A002162 -id:A002162 - cp's OEIS frontend

A230476 a(n) = Sum_{i=1..n} d(8i+1) - Sum_{i=1..n} d(2i+1), where d(n) = A000005(n) is the number of divisors of n.

1, 1, 2, 3, 3, 4, 4, 6, 6, 7, 7, 6, 10, 10, 11, 11, 9, 11, 13, 15, 16, 14, 16, 15, 15, 17, 17, 22, 22, 22, 20, 18, 20, 24, 24, 25, 27, 27, 27, 26, 28, 26, 30, 30, 29, 31, 31, 37, 35, 35, 35, 31, 35, 35, 40, 40, 38, 40, 40, 41, 41, 41, 43, 47, 47, 46, 42, 44, 46, 50, 48, 46, 52, 52, 52, 54, 52, 55, 55, 53, 55, 53, 59, 58, 56, 58

Offset: 1

Jonathan Sondow, Oct 20 2013

nonn

Cimadevilla proved that a(n) >= 0. That is surprising because d(8*i+1) - d(2*i+1) < 0 for i = 12, 17, 22, 24, 31, 32, 40, 42, 45, 49, 52, 57, 66, 67, 71, 72, 77, 80, 82, 84, 85, ...

			The divisors of 8*1 + 1 = 9 are 1, 3, 9 and those of 2*1 + 1 = 3 are 1, 3, so a(1) = d(9) - d(3) = 3 - 2 = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jorge Luis Cimadevilla Villacorta, Certain inequalities associated with the divisor function, Amer. Math. Monthly, 120 (2013), 832-837. See inequalities (1.5).

Cf. A000005, A002162, A230290, A230291, A230292, A230293, A230295, A230296.

Table[Sum[ DivisorSigma[0, 8 i + 1] - DivisorSigma[0, 2 i + 1], {i, n}], {n, 100}]

a(n) = sum(i=1, n, numdiv(8*i+1) - numdiv(2*i+1)); \\ Michel Marcus, Jun 19 2015

a(n) = Sum_{i=1..n} (d(8*i+1) - d(2*i+1)) = A230293(n) + A230294(n).

a(n) = log(2) * n + O(n^(1/3)*log(n)). - Amiram Eldar, Apr 12 2024

A244647 Decimal expansion of the sum of the reciprocals of the decagonal numbers (A001107).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

For the partial sums of the reciprocals of the (positive) decagonal numbers see A250551(n+1)/A294515(n), n >= 0. - Wolfdieter Lang, Nov 07 2017

			1.216745956158244182494339352004760382108361700922772890949837441544696356350....

Wikipedia, Polygonal number

Cf. A001107, A000038, A013661, A244639, A016627, A244645, A244646, A244648, A244649, A250551(n+1)/A294515(n).

RealDigits[ Log[2] + Pi/6, 10, 111][[1]] (* or *)
RealDigits[ Sum[1/(4n^2 - 3n), {n, 1 , Infinity}], 10, 111][[1]]

log(2)+Pi/6 \\ Charles R Greathouse IV, Feb 08 2023

Sum_{n>0} 1/(4n^2 - 3n) = log(2) + Pi/6, (A002162 + A019673).

A016635 Decimal expansion of log(12).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.484906649788000310229709479838878840798490826543259959976...

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for transcendental numbers

Cf. A016740 Continued fraction.

RealDigits[Log[12], 10, 120][[1]] (* Alonso del Arte, Mar 12 2015 *)

default(realprecision, 20080); x=log(12); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016635.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009, corrected May 19 2009

Equals 2*A002162 + A002391. - R. J. Mathar, Jun 10 2024

A097064 Expansion of (1 - 4x + 6x^2)/(1 - 2*x)^2.

Original entry on oeis.org

1, 0, 2, 8, 24, 64, 160, 384, 896, 2048, 4608, 10240, 22528, 49152, 106496, 229376, 491520, 1048576, 2228224, 4718592, 9961472, 20971520, 44040192, 92274688, 192937984, 402653184, 838860800, 1744830464, 3623878656, 7516192768, 15569256448, 32212254720, 66571993088

Offset: 0

Paul Barry, Jul 22 2004

Binomial transform of A097062.

