A111003 -id:A111003 - cp's OEIS frontend

A298735 Number of odd squares dividing n.

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

Offset: 1

Ilya Gutkovskiy, Jan 25 2018

The smallest integer with exactly m odd square divisors is A357450(m). - Bernard Schott, Oct 03 2022

			a(81) = 3 because 81 has 5 divisors {1, 3, 9, 27, 81} among which 3 are odd squares {1, 9, 81}.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000265, A001227, A016754, A046951, A056170, A071325, A111003, A122132 (positions of ones), A357450.

nmax = 105; Rest[CoefficientList[Series[Sum[x^(2 k - 1)^2/(1 - x^(2 k - 1)^2), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Length[Select[Divisors[n], IntegerQ[Sqrt[#]] && OddQ[#] &]]; Table[a[n], {n, 1, 105}]
f[2, e_] := 1; f[p_, e_] := Floor[e/2] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

a(n)=factorback(apply(e->e\2+1, factor(n/2^valuation(n,2))[, 2])) \\ Rémy Sigrist, Jan 26 2018

G.f.: Sum_{k>=1} x^((2*k-1)^2)/(1 - x^((2*k-1)^2)).
Multiplicative with a(2^e) = 1 and a(p^e) = floor(e/2) + 1 for p > 2. - Amiram Eldar, Sep 11 2020
a(n) = A046951(4*n) - A046951(n) = A046951(A000265(n)). - Velin Yanev, Antti Karttunen, Dec 06 2021
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/8 (A111003). - Amiram Eldar, Sep 25 2022

Keyword mult added by Rémy Sigrist, Jan 26 2018

A379481 Square of prime-shifted n, or equally, n squared, then prime-shifted one step towards larger primes.

Original entry on oeis.org

1, 9, 25, 81, 49, 225, 121, 729, 625, 441, 169, 2025, 289, 1089, 1225, 6561, 361, 5625, 529, 3969, 3025, 1521, 841, 18225, 2401, 2601, 15625, 9801, 961, 11025, 1369, 59049, 4225, 3249, 5929, 50625, 1681, 4761, 7225, 35721, 1849, 27225, 2209, 13689, 30625, 7569, 2809, 164025, 14641, 21609, 9025, 23409, 3481, 140625

Offset: 1

Antti Karttunen, Dec 27 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to prime indices in the factorization of n.

Cf. A000290, A003961, A016754, A048673, A111003, A337336, A378231, A379482 [= sigma(a(n))], A379484 [= A379473(a(n))].

{1}~Join~Array[Apply[Times, Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]] ]^2 &, 53, 2] (* Michael De Vlieger, Dec 27 2024 *)

A379481(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1); f[i, 2] *= 2); factorback(f); };

Fully multiplicative with a(prime(i)) = prime(i+1)^2.
a(n) = A003961(n^2) = A003961(n)^2.
a(n) = A016754(A048673(n)-1).
a(n) = (1/2)*(A378231(n)+A379482(n)).
From Amiram Eldar, Dec 28 2024: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/8 (A111003).
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^2/72. (End)

A242822 Decimal expansion of B. Davis' constant Pi^2/(8*G), a Riesz-Kolmogorov constant, where G is Catalan's constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 23 2014

			1.3468852519994065951820075554411...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 7.7 Riesz-Kolmogorov Constants, p. 474.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001221, A002145, A004614, A006752, A111003, A330890, A377753.

SetDefaultRealField(RealField(100)); R:=RealField(); Pi(R)^2/(8*Catalan(R)); // G. C. Greubel, Aug 25 2018

s:= convert(evalf(Pi^2/(8*Catalan), 140), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014

RealDigits[Pi^2/(8*Catalan), 10, 100] // First

default(realprecision, 100); Pi^2/(8*Catalan) \\ G. C. Greubel, Aug 25 2018

(Sum_{n>=0} 1/(2*n + 1)^2) / (Sum_{n>=0} (-1)^n/(2*n + 1)^2) = A111003/A006752.
Equals Product_{k>=1} (1 + 1/A002145(k)^2)/(1 - 1/A002145(k)^2) = A243381 / A243379. - Vaclav Kotesovec, Apr 30 2020
Equals Sum_{q in A004614} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021
Equals 1/A377753. - Hugo Pfoertner, Nov 22 2024

A300690 Decimal expansion of sqrt(Pi^2/8 - 1).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 11 2018

Also the total harmonic distortion (THD) of a square wave, see formula (11) in the Blagouchine & Moreau link.

			0.4834258476086790990137326370639317022328017276651459...

I. V. Blagouchine and E. Moreau, Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues. IEEE Trans. Commun., vol. 59, no. 9, pp. 2478-2491, 2011. PDF file.
Index entries for transcendental numbers

Cf. A002388, A111003, A300713, A300714, A300727, A300731.

