A016627 -id:A016627 - cp's OEIS frontend

A319232 Decimal expansion of Sum_{p = prime} 1/(p*log p)^2.

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

Offset: 0

R. J. Mathar, Sep 14 2018

Obtained by expanding the formalism of arXiv:0811.4739 to double integrals over the Riemann zeta function.

			1/A016627^2 + 1/A016650^2 + 1/8.047189^2 + ... = 0.637056184074676....

R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 (2008-2009).

Cf. A137245, A115563, A221711, A319231.

digits = 106; precision = digits + 10;
tmax = 500; (* integrand considered negligible beyond tmax *)
kmax = 300; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[(t - 2k) Log[Zeta[t]], {t, 2k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20, AccuracyGoal -> precision];
f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^3)*InLogZeta[k]]];
s = 0;
Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
RealDigits[s][[1]][[1 ;; digits]] (* Jean-François Alcover, Jun 21 2022, after Vaclav Kotesovec *)

default(realprecision, 200); s=0; for(k=1, 300, s = s + moebius(k)/k^3 * intnum(x=2*k,[[1], 1], (x-2*k)*log(zeta(x))); print(s)); \\ Vaclav Kotesovec, Jun 12 2022

More terms from Vaclav Kotesovec, Jun 12 2022

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)

A097448 If n is square, replace it with sqrt(n).

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 12, 13, 14, 15, 4, 17, 18, 19, 20, 21, 22, 23, 24, 5, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 8, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 0

Cino Hilliard, Aug 23 2004

			Among the first five integers 0, 1, 2, 3, and 4 the squares are 0, 1, and 4. Thus the first five terms in the sequence are 0, 1, 2, 3, and 2.

Cf. A016627, A072691, A097449.

Table[If[IntegerQ[Sqrt[n]],Sqrt[n],n],{n,0,100}] (* Harvey P. Dale, Jul 09 2017 *)

g(n) = for(x=0,n,if(issquare(x),y=sqrt(x),y=x);print1(floor(y)","))

for(n=0,74,print1(if(issquare(n,&m),m,n)", ")) \\ Zak Seidov, Feb 21 2013

Sum_{n>=1} (-1)^(n+1)/n = 2*log(2) - Pi^2/12 = A016627 - A072691. - Amiram Eldar, Jul 07 2024

A191907 Square array read by antidiagonals up: T(n,k) = -(n-1) if n divides k, else 1.

Original entry on oeis.org

0, 1, 0, 1, -1, 0, 1, 1, 1, 0, 1, 1, -2, -1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, -3, 1, -1, 0, 1, 1, 1, 1, 1, -2, 1, 0, 1, 1, 1, 1, -4, 1, 1, -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, -5, 1, -3, -2, -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, -6, 1, 1, 1, 1, -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, -4, 1, -2, 1, 0, 1, 1, 1, 1, 1, 1, 1, -7, 1, 1, 1, -3, 1, -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 1

Mats Granvik, Jun 19 2011

Apart from the top row, the same as A177121.

Sum_{k>=1} T(n,k)/k = log(n); this has been pointed out by Jaume Oliver Lafont in A061347 and A002162.

			Table starts:
0..0..0..0..0..0..0..0..0...
1.-1..1.-1..1.-1..1.-1..1...
1..1.-2..1..1.-2..1..1.-2...
1..1..1.-3..1..1..1.-3..1...
1..1..1..1.-4..1..1..1..1...
1..1..1..1..1.-5..1..1..1...
1..1..1..1..1..1.-6..1..1...
1..1..1..1..1..1..1.-7..1...
1..1..1..1..1..1..1..1.-8...

Cf. A002162, A002391, A016627, A191904.

Clear[t, n, k];
nn = 30;
t[n_, k_] := t[n, k] = If[Mod[n, k] == 0, -(k - 1), 1]
MatrixForm[Transpose[Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]]]

N=20; M=matrix(N,N,n,k, if(n%k==0,1-k,1))~

If n divides k then T(n,k) = -(n-1) else 1.

A244641 Decimal expansion of the sum of the reciprocals of the pentagonal numbers (A000326).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

			1.482037501770111223591657453125421381658405425375507779634198065524359698529473...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Hongwei Chen and G. C. Greubel, Sum of the Reciprocals of Polygonal Numbers (Solved), SIAM Problems and solutions.
Hongwei Chen and G. C. Greubel, Siam, Problems and Solutions, problem 07-003 and the solution

Cf. A000326, A016650, A093602, A294514.

Decimal expansion of the sum of the reciprocals of the m-gonal numbers: A000038 (m=3), A013661 (m=4), this sequence (m=5), A016627 (m=6), A244639 (m=7), A244645 (m=8), A244646 (m=9), A244647 (m=10), A244648 (m=11), A244649 (m=12), A275792 (m=14).

