A111003 -id:A111003 - cp's OEIS frontend

A086729 Decimal expansion of Pi^2/72.

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

Offset: 0

N. J. A. Sloane, Jul 31 2003

The original name was: Decimal expansion of Sum_{m=0..infinity} 1/(6*m+3)^2.

			0.1370778389040188697...

L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

Index entries for transcendental numbers

Cf. A086722, A086723, A086724, A086725, A086726, A086727, A086728, A086730, A086731, A111003, A353908.

evalf(Pi^2/72) ;  # R. J. Mathar, Dec 18 2010

PARI
```
Pi^2/72 \\ Omar E. Pol, May 12 2022
```

Equals A111003/9. - R. J. Mathar, Dec 18 2010
From Amiram Eldar, Jul 19 2020: (Start)
Sum_{k>=0} (1/(12*k+3)^2 + 1/(12*k+9)^2).
Equals Integral_{x=1..oo} log(1 + 1/x^6)/x dx. (End)
Equals A353908/2. - Omar E. Pol, May 12 2022

New name after R. J. Mathar's Maple program. - Omar E. Pol, May 12 2022

A168277 a(n) = 2*n - (-1)^n - 2.

Original entry on oeis.org

1, 1, 5, 5, 9, 9, 13, 13, 17, 17, 21, 21, 25, 25, 29, 29, 33, 33, 37, 37, 41, 41, 45, 45, 49, 49, 53, 53, 57, 57, 61, 61, 65, 65, 69, 69, 73, 73, 77, 77, 81, 81, 85, 85, 89, 89, 93, 93, 97, 97, 101, 101, 105, 105, 109, 109, 113, 113, 117, 117, 121, 121, 125, 125, 129, 129

Offset: 1

Vincenzo Librandi, Nov 22 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A016813, A163980, A168276.

Cf. A006752, A111003 (Pi^2/8).

[n eq 1 select 1 else 4*n-Self(n-1)-6: n in [1..70]]; // Vincenzo Librandi, Sep 16 2013

CoefficientList[Series[(1 + 3 x^2) / ((1 + x) (x - 1)^2), {x, 0, 80}], x] (* Vincenzo Librandi, Sep 16 2013 *)
Table[2 n - (-1)^n - 2, {n, 70}] (* Bruno Berselli, Sep 17 2013 *)
LinearRecurrence[{1,1,-1},{1,1,5},70] (* Harvey P. Dale, Aug 25 2015 *)

a(n)=2*n-(-1)^n-2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 4*n - a(n-1) - 6, with n>1, a(1)=1.
a(n) = A163980(n-1), n>1. - R. J. Mathar, Nov 25 2009
G.f.: x*(1 + 3*x^2)/( (1+x)*(x-1)^2 ). - R. J. Mathar, Jul 15 2013
a(n) = A168276(n) - 1. - Vincenzo Librandi, Sep 17 2013
a(n) = a(n-1) +a(n-2) -a(n-3). - Vincenzo Librandi, Sep 17 2013
E.g.f.: (-1 + 3*exp(x) + 2*(x - 1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 16 2016
Sum_{n>=1} 1/a(n)^2 = Pi^2/8 + G, where G is Catalan's constant (A006752). - Amiram Eldar, Aug 21 2022

New definition from Bruno Berselli, Sep 17 2013

A275647 Decimal expansion of Pi^2/6 - Sum_{k>=1} 1/prime(k)^2.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 04 2016

Decimal expansion of sum of squares of reciprocals of nonprime numbers.
Decimal expansion of the nonprime zeta function at 2.
Continued fraction [1; 5, 5, 3, 1, 2, 2, 6, 2, 2, 4, 1, 1, 93, 2, 1, 1, 5, 3, 5, 3, 2, 1, 2, 6, 1, 4, 5, 1, 34, 1, ...]
More generally, the nonprime zeta function at s equals Sum_{k>=1} (1/k^s - 1/prime(k)^s) = Product_{k>=1} 1/(1 - prime(k)^(-s)) - Sum_{k>=1} 1/prime(k)^s.
Floor(1/(zeta(s)-prime zeta(s)-1)) gives second term in continued fraction for nonprime zeta(s): 5, 36, 187, 852, 3663, 15280, 62692, 254760, 1029279, 4143617, ...
Dirichlet g.f. of A005171: nonprime zeta(s).