MATLAB
```
format long; sqrt(pi^2/8-1)
```
Maple
```
evalf(sqrt((1/8)*Pi^2-1), 120)
```

RealDigits[Sqrt[Pi^2/8 - 1], 10, 120][[1]]

default(realprecision, 120); sqrt(Pi^2/8-1)

A300710 Decimal expansion of 17*Pi^8/161280.

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 11 2018

Also the sum of the series Sum_{n>=0} (1/(2n+1)^8), whose value is obtained from zeta(8) given by L. Euler in 1735: Sum_{n>=0} (2n+1)^(-s)=(1-2^(-s))*zeta(s).

			1.0001551790252961193029872492957280415665429750613740...

Jason Bard, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21.

Cf. A092736, A111003, A300707, A300709.

MATLAB
```
format long; (17/161280)*pi^8
```
Maple
```
evalf((17/161280)*Pi^8, 120);
```

RealDigits[(17/161280)*Pi^8, 10, 120][[1]]

default(realprecision, 120); (17/161280)*Pi^8

Equals 17*A092736/161280. - Omar E. Pol, Mar 11 2018
From Artur Jasinski, Jun 24 2025: (Start)
Equals DirichletL(2,1,8).
Equals DirichletL(4,1,8).
Equals DirichletL(8,1,8).
Equals DirichletL(16,1,8). (End)
Equals 255*Zeta(8)/256. - Jason Bard, Aug 21 2025

A276712 Decimal expansion of zeta(3)/8.

Original entry on oeis.org

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

Offset: 0

Terry D. Grant, Sep 15 2016

			0.150257112894949285674967270188...

James Dodson, The Mathematical Repository Containing Analytical Solutions of Five Hundred Questions: Mostly Selected from Scarce and Valuable Authors, (1748), page 375.

G. C. Greubel, Table of n, a(n) for n = 0..5000
R. Barbieri, J. A. Mignaco and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864, table II (5)
Nick Lord, Problen 89.D, Problem Corner, The Mathematical Gazette, Vol. 89, No. 514 (2005), pp. 115-119; Solution, ibid., Vol. 89, No. 516 (2005), pp. 539-542.
Michael Penn, The solution is an important constant, YouTube video, 2021.

Cf. A001008, A002117, A002805, A013661, A111003, A222171, A233091.

SetDefaultRealField(RealField(120)); L:=RiemannZeta();  Evaluate(L,3)/8; // G. C. Greubel, Nov 24 2021

Mathematica
```
RealDigits[(Zeta[3])/8, 10, 100][[1]]
```

zeta(3)/8 \\ Michel Marcus, Sep 16 2016

Sage
```
(zeta(3)/8).n(100)
```

Equals Sum_{n>=1} 1/(2n)^3 = 1/8 + 1/64 + 1/216 + 1/512 + ...
Equals A002117/8.
zeta(3)/8 + A233091 = Sum_{n>=1} 1/(2n+1)^3 + Sum_{n>=1} 1/(2n)^3 = zeta(3).
Equals Sum_{k>=1} (-1)^(k+1) * H(k)/(k+1)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jul 22 2020
Equals Integral_{x=0..Pi/4} log(sin(x))*log(cos(x))/(sin(x)*cos(x)) dx (Lord, 2005). - Amiram Eldar, Jun 23 2023
Equals -integral_{x=0..1} log(x) log(1+x)/(1+x). [Barbieri] - R. J. Mathar, Jun 07 2024

A300709 Decimal expansion of Pi^6/960.

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 11 2018

Also the sum of the series Sum_{n>=0} (1/(2n+1)^6), whose value is obtained from zeta(6) given by L. Euler in 1735: Sum_{n>=0}(2n+1)^(-s) = (1-2^(-s))*zeta(s).

			1.0014470766409421219064785871379373946533515917510902...

Index entries for transcendental numbers

Cf. A092732, A111003, A300707, A300710.

MATLAB
```
format long; pi^6/960
```
Maple
```
evalf((1/960)*Pi^6, 120)
```
Mathematica
```
RealDigits[Pi^6/960, 10, 120][[1]]
```
PARI
```
default(realprecision, 120); Pi^6/960
```

Equals A092732/960. - Omar E. Pol, Mar 11 2018
From Artur Jasinski, Jun 24 2025: (Start)
Equals DirichletL(2,1,6).
Equals DirichletL(4,1,6).
Equals DirichletL(8,1,6).
Equals DirichletL(16,1,6). (End)

A354246 Indices of coefficients of x^(2k-1) in Integral exp(-xtan(x))/cos(x) dx at which the signs of the coefficients change: list of k such that sign(A354245(k)) != sign(A354245(k-1)), starting with 1.

