).

Original entry on oeis.org

1, 3, 5, 11, 7, 15, 9, 31, 18, 21, 13, 55, 15, 27, 35, 75, 19, 54, 21, 77, 45, 39, 25, 155, 38, 45, 58, 99, 31, 105, 33, 167, 65, 57, 63, 198, 39, 63, 75, 217, 43, 135, 45, 143, 126, 75, 49, 375, 66, 114, 95, 165, 55, 174, 91, 279, 105, 93, 61, 385, 63, 99, 162, 355, 105, 195

Offset: 1

N. J. A. Sloane, Mar 13 2009

Dirichlet convolution of [1,-1,0,4,0,0,...] with A007429.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. A48 (1992), 500-508. Table 1, symmetry C2/m.

Cf. A007429, A145378, A152649.

read("transforms") ;  s1 := [1,-1,0,4,seq(0,n=1..40)] ; s2 := [seq(add(sigma(d),d=divisors(n)),n=1..40)] ; DIRICHLET(s1,s2) ; # R. J. Mathar, Feb 07 2011

f[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; f[2, e_] := 3*2^(e + 1) - 4*e - 5; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 25 2022 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 3*2^(f[i,2]+1) - 4*f[i,2] - 5,  (f[i,1]*(f[i,1]^(f[i,2]+1)-1) - (f[i,1]-1)*(f[i,2]+1))/(f[i,1]-1)^2)); } \\ Amiram Eldar, Oct 25 2022

Dirichlet g.f.: (1-1/2^s+4/4^s)*(zeta(s))^2*zeta(s-1).
From Amiram Eldar, Oct 25 2022: (Start):
Multiplicative with a(2^e) = 3*2^(e+1)-4*e-5, and a(p^e) = (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2 if p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/72 = 1.352904... (A152649). (End)

A238183 Decimal expansion of sum_(n>=1) H(n)^2/n^7 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,7)).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Feb 19 2014

			1.019483497494382286206496675928126515...

Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 24.

Cf. A152648, A152649, A152651, A218505, A238168, A238181, A238182.

Zeta[3]^3/3 - 5/2*Zeta[4]*Zeta[5] - 7/2*Zeta[3]*Zeta[6] - Zeta[2]*Zeta[7] + 55/6*Zeta[9] // RealDigits[#, 10, 100]& // First

zeta(3)^3/3-5/2*zeta(4)*zeta(5)-7/2*zeta(3)*zeta(6)-zeta(2)*zeta(7)+55/6*zeta(9).

A233033 Decimal expansion of sum_(n=1..infinity) (-1)^(n-1)*H(n)/n^3 where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Dec 03 2013

			0.859247157928590615539909939475759980712884350860414926760520689766...

Philippe Flajolet, Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998) page 32.

Cf. A076788 (same alternating sum with denominator n), A152648 (non-alternating sum with denominator n^2), A152649 (non-alternating sum with denominator n^3).

RealDigits[ 11*Pi^4/360 + 1/12*Pi^2*Log[2]^2 - Log[2]^4/12 - 2*PolyLog[4, 1/2] - 7/4*Log[2]*Zeta[3], 10, 100] // First

11*Pi^4/360 + Pi^2*log(2)^2/12 - log(2)^4/12 - 2*polylog(4, 1/2) - 7*log(2)*zeta(3)/4 \\ Charles R Greathouse IV, Aug 27 2014

Equals 11*Pi^4/360 +1/12*Pi^2*log(2)^2 -log(2)^4/12 -2*Li4(1/2) -7/4*log(2)*zeta(3).

Also, equals 1/2*integral_{z=0..1} (log(z)^2*log(1+z)) / (z*(1+z)) dz.

A333645 a(n) = Sum_{d|n} uphi(d).