			1/1^2 + 1/4^2 + 1/6^2 + 1/8^2 + 1/9^2 + 1/10^2 + ... = 1.192686646807160937965871801813777255045718557966906015999...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Ilya Gutkovskiy, Nonprime zeta function.
Eric Weisstein's World of Mathematics, Riemann Zeta Function 2.
Eric Weisstein's World of Mathematics, Prime Zeta Function.

Cf. A002808, A008683, A013661, A018252, A062312, A085548, A111003, A222171.

RealDigits[Pi^2/6 - PrimeZetaP[2], 10, 120][[1]]
RealDigits[Zeta[2] - PrimeZetaP[2], 10, 120][[1]]

eps()=2.>>bitprecision(1.)
primezeta(s)=my(lm=s*log(2)); lm=lambertw(lm/eps())\lm; sum(k=1,lm, moebius(k)*log(abs(zeta(k*s)))/k)
zeta(2) - primezeta(2) \\ Charles R Greathouse IV, Aug 05 2016

Pi^2/6 - sumeulerrat(1/p, 2) \\ Amiram Eldar, Mar 19 2021

Equals zeta(2) - prime zeta(2) = A013661 - A085548.
Equals Sum_{k>=1} (1 - k*mu(k)*log(zeta(2*k)))/k^2, where mu(k) is the Moebius function (A008683).
Equals Sum_{k>=1} 1/A062312(k).
Equals Sum_{k>=1} 1/A018252(k)^2.
Equals 1 + Sum_{k>=1} 1/A002808(k)^2.
Equals A222171 + A111003 - A085548.

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)

A123092 Decimal expansion of Sum_{k>=1} 1/((2k-1)^2(2k+1)^2) = (Pi^2-8)/16.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Sep 27 2006

			0.116850275068084913677155687492259445957106212952549414150834336...

Erwin Kreyszig, Advanced Engineering Mathematics, 9th Edition, John Wiley and Sons, Inc., NJ, 2006, page 506.

Index entries for transcendental numbers.

Cf. A000796, A002388, A069075, A111003.

Cf. A248895, A248896.

RealDigits[Sum[1/((2k - 1)^2(2k + 1)^2), {k, Infinity}], 10, 111][[1]]

(Pi^2-8)/16 \\ Charles R Greathouse IV, Sep 30 2022

Equals (A111003-1)/2. - Hugo Pfoertner, Aug 20 2024

Equals Sum_{k>=1} 1/(4*k^2-1)^2. - Sean A. Irvine, Mar 29 2025

A215947 Difference between the sum of the even divisors and the sum of the odd divisors of 2n.

Original entry on oeis.org

1, 5, 4, 13, 6, 20, 8, 29, 13, 30, 12, 52, 14, 40, 24, 61, 18, 65, 20, 78, 32, 60, 24, 116, 31, 70, 40, 104, 30, 120, 32, 125, 48, 90, 48, 169, 38, 100, 56, 174, 42, 160, 44, 156, 78, 120, 48, 244, 57, 155, 72, 182, 54, 200, 72, 232, 80, 150, 60, 312, 62, 160

Offset: 1

Michel Lagneau, Aug 28 2012

Multiplicative because a(n) = -A002129(2*n), A002129 is multiplicative and a(1) = -A002129(2) = 1. - Andrew Howroyd, Jul 31 2018

			a(6) = 20 because the divisors of 2*6 = 12 are {1, 2, 3, 4, 6, 12} and (12 + 6 + 4 +2) - (3 + 1) = 20.