Original entry on oeis.org

1, 2, 5, 10, 18, 29, 42, 57, 75, 95, 118, 143, 171, 201, 234, 269, 307, 347, 390, 435, 482, 532, 585, 639, 697, 757, 819, 884, 951, 1021, 1093, 1167, 1245, 1324, 1406, 1491, 1578, 1667, 1759, 1853, 1950, 2050, 2151, 2256, 2362, 2471, 2583, 2697, 2814, 2933, 3054, 3178, 3305, 3434, 3565, 3699, 3835, 3974, 4115, 4259, 4405, 4554, 4705, 4859

Offset: 1

Paul D. Hanna, May 20 2022

nonn

The e.g.f. of A354245 is Integral exp(-x*tan(x))/cos(x) dx.
What is the limit of a(n)/n^2 ?
Conjecture: lim_{n->oo} a(n)/n^2 = Pi^2/8 = A111003 = 1.2337... - Vaclav Kotesovec, May 26 2022

			The expansion of Integral exp(-x*tan(x)) / cos(x) dx = x - x^3/3! - 3*x^5/5! - 5*x^7/7! + 441*x^9/9! + 25911*x^11/11! + 1384757*x^13/13! + 74436531*x^15/15! + 3175224945*x^17/17! - 135369432209*x^19/19! + ... + A354245(n)*x^(2*n-1)/(2*n-1)! + ...
The signs (+-1) of the coefficients A354245 begin:
[+, -, -, -, +, +, +, +, +, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, +, +, +, +, +, +, +, +, +, +, +, +, +, +, +, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, +, ...].
This sequence gives the positions in A354245 at which the signs of the coefficients change.

Cf. A354245, A354399, A354425.

nmax = 500; A354245 = Table[(CoefficientList[Series[1/(E^(x*Tan[x])*Cos[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[k]], {k, 1, 2*nmax, 2}]; Join[{1}, Select[Range[nmax], A354245[[#]]*A354245[[#-1]] < 0 &]] (* Vaclav Kotesovec, May 24 2022 *)

a(39)-a(64) from Vaclav Kotesovec, May 26 2022

A354399 List of k such that sign(A009273(k)) = sign(A009273(k+1)).

Original entry on oeis.org

0, 1, 5, 12, 21, 33, 47, 64, 83, 105, 129, 155, 184, 216, 250, 286, 325, 366, 410, 456, 505, 556, 610, 666, 725, 786, 849, 915, 984, 1055, 1128, 1204, 1282, 1363, 1446, 1532, 1620, 1711, 1804, 1900, 1998, 2098, 2201, 2307, 2415, 2525, 2638, 2753, 2871, 2991, 3114, 3239, 3367, 3497, 3630, 3765, 3903, 4043, 4185, 4330, 4477, 4627, 4780, 4935

Offset: 1

Vaclav Kotesovec, May 25 2022

nonn

Conjecture: lim_{n->oo} a(n)/n^2 = Pi^2/8 = A111003 = 1.2337...

			12 is in the sequence because A009273(12) = -454930757753597952 and A009273(13) = -94991612229069430784 have the same sign.

Cf. A009273, A354246, A354425.

nmax = 400; A009273 = Table[(CoefficientList[Series[E^(x*Tanh[x]), {x, 0, 2*nmax}], x]*Range[0, 2*nmax]!)[[k]], {k, 1, 2*nmax, 2}]; Join[{0}, Select[Range[nmax-1], A009273[[#]]*A009273[[#-1]] > 0 &] - 1]

A195056 Decimal expansion of Pi^2/7.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 04 2011

			1.409943485869908374119212999982307305045...

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.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A000265, A002388, A013661, A019674, A100044, A111003, A164102.

Pi(RealField(128))^2/7; // G. C. Greubel, Jun 02 2021

RealDigits[Pi^2/7, 10, 105][[1]] (* T. D. Noe, Oct 05 2011 *)

PARI
```
Pi^2/7 \\ Michel Marcus, Feb 04 2022
```

numerical_approx(pi^2/7, digits=128) # G. C. Greubel, Jun 02 2021

Equals Sum_{k>=1} A000265(k)/k^3. - Amiram Eldar, Jun 27 2020

Equals Integral_{x=0..1} log(1+x+x^2+x^3+x^4+x^5+x^6)/x dx (Aubonnet). - Bernard Schott, Feb 04 2022

Extended by T. D. Noe, Oct 05 2011

cp's OEIS Frontend

A298735 Number of odd squares dividing n.

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A379481 Square of prime-shifted n, or equally, n squared, then prime-shifted one step towards larger primes.

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A242822 Decimal expansion of B. Davis' constant Pi^2/(8*G), a Riesz-Kolmogorov constant, where G is Catalan's constant.

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Magma

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A300690 Decimal expansion of sqrt(Pi^2/8 - 1).

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MATLAB

Maple

Mathematica

PARI

A300710 Decimal expansion of 17*Pi^8/161280.

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MATLAB

Maple

Mathematica

PARI

Formula

A276712 Decimal expansion of zeta(3)/8.

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Programs

Magma

Mathematica

PARI

Sage

Formula

A300709 Decimal expansion of Pi^6/960.

A354246 Indices of coefficients of x^(2k-1) in Integral exp(-xtan(x))/cos(x) dx at which the signs of the coefficients change: list of k such that sign(A354245(k)) != sign(A354245(k-1)), starting with 1.