Index entries for linear recurrences with constant coefficients, signature (4,-4).

Essentially the same as A036289.

Cf. A001787, A002162, A016578, A097062.

CoefficientList[Series[(1-4x+6x^2)/(1-2x)^2,{x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{4,-4},{0,2},30]] (* Harvey P. Dale, May 26 2011 *)

a(n) = (n-1)*2^(n-1) + 3*0^n/2.
a(n) = 4*a(n-1) - 4*a(n-2), n>2.
a(n) = Sum_{k=0..n} binomial(n, k)*((2k-1)/2 + 3*(-1)^k/2).
a(n+1)/2 = A001787(n).
From Amiram Eldar, Oct 01 2022: (Start)
Sum_{n>=2} 1/a(n) = log(2) (A002162).
Sum_{n>=2} (-1)^n/a(n) = log(3/2) (A016578). (End)
E.g.f.: (3 - exp(2*x)*(1 - 2*x))/2. - Stefano Spezia, Feb 12 2023

A101550 Lopsided (or biased) numbers: numbers n such that the largest prime factor of n is > 2*sqrt(n).

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 74, 76, 79, 82, 83, 86, 87, 89, 92, 93, 94, 97, 101, 103, 106, 107, 109, 111, 113, 115, 116, 118, 122, 123, 124, 127, 129, 131, 134, 137, 139, 141

Offset: 1

T. D. Noe, Dec 06 2004

nonn

Note that all primes > 3 are here. See A101549 for composite lopsided numbers.
First differs from A320048 at a(51). - (After R. J. Mathar), - Omar E. Pol, Oct 04 2018
The asymptotic density of this sequence is log(2) (Chowla and Todd, 1949). - Amiram Eldar, Jul 09 2020

T. D. Noe, Table of n, a(n) for n = 1..1000
S. D. Chowla and John Todd, The Density of Reducible Integers, Canadian Journal of Mathematics, Vol. 1, No. 3 (1949), pp. 297-299.
G. Everest, S. Stevens, D. Tamsett and T. Ward, Primitive Divisors of Quadratic Polynomial Sequences, arXiv:math/0412079 [math.NT], 2004-2006.
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.

Cf. A002162, A063763 (composite n such that the largest prime factor > sqrt(n)), A064052 (n such that the largest prime factor > sqrt(n)).

with(numtheory): a:=proc(n) if max((seq(factorset(n)[j],j=1..nops(factorset(n)))))^2>4*n then n else fi end: seq(a(n),n=2..170); # Emeric Deutsch, May 27 2007

Select[Range[2, 200], FactorInteger[ # ][[ -1, 1]]>2Sqrt[ # ]&]

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A115252 Decimal expansion of -(Pilog((sqrt(2Pi)*Gamma(3/4))/Gamma(1/4)))/2.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 17 2006

This sequence (its negated version) is also the decimal expansion of the first Malmsten integral int_{x=1..infinity} log(log(x))/(1 + x^2) dx = int_{x=0..1} log(log(1/x))/(1 + x^2) dx = int_{x=0..infinity} 0.5*log(x)/cosh(x) dx = int_{x=Pi/4..Pi/2} log(log(tan(x))) dx = (1/2)*Pi*log(2) + (3/4)*Pi*log(Pi) - Pi*log(Gamma(1/4)). - Iaroslav V. Blagouchine, Mar 29 2015

			0.26044280630098844554009386878972721953181917772314...

G. C. Greubel, Table of n, a(n) for n = 0..5000
I. V. Blagouchine, Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal, Volume 35, Issue 1, pp. 21-110, 2014, DOI: 10.1007/s11139-013-9528-5. PDF file
Eric Weisstein's World of Mathematics, Vardi's Integral

Cf. A256127 (second Malmsten integral), A256128 (third Malmsten integral), A256129 (fourth Malmsten integral), A068466 (Gamma(1/4)), A256166 (log(Gamma(1/4))), A002162 (log 2), A053510 (log Pi).

