A191898 -id:A191898 - cp's OEIS frontend

A007917 Version 1 of the "previous prime" function: largest prime <= n.

2, 3, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 29, 29, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 59, 59, 61, 61, 61, 61, 61, 61, 67, 67, 67, 67, 71, 71, 73, 73, 73, 73

Offset: 2

R. Muller

Version 2 of the "previous prime" function (see A151799) is "largest prime < n". This produces the same sequence of numerical values, except the offset (or indexing) starts at 3 instead of 2.
Maple's "prevprime" function uses version 2.
Also the largest prime dividing n! or lcm(1,...,n). - Labos Elemer, Jun 22 2000
Also largest prime among terms of (n+1)st row of Pascal's triangle. - Jud McCranie, Jan 17 2000
Also largest integer k such that A000203(k) <= n+1. - Benoit Cloitre, Mar 17 2002. - Corrected by Antti Karttunen, Nov 07 2017
Also largest prime factor of A061355(n) (denominator of Sum_{k=0..n} 1/k!). - Jonathan Sondow, Jan 09 2005
Also prime(pi(x)) where pi(x) is the prime counting function = number of primes <= x. - Cino Hilliard, May 03 2005
Also largest prime factor, occurring to the power p, in denominator of Sum_{k=1..n} 1/k^p, for any positive integer p. - M. F. Hasler, Nov 10 2006
For n > 10, these values are close to the most negative eigenvalues of A191898 (conjecture). - Mats Granvik, Nov 04 2011

K. Atanassov, On the 37th and the 38th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 83-85.
J. Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 202-204.

N. J. A. Sloane, Table of n, a(n) for n = 2..10000
Marc Deleglise and Jean-Louis Nicolas, Maximal product of primes whose sum is bounded, arXiv preprint arXiv:1207.0603 [math.NT], 2012. See Fig. 1. - From _N. J. A. Sloane_, Dec 17 2012
Hans Gunter, Puzzle 145. The Inferior Smarandache Prime Part and Superior Smarandache Prime Part functions; Solutions by Jean Marie Charrier, Teresinha DaCosta, Rene Blanch, Richard Kelley and Jim Howell. F. Smarandache, Only Problems, Not Solutions!.
Eric Weisstein's World of Mathematics, Previous Prime

Cf. A000040, A000203, A005145, A007918, A070801, A113523, A151799, A179278, A175851, A000720.

a007917 n = if a010051' n == 1 then n else a007917 (n-1)
-- Reinhard Zumkeller, May 01 2013, Jul 26 2012

[NthPrime(#PrimesUpTo(n)): n in [2..100]]; // Vincenzo Librandi, Nov 25 2015

Maple
```
A007917 := n-> prevprime(n+1);
```

Table[Prime[PrimePi[n]], {n, 2, 70}] (* Stefan Steinerberger, Apr 06 2006 *)
NextPrime[Range[3,80],-1] (* Harvey P. Dale, Jan 23 2011 *)

a=precprime \\ In older versions, use a(n)=precprime(n)
\\ Charles R Greathouse IV, Jun 15 2011

Equals A006530(A000142(n)). - Jonathan Sondow, Jan 09 2005
Equals A006530(A056040(n)). - Peter Luschny, Mar 04 2011
a(n) = A000040(A049084(A007918(n)) + 1 - A010051(n)). - Reinhard Zumkeller, Jul 26 2012
From Wesley Ivan Hurt, May 22 2013: (Start)
omega( Product_{i=2..n} a(i) ) = pi(n).
Omega( Product_{i=2..n} a(i) ) = n - 1. (End)
For n >= 2, a(A000203(n)) = A070801(n). - Antti Karttunen, Nov 07 2017
a(n) = n + 1 - Sum_{i=1..n} floor(pi(i)/pi(n)) = n + 1 - A175851(n). - Ridouane Oudra, Jun 24 2024
Conjecture: a(n) = floor(log(Sum_{k=2..n} exp(A000010(k)+1))). - Joseph M. Shunia, Aug 09 2024
a(n) = A000040(A000720(n)). - Ridouane Oudra, Oct 04 2024