Michel Lagneau, Table of n, a(n) for n = 1..10000
Hartosh Singh Bal and Gaurav Bhatnagar, Two curious congruences for the sum of divisors function, arXiv:2102.10804 [math.NT], 2021. Mentions this sequence.
H. Movasati and Y. Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411 [math.AG], 2016-2017.

Cf. A000593, A002129, A022998 (Moebius transform), A074400, A195382, A195690.

Cf. A002513, A062731, A111003, A239050.

with(numtheory):for n from 1 to 100 do:x:=divisors(2*n):n1:=nops(x):s0:=0:s1:=0:for m from 1 to n1 do: if irem(x[m],2)=0 then s0:=s0+x[m]:else s1:=s1+x[m]:fi:od:if s0>s1  then printf(`%d, `,s0-s1):else fi:od:

a[n_] := DivisorSum[2n, (1 - 2 Mod[#, 2]) #&];
Array[a, 62] (* Jean-François Alcover, Sep 13 2018 *)
edod[n_]:=Module[{d=Divisors[2n]},Total[Select[d,EvenQ]]-Total[ Select[ d,OddQ]]]; Array[edod,70] (* Harvey P. Dale, Jul 30 2021 *)

a(n) = 4*sigma(n) - sigma(2*n); \\ Andrew Howroyd, Jul 28 2018

From Andrew Howroyd, Jul 28 2018: (Start)
a(n) = 4*sigma(n) - sigma(2*n).
a(n) = -A002129(2*n). (End)
G.f.: Sum_{k>=1} x^k*(1 + 4*x^k + x^(2*k))/(1 - x^(2*k))^2. - Ilya Gutkovskiy, Sep 14 2019
a(p) = p + 1 for p prime >= 3. - Bernard Schott, Sep 14 2019
a(n) = A239050(n) - A062731(n) - Omar E. Pol, Mar 06 2021 (after Andrew Howroyd)
From Amiram Eldar, Nov 18 2022: (Start)
Multiplicative with a(2^e) = 2^(e+2) - 3, and a(p^e) = sigma(p^e) = (p^(e+1) - 1)/(p-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/8 = 1.2337005... (A111003). (End)
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1+2^(1-s)). - Amiram Eldar, Jan 05 2023
From Peter Bala, Sep 25 2023: (Start)
a(2*n) = sigma(2*n) + 2*sigma(n); a(2*n+1) = sigma(2*n+1) = A008438(n)
G.f.: A(q) = Sum_{n >= 1} n*q^n*(1 + 3*q^n)/(1 - q^(2*n)).
Logarithmic g.f.: Sum_{n >= 1} a(n)*q^n/n = Sum_{n >= 1} log(1/(1 - q^n)) + Sum_{n >= 1} log(1/(1 - q^(2*n))) = log (G(q)), where G(q) is the g.f. of A002513. (End)

A222183 Decimal expansion of Sum_{k >= 0} 1/(4*k+1)^2.

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Feb 11 2013

			1.074833072156694421204457444958451501344... = 1 + 1/25 + 1/81 + 1/169 + 1/289 + ...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
E. D. Krupnikov and K. S. Kolbig, Some special cases of the generalized hypergeometric function (q+1)Fq, J. Comp. Appl. Math. 78 (1997) 79-95.
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 112.

Cf. A006752, A222068.

Cf. A013661, A111003, A214550.

SetDefaultRealField(RealField(100)); R:= RealField(); (8*Catalan(R) + Pi(R)^2)/16; // G. C. Greubel, Aug 23 2018

RealDigits[Catalan/2 + Pi^2/16, 10, 90][[1]] (* or *) RealDigits[PolyGamma[1, 1/4]/16, 10, 90]

(8*Catalan + Pi^2)/16 \\ G. C. Greubel, Aug 23 2018

Equals A006752/2 + A222068.
Equals -Integral_{x=0..1} log(x)/(1 - x^4) dx. - Amiram Eldar, Jul 17 2020
Equals 3F2(1/4,1/4,1;5/4,5/4;1). [Krupnikov] - R. J. Mathar, Jun 12 2024
Equals psi'(1/4)/16 (see Shamos). - Stefano Spezia, Nov 12 2024

A073579 Signed primes: if prime(n) even, a(n) = 0; if prime(n) == 1 (mod 4), a(n) = prime(n); if prime(n) == -1 (mod 4), a(n) = -prime(n).