SetDefaultRealField(RealField(139)); R:= RealField(); 3*Log(3)-Pi(R)*Sqrt(3)/3; // G. C. Greubel, Mar 24 2024

RealDigits[Sum[2/(3*n^2-n), {n,1,Infinity}], 10, 111][[1]]
RealDigits[3*Log[3] - Pi*Sqrt[3]/3, 10, 140][[1]] (* G. C. Greubel, Mar 24 2024 *)

numerical_approx(3*log(3)-pi*sqrt(3)/3, digits=139) # G. C. Greubel, Mar 24 2024

Sum_{n>=1} 2/(3*n^2 - n).
Equals 3*log(3) - Pi*sqrt(3)/3 = A016650 - A093602. - Michel Marcus, Jul 03 2014
Equals 2*A294514. - Hugo Pfoertner, Apr 24 2025

A274540 Decimal expansion of exp(sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Johannes W. Meijer, Jun 27 2016

Define P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k) for n >= 1, and P(0) = 1, with x(q) = C1 and x(n) = 1 for all other n. We find that C2 = lim_{n -> infinity} P(n) = exp((C1-1)/q).
The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.
Some transform pairs: C1 = A002162 (log(2)) and C2 = A135002 (2/exp(1)); C1 = A016627 (log(4)) and C2 = A135004 (4/exp(1)); C1 = A001113 (exp(1)) and C2 = A234473 (exp(exp(1)-1)).
From Peter Bala, Oct 23 2019: (Start)
The constant is irrational: Henry Cohn gives the following proof in Todd and Vishals Blog - "By the way, here's my favorite application of the tanh continued fraction: exp(sqrt(2)) is irrational.
Consider sqrt(2)*(exp(sqrt(2))-1)/(exp(sqrt(2))+1). If exp(sqrt(2)) were rational, or even in Q(sqrt(2)), then this expression would be in Q(sqrt(2)). However, it is sqrt(2)*tanh(1/sqrt(2)), and the tanh continued fraction shows that this equals [0,1,6,5,14,9,22,13,...]. If it were in Q(sqrt(2)), it would have a periodic simple continued fraction expansion, but it doesn't." (End)

			c = 4.113250378782927517173581815140304502401663943151...

G. C. Greubel, Table of n, a(n) for n = 1..10000
The Dev Team and Simon Plouffe, The Inverse Symbolic Calculator (ISC).
Todd Trimble and Vishal Lama, Continued fraction for e, Todd and Vishal’s blog 2008/08/04
Index entries for transcendental numbers

Cf. A274541, A274542, A036039, A135002, A135004, A234473.

Cf. A014176, A002193, A010503, A131594, A020765, A010466, A020775.

Digits := 80: evalf(exp(sqrt(2))); # End program 1.
P := proc(n) : if n=0 then 1 else P(n) := expand((1/n)*(add(x(n-k)*P(k), k=0..n-1))) fi; end: x := proc(n): if n=1 then (1 + sqrt(2)) else 1 fi: end: Digits := 49; evalf(P(120)); # End program 2.

First@ RealDigits@ N[Exp[Sqrt@ 2], 80] (* Michael De Vlieger, Jun 27 2016 *)

my(x=exp(sqrt(2))); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", ")) \\ Felix Fröhlich, Jun 27 2016

c = exp(sqrt(2)).

c = lim_{n -> infinity} P(n) with P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k) for n >= 1, and P(0) = 1, with x(1) = (1 + sqrt(2)), the silver mean A014176, and x(n) = 1 for all other n.

More terms from Jon E. Schoenfield, Mar 15 2018

A379324 Decimal expansion of log(6)^(log(5)^(log(4)^(log(3)^log(2)))).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 20 2024

This is an approximation to Pi accurate to 5 digits.

			3.1415773871691905336574449813486768105453105619...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pi Approximations.
Index entries for sequences related to the number Pi

Cf. A000796, A002162, A002391, A016627, A016628, A016629.

Cf. A221185, A379276, A379321, A379322, A379323.

First[RealDigits[Log[6]^Log[5]^Log[4]^Log[3]^Log[2], 10, 100]]

Equals A016629^(A016628^(A016627^(A002391^A002162))).