RealDigits[-Pi/2*Log[Sqrt[2 Pi] Gamma[3/4]/Gamma[1/4]], 10, 111][[1]] (* Robert G. Wilson v, Dec 06 2014 *)

(-Pi*log((sqrt(2*Pi)*gamma(3/4))/gamma(1/4)))/2 \\ Michel Marcus, Dec 06 2014

Equals integral_[0..1] log(1/log(1/x))/(1+x^2) dx. - Jean-François Alcover, Jan 28 2015

A163973 Decimal expansion of Van der Pauw's constant = Pi/log(2).

Original entry on oeis.org

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

Offset: 1

Johannes W. Meijer, Aug 13 2009

Van der Pauw developed a method for measuring the sheet resistance of a four-terminal conducting sheet of arbitrary shape. Assuming the terminals to be point contacts at the periphery of the structure, he proved a general theorem that yields an analytical expression for the sheet resistance Rs. In the special case that the structure is invariant for a rotation of ninety degrees, the formula of Van der Pauw is Rs = (Pi/log(2))*(V/I).
A general theorem for the sheet resistance Rs of a Van der Pauw structure with finite contacts that is invariant for a rotation of ninety degrees was proved by Versnel. His theorem states that Rs = [K(k1)/K'(k1) - K(k2)/(2*K'(k2))]^(-1)*(V/I) with K(k) and K'(k) complete elliptic integrals with modulus k (Abramowitz and Stegun use parameter m = k^2).
Versnel found, with a little help from the author, expressions for Rs = C(d)*(V/I) for several Van der Pauw structures if d, the ratio of the sum of the lengths of the contacts and the length of the boundary of the sheet, tends to zero, see the formulas (first two terms are given). For point contacts, i.e., d = 0, Van der Pauw's constant appears.

			4.5323601418271938

G. C. Greubel, Table of n, a(n) for n = 1..5000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 17, pp. 589-626.
L.J. van der Pauw, A method of measuring specific resistivity and Hall effect of disc of arbitrary shape, Philips Research Reports, Vol. 13. no. 1, pp 1-9, February 1958.
W. Versnel, Analysis of symmetrical Van der Pauw structures with finite contacts, Solid State Electronics, Vol. 21, pp. 1261-1268, 1978.
W. Versnel, Analysis of the Greek cross, a Van der Pauw structure with finite contacts, Solid State Electronics, Vol. 22, pp. 911-914, 1979.
Eric. W. Weisstein, Elliptic Integral, from Wolfram MathWorld.
Wikipedia, Van der Pauw method

Cf. A000796 (Pi), A002162 (log(2)), A093341 (K), A131223 (2*Pi/log(2)), A259679 (log(2)/(4*Pi^2)).

RealDigits[N[Pi/Log[2], 103]][[1]] (* Mats Granvik, Apr 04 2012 *)

Pi/log(2) \\ Charles R Greathouse IV, Jan 30 2016

1) Circle with contacts in the middle of each side:
C(d) = Pi/log(2) + (Pi^3/(64*(log(2))^2))*d^2
2) Square with contacts in the middle of each side:
C(d) = Pi/log(2) + (Pi*K^2/(8*(log(2))^2))*d^2
3) Square with complementary contacts:
C(d) = Pi/log(2) + (Pi*K^4/(64*(log(2))^2))*d^4
with K = K(sqrt(2)/2) = 1.8540746773.
4) Greek cross with contacts at the cross ends:
C(d) = Pi/log(2) + 2*Pi/(log(2))^2*exp(Pi/2-Pi/d)
5) Greek cross with contacts between the cross ends:
C(d) = Pi/log(2) + ((Pi/(2^12*log(2)^2)*((-3/4)!/(-1/4)!)^8))*d^4

A275703 Decimal expansion of the Dirichlet eta function at 6.

Original entry on oeis.org

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

Offset: 0

Terry D. Grant, Aug 05 2016

It appears that each sum of a Dirichlet eta function is 1/2^(x-1) less than the zeta(x), where x is a positive integer > 1. In this case, eta(x) = eta(6) = (31/32)*zeta(6) = 31*(Pi^6)/30240. Therefore eta(6) = 1/2^(6-1) or 1/32nd less than zeta(6) (see A013664). [Edited by Petros Hadjicostas, May 07 2020]

			31*(Pi^6)/30240 = 0.9855510912974...