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 7, 12, 11, 10, 11, 15, 13, 14, 15, 27, 17, 22, 19, 25, 21, 22, 23, 36, 29, 26, 37, 35, 29, 30, 31, 58, 33, 34, 35, 55, 37, 38, 39, 60, 41, 42, 43, 55, 55, 46, 47, 81, 55, 58, 51, 65, 53, 74, 55, 84, 57, 58, 59, 75, 61, 62, 77, 121, 65, 66, 67, 85, 69, 70

Offset: 1

Ilya Gutkovskiy, Mar 31 2020

Inverse Moebius transform of A047994.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117 (fixed points), A023900, A047994, A055653, A062949, A065464, A152649.

uphi[1] = 1; uphi[n_] := Times @@ (#[[1]]^#[[2]] - 1 & /@ FactorInteger[n]); a[n_] := Sum[uphi[d], {d, Divisors[n]}]; Table[a[n], {n, 70}]
A023900[n_] := Sum[MoebiusMu[d] d, {d, Divisors[n]}]; A062949[n_] := Sum[EulerPhi[d] DivisorSigma[0, d], {d, Divisors[n]}]; a[n_] := Sum[A023900[d] A062949[n/d], {d, Divisors[n]}]; Table[a[n], {n, 70}]
f[p_,e_] := (p^(e+1) - e*p + e - 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100] (* Amiram Eldar, Nov 12 2022 *)

uphi(n)=my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); \\ A047994
a(n) = sumdiv(n, d, uphi(d)); \\ Michel Marcus, Mar 31 2020

G.f.: Sum_{k>=1} uphi(k) * x^k / (1 - x^k).
a(n) = Sum_{d|n} A023900(d) * A062949(n/d).
From Amiram Eldar, Nov 12 2022: (Start)
Multiplicative with a(p^e) = (p^(e+1) - e*p + e - 1)/(p-1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - (2*p-1)/p^3) = A152649 * A065464 = 0.5793804872... . (End)

A241215 Decimal expansion of Sum_{n>=1} H(n)^4/(n+1)^3 where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 17 2014

			1.80161326804341290372948894202088843...

Philippe Flajolet and Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998), page 16.

Cf. A016627, A083680, A102886, A152648, A152649, A152651, A233033, A233090, A238166, A238167, A239168, A238169, A238181, A238182, A238183, A240264.

37/180*Pi^4*Zeta[3] - 5/6*Pi^2*Zeta[5] - 109/8*Zeta[7] // RealDigits[#, 10, 100]& // First

37/2*zeta(3)*zeta(4) - 5*zeta(2)*zeta(5) - 109/8*zeta(7) \\ Stefano Spezia, Jan 19 2025

Equals (37/2)*zeta(3)*zeta(4) - 5*zeta(2)*zeta(5) - (109/8)*zeta(7).

Equals (37/180)*Pi^4*zeta(3) - (5/6)*Pi^2*zeta(5) - (109/8)*zeta(7).

A384457 Decimal expansion of Sum_{k>=1} H(k)^3/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 30 2025

			3.59342794177494296025518240703339219591695480351933...

K. Ramachandra and R. Sitaramachandrarao, On series, integrals and continued fractions - II, Madras Univ. J., Sect. B, 51 (1988), pp. 181-198.

K. Ramachandra, On series integrals and continued fractions I, Hardy-Ramanujan Journal, Vol. 4 (1981), pp. 1-11.
K. Ramachandra, On series, integrals and continued fractions, III, Acta Arithmetica, Vol. 99, No. 3 (2001), pp. 257-266.

Cf. A001008, A002805.
Cf. A000796, A002117, A002162, A352769.
Related constants: A152648, A152649, A152651, A218505, A233090, A238168, A238181, A238182, A241753, A244667, A253191, A256988, A345203.

RealDigits[Zeta[3] + (Pi^2*Log[2] + Log[2]^3)/3, 10, 120][[1]]

PARI
```
zeta(3) + (Pi^2*log(2) + log(2)^3)/3
```

Equals zeta(3) + (Pi^2*log(2) + log(2)^3)/3.

cp's OEIS Frontend

A238167 Decimal expansion of sum_(n>=1) H(n,3)/n^5 where H(n,3) = A007408(n)/A007409(n) is the n-th harmonic number of order 3.

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A238169 Decimal expansion of sum_(n>=1) H(n)^3/n^4 where H(n) is the n-th harmonic number.

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A304407 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*k_j).

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A158274 Numerators of antiharmonic means of divisors of n.

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A145398 a(n) = Sum_{d|n} sigma(d) - Sum_{2c|n} sigma(c) + 4*Sum_{4b|n} sigma(b).

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A238183 Decimal expansion of sum_(n>=1) H(n)^2/n^7 where H(n) is the n-th harmonic number (Quadratic Euler Sum S(2,7)).

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A233033 Decimal expansion of sum_(n=1..infinity) (-1)^(n-1)*H(n)/n^3 where H(n) is the n-th harmonic number.

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