Original entry on oeis.org

0, -3, 5, -7, -11, 13, 17, -19, -23, 29, -31, 37, 41, -43, -47, 53, -59, 61, -67, -71, 73, -79, -83, 89, 97, 101, -103, -107, 109, 113, -127, -131, 137, -139, 149, -151, 157, -163, -167, 173, -179, 181, -191, 193, 197, -199, -211, -223, -227, 229, 233, -239

Offset: 1

Miklos Kristof, Aug 28 2002

Product_{i>1} (1/(1 - 1/a(i))) = 1 - 1/3 + 1/5 - 1/7 - 1/9 + 1/11 ...
= (Pi/4)*Product_{i>1} (1/(1 + 1/a(i)))
= (Pi/2)*Product_{i>1} (1/(1 - 1/a(i)))*Product_{i>1} (1/(1 + 1/a(i)))
= Product_{i>1} (1/(1 - 1/prime(i)^2))
= 1 + 1/3^2 + 1/5^2 + 1/7^2 + 1/9^2 + ...
= Pi^2/8.
Also prime(n)*(2 - prime(n) mod 4) = A000040(n)*A070750(n). - Reinhard Zumkeller, Oct 21 2002
This is a signed version of A160656. - T. D. Noe, Feb 28 2012

			a(1) = 0 because prime(1)=2 is neither 4k+1 nor 4k-1.
a(6) = 13 = prime(6) because 13 = 4*3 + 1.
a(8) = -19 = -prime(8) because 19 = 4*5 - 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A160656, A111003 (Pi^2/8).

a073579 n = p * (2 - p `mod` 4) where p = a000040 n
-- Reinhard Zumkeller, Feb 28 2012

C := ComplexField(); [0] cat [Round(i^(NthPrime(n)-1)*NthPrime(n)): n in [2..100]]; // G. C. Greubel, Dec 31 2019

Maple

0, seq(I^(ithprime(n)-1)*ithprime(n), n = 2..100); # G. C. Greubel, Dec 31 2019

Mathematica

Join[{0},If[Mod[#,4]==1,#,-#]&/@Prime[Range[2,60]]] (* Harvey P. Dale, Feb 27 2012 *) Join[{0}, Table[p = Prime[n]; If[Mod[p, 4] == 1, p, -p], {n, 2, 100}]] (* T. D. Noe, Feb 28 2012 *)

PARI

forprime(p=2,239,print1(p*(2-p%4),", ")) \\ Hugo Pfoertner, Dec 17 2019

Sage

[0]+[I^(nth_prime(n)-1)*nth_prime(n) for n in (2..100)] # G. C. Greubel, Dec 31 2019

Formula

a(1)=0 and for i>1: a(i) = (-1)^((prime(i)-1)/2)*prime(i).

Extensions

Corrected (sign changed on 179) by Harvey P. Dale, Feb 27 2012

cp's OEIS Frontend

A086729 Decimal expansion of Pi^2/72.

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A168277 a(n) = 2*n - (-1)^n - 2.

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A275647 Decimal expansion of Pi^2/6 - Sum_{k>=1} 1/prime(k)^2.

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

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A123092 Decimal expansion of Sum_{k>=1} 1/((2k-1)^2(2k+1)^2) = (Pi^2-8)/16.

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A215947 Difference between the sum of the even divisors and the sum of the odd divisors of 2n.

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A222183 Decimal expansion of Sum_{k >= 0} 1/(4*k+1)^2.

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A173624 Decimal expansion of Pi^2log(2)/8 - 7zeta(3)/16, where zeta is the Riemann zeta function.