Edited by N. J. A. Sloane, Apr 06 2008

A036987 Fredholm-Rueppel sequence.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Binary representation of the Kempner-Mahler number Sum_{k>=0} 1/2^(2^k) = A007404.
Also a(n) = A(n) mod 2 where A is any of A001700, A005573, A007854, A026641, A049027, A064063, A064088, A064090, A064092, A064325, A064327, A064329, A064331, A064613, A076026, A105523, A123273, A126694, A126930, A126931, A126982, A126983, A126987, A127016, A127053, A127358, A127360, A127361, A127363. - Philippe Deléham, May 26 2007
a(n) = (product of digits of n; n in binary notation) mod 2. This sequence is a transformation of the Thue-Morse sequence (A010060), since there exists a function f such that f(sum of digits of n) = (product of digits of n). - Ctibor O. Zizka, Feb 12 2008
a(n-1), n >= 1, the characteristic sequence for powers of 2, A000079, is the unique solution of the following formal product and formal power series identity: Product_{j>=1} (1 + a(j-1)*x^j) = 1 + Sum_{k>=1} x^k = 1/(1-x). The product is therefore Product_{l>=1} (1 + x^(2^l)). Proof. Compare coefficients of x^n and use the binary representation of n. Uniqueness follows from the recurrence relation given for the general case under A147542. - Wolfdieter Lang, Mar 05 2009
a(n) is also the number of orbits of length n for the map x -> 1-cx^2 on [-1,1] at the Feigenbaum critical value c=1.401155... . - Thomas Ward, Apr 08 2009
A054525 (Mobius transform) * A001511 = A036987 = A047999^(-1) * A001511 = the inverse of Sierpiński's gasket * the ruler sequence. - Gary W. Adamson, Oct 26 2009 [Of course this is only vaguely correct depending on how the fuzzy indexing in these formulas is made concrete. - R. J. Mathar, Jun 20 2014]
Characteristic function of A000225. - Reinhard Zumkeller, Mar 06 2012
Also parity of the Catalan numbers A000108. - Omar E. Pol, Jan 17 2012
For n >= 2, also the largest exponent k >= 0 such that n^k in binary notation does not contain both 0 and 1. Unlike for the decimal version of this sequence, A062518, where the terms are only conjectural, for this sequence the values of a(n) can be proved to be the characteristic function of A000225, as follows: n^k will contain both 0 and 1 unless n^k = 2^r-1 for some r. But this is a special case of Catalan's equation x^p = y^q-1, which was proved by Preda Mihăilescu to have no nontrivial solution except 2^3 = 3^2 - 1. - Christopher J. Smyth, Aug 22 2014
Image, under the coding a,b -> 1; c -> 0, of the fixed point, starting with a, of the morphism a -> ab, b -> cb, c -> cc. - Jeffrey Shallit, May 14 2016
Number of nonisomorphic Boolean algebras of order n+1. - Jianing Song, Jan 23 2020

			G.f. = 1 + x + x^3 + x^7 + x^15 + x^31 + x^63 + x^127 + x^255 + x^511 + ...
a(7) = 1 since 7 = 2^3 - 1, while a(10) = 0 since 10 is not of the form 2^k - 1 for any integer k.

Antti Karttunen, Table of n, a(n) for n = 0..65537
D. Bailey et al., On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518.
Paul Barry, Some observations on the Rueppel sequence and associated Hankel determinants, arXiv:2005.04066 [math.CO], 2020.
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Mats Granvik, Mathematica program for computing Fredholm Rueppel sequences
D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999.
Preda Mihăilescu, Primary Cyclotomic Units and a Proof of Catalan's Conjecture, J. Reine angew. Math. 572 (2004): 167-195. doi:10.1515/crll.2004.048. MR 2076124.
H. Niederreiter and M. Vielhaber, Tree Complexity and a Doubly Exponential Gap between Structured and Random Sequences, J. Complexity, 12 (1996), 187-198.
Apisit Pakapongpun and Thomas Ward, Functorial orbit counting, Journal of Integer Sequences, 12 (2009) Article 09.2.4. [From _Thomas Ward_, Apr 08 2009]
Eric Rowland and Reem Yassawi, Profinite automata, arXiv:1403.7659 [math.DS], 2014. See p. 8.
E. Sheppard, net.math post (1985)
Stephen Wolfram, [Page 1092] A New Kind of Science | Online.
Index entries for characteristic functions