A275792 Decimal expansion of the sum of the reciprocals of the tetradecagonal numbers A051866.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Sep 12 2016

See Table 1 of the Downey et al. link.
From Wolfdieter Lang, Nov 09 2017: (Start)
The general formula for S_{2*(k+1)} = Sum_{n>=0} 1/((n+1)*(k*n+1)) given in the Downey et al. link is a special case of the simpler formula for V(m,r) = Sum_{n>=0} 1/((n+1)*(m*n + r)), r = 1,2, ... ,m -1. V(m,r) = (m/(m-r))*v_m(r) in Koecher's notation. For this formula for m*v_m(r) see a comment in A294512.
The special case is m = k and r = 1, leading to S_{2*(k+1)} = V(k,1) = (log(k) + (Pi/2)*cot(Pi/k) - Sum_{j=1..k-1} cos(2*Pi*j/k)*log(2*sin(Pi*j/k)))/(k-1), for k >= 2.
S_14, for k=6, is then given by the formula below (also obtained from the more complicated formula of Downey et al.).
The partial sums are given in A294834/A294835.
(End)

			1.150982368094676386363689896952675058309...

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193. See (6/5)*v_6(1) on p. 192.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.

Cf. A013661, A016627, A051866, A244641, A244645, A294512, A294834, A294835.

SetDefaultRealField(RealField(139)); R:= RealField(); (4*Log(2) + 3*Log(3) + Pi(R)*Sqrt(3))/10; // G. C. Greubel, Mar 25 2024

RealDigits[2*Log[2]/5 + 3*Log[3]/10 + Sqrt[3]*Pi/10, 10, 120][[1]] (* Amiram Eldar, Jun 25 2023 *)

2*log(2)/5 + 3*log(3)/10 + sqrt(3)*Pi/10 \\ Michel Marcus, Nov 09 2017

numerical_approx((4*log(2) + 3*log(3) + pi*sqrt(3))/10, digits=139) # G. C. Greubel, Mar 25 2024

Sum_{n >= 1} 1/(n*(6*n - 5)) = 2*log(2)/5 + 3*log(3)/10 + sqrt(3)*Pi/10.

A016732 Continued fraction for log(4).

Original entry on oeis.org

1, 2, 1, 1, 2, 3, 7, 6, 4, 1, 1, 21, 3, 1, 3, 3, 1, 1, 2, 1, 1, 2, 6, 1, 1, 3, 9, 3, 3, 1, 2, 1, 1, 1, 3, 1, 10, 7, 2, 5, 2, 2, 4, 9, 7, 1, 1, 1, 13, 1, 14, 1, 1, 1, 1, 2, 6, 1, 1, 1, 2, 2, 9, 1, 1, 3, 3, 1, 34, 1, 1, 5, 16, 3, 3, 1, 1, 9, 2, 1, 3, 2, 2, 1, 1, 1

Offset: 0

N. J. A. Sloane

			1.386294361119890618834464242... = 1 + 1/(2 + 1/(1 + 1/(1 + 1/(2 + ...)))). - _Harry J. Smith_, May 16 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
G. Xiao, Contfrac

Cf. A016627 (decimal expansion).

ContinuedFraction(2*Log(2)); // G. C. Greubel, Sep 15 2018

ContinuedFraction[2*Log[2], 100] (* G. C. Greubel, Sep 15 2018 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(log(4)); for (n=1, 20000, write("b016732.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 16 2009

Offset changed by Andrew Howroyd, Jul 10 2024

A097449 If n is a cube, replace it with the cube root of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 3, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 0

Cino Hilliard, Aug 23 2004

			The 9th integer is 8 so a(9) = 8^(1/3) = 2.

Cf. A016627, A097448, A197070.

rcr[n_]:=Module[{crn=Power[n, (3)^-1]},If[IntegerQ[crn],crn,n]]; Array[ rcr,80,0] (* Harvey P. Dale, Jan 28 2012 *)

iscube(n) = { local(r); r = n^(1/3); if(floor(r+.5)^3== n,1,0) }
replcube(n) = { for(x=0,n, if(iscube(x),y=x^(1/3),y=x); print1(floor(y)",")) }

a(n)=ispower(n,3,&n);n \\ Charles R Greathouse IV, Oct 27 2011

Sum_{n>=1} (-1)^(n+1)/n = 2*log(2) - 3*zeta(3)/4 = A016627 - A197070. - Amiram Eldar, Jul 07 2024

Corrected by T. D. Noe, Oct 25 2006

cp's OEIS Frontend

A319232 Decimal expansion of Sum_{p = prime} 1/(p*log p)^2.

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

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A097448 If n is square, replace it with sqrt(n).

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A191907 Square array read by antidiagonals up: T(n,k) = -(n-1) if n divides k, else 1.

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A244641 Decimal expansion of the sum of the reciprocals of the pentagonal numbers (A000326).

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A274540 Decimal expansion of exp(sqrt(2)).

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A379324 Decimal expansion of log(6)^(log(5)^(log(4)^(log(3)^log(2)))).

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