Cf. A002162 (decimal expansion of value at 1), A072691 (value at 2), A197070 (value at 3), A267315 (value at 4), A267316 (value at 5), A275710 (value at 7).

Cf. A013664, A136677, A334605.

Mathematica
```
RealDigits[31*(Pi^6)/30240,10,100]
```

s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s -= (-1)^i / i^6
print(s) # Terry D. Grant, Aug 05 2016

eta(6) = 31*(Pi^6)/30240 = 31*A092732/30240 = Sum_{n>=1} (-1)^(n+1)/n^6.

eta(6) = lim_{n -> infinity} A136677(n)/A334605(n). - Petros Hadjicostas, May 07 2020

A275710 Decimal expansion of the Dirichlet eta function at 7.

Original entry on oeis.org

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

Offset: 0

Terry D. Grant, Aug 06 2016

			0.99259381992283028267...

Cf. A002162 (value at 1), A013665, A072691 (value at 2), A197070 (value at 3), A267315 (value at 4), A267316 (value at 5), A275703 (value at 6), A334668, A334669, A347150, A347059.

RealDigits[63 Zeta[7]/64, 10, 100] [[1]]

-polylog(7, -1) \\ Michel Marcus, Aug 20 2021

s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s -= (-1)^i / i^7
print(s) # Terry D. Grant, Aug 06 2016

eta(7) = 63*zeta(7)/64 = (63*A013665)/64.
eta(7) = Lim_{n -> infinity} A334668(n)/A334669(n). - Petros Hadjicostas, May 07 2020
Equals Sum_{k>=1} (-1)^(k+1) / k^7. - Sean A. Irvine, Aug 19 2021

A016687 Decimal expansion of log(64) = 6*log(2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			4.158883083359671856503392728749059408453000806161531524724080056960361...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for transcendental numbers

Cf. A002162, A005900, A016492 (continued fraction), A016627, A016631.

RealDigits[Log[64],10,120][[1]] (* Harvey P. Dale, May 06 2022 *)

default(realprecision, 20080); x=log(64); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016687.txt", n, " ", d)); \\ Harry J. Smith, May 22 2009

Equals 2*A016631 = 3*A016627 = 6*A002162. - Alois P. Heinz, Aug 07 2023
From Peter Bala, Mar 05 2024: (Start)
log(64) = 4 + Sum_{n >= 1} (-1)^(n+1)/(p(n)*p(n+1)), where p(n) = n*(2*n^2 + 1)/3 = A005900.
Continued fraction: log(64) = 4 + 1/(6 + (1*2)/(6 + (2*3)/(6 + (3*4)/(6 + (4*5)/(6 + ... ))))). See A142983. Cf. A016627. (End)

cp's OEIS Frontend

A230476 a(n) = Sum_{i=1..n} d(8*i+1) - Sum_{i=1..n} d(2*i+1), where d(n) = A000005(n) is the number of divisors of n.

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A244647 Decimal expansion of the sum of the reciprocals of the decagonal numbers (A001107).

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A016635 Decimal expansion of log(12).

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A097064 Expansion of (1 - 4*x + 6*x^2)/(1 - 2*x)^2.

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A101550 Lopsided (or biased) numbers: numbers n such that the largest prime factor of n is > 2*sqrt(n).

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A115252 Decimal expansion of -(Pi*log((sqrt(2*Pi)*Gamma(3/4))/Gamma(1/4)))/2.

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A163973 Decimal expansion of Van der Pauw's constant = Pi/log(2).

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A230476 a(n) = Sum_{i=1..n} d(8i+1) - Sum_{i=1..n} d(2i+1), where d(n) = A000005(n) is the number of divisors of n.

A097064 Expansion of (1 - 4x + 6x^2)/(1 - 2*x)^2.

A115252 Decimal expansion of -(Pilog((sqrt(2Pi)*Gamma(3/4))/Gamma(1/4)))/2.