Cf. A007404, A043545, A062518, A078885, A078585, A078886, A078887, A078888, A078889, A078890.
The first row of A073346. Occurs for first time in A073202 as row 6 (and again as row 8).
Congruent to any of the sequences A000108, A007460, A007461, A007463, A007464, A061922, A068068 reduced modulo 2. Characteristic function of A000225.
If interpreted with offset=1 instead of 0 (i.e., a(1)=1, a(2)=1, a(3)=0, a(4)=1, ...) then this is the characteristic function of 2^n (A000079) and as such occurs as the first row of A073265. Also, in that case the INVERT transform will produce A023359.
This is Guy Steele's sequence GS(1, 3), also GS(3, 1) (see A135416).
Cf. A054525, A047999. - Gary W. Adamson, Oct 26 2009
Cf. A043529, A127802.

a036987 n = ibp (n+1) where
   ibp 1 = 1
   ibp n = if r > 0 then 0 else ibp n' where (n',r) = divMod n 2
a036987_list = 1 : f [0,1] where f (x:y:xs) = y : f (x:xs ++ [x,x+y])
-- Same list generator function as for a091090_list, cf. A091090.
-- Reinhard Zumkeller, May 19 2015, Apr 13 2013, Mar 13 2013

A036987:= n-> `if`(2^ilog2(n+1) = n+1, 1, 0):
seq(A036987(n), n=0..128);

RealDigits[ N[ Sum[1/10^(2^n), {n, 0, Infinity}], 110]][[1]]
(* Recurrence: *)
t[n_, 1] = 1; t[1, k_] = 1;
t[n_, k_] := t[n, k] =
  If[n < k, If[n > 1 && k > 1, -Sum[t[k - i, n], {i, 1, n - 1}], 0],
   If[n > 1 && k > 1, Sum[t[n - i, k], {i, 1, k - 1}], 0]];
Table[t[n, k], {k, n, n}, {n, 104}]
(* Mats Granvik, Jun 03 2011 *)
mb2d[n_]:=1 - Module[{n2 = IntegerDigits[n, 2]}, Max[n2] - Min[n2]]; Array[mb2d, 120, 0] (* Vincenzo Librandi, Jul 19 2019 *)
Table[PadRight[{1},2^k,0],{k,0,7}]//Flatten (* Harvey P. Dale, Apr 23 2022 *)

{a(n) =( n++) == 2^valuation(n, 2)}; /* Michael Somos, Aug 25 2003 */

a(n) = !bitand(n, n+1); \\ Ruud H.G. van Tol, Apr 05 2023

from sympy import catalan
def a(n): return catalan(n)%2 # Indranil Ghosh, May 25 2017

def A036987(n): return int(not(n&(n+1))) # Chai Wah Wu, Jul 06 2022

1 followed by a string of 2^k - 1 0's. Also a(n)=1 iff n = 2^m - 1.
a(n) = a(floor(n/2)) * (n mod 2) for n>0 with a(0)=1. - Reinhard Zumkeller, Aug 02 2002 [Corrected by Mikhail Kurkov, Jul 16 2019]
Sum_{n>=0} 1/10^(2^n) = 0.110100010000000100000000000000010...
1 if n=0, floor(log_2(n+1)) - floor(log_2(n)) otherwise. G.f.: (1/x) * Sum_{k>=0} x^(2^k) = Sum_{k>=0} x^(2^k-1). - Ralf Stephan, Apr 28 2003
a(n) = 1 - A043545(n). - Michael Somos, Aug 25 2003
a(n) = -Sum_{d|n+1} mu(2*d). - Benoit Cloitre, Oct 24 2003
Dirichlet g.f. for right-shifted sequence: 2^(-s)/(1-2^(-s)).
a(n) = A000108(n) mod 2 = A001405(n) mod 2. - Paul Barry, Nov 22 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*Sum_{j=0..k} binomial(k, 2^j-1). - Paul Barry, Jun 01 2006
A000523(n+1) = Sum_{k=1..n} a(k). - Mitch Harris, Jul 22 2011
a(n) = A209229(n+1). - Reinhard Zumkeller, Mar 07 2012
a(n) = Sum_{k=1..n} A191898(n,k)*cos(Pi*(n-1)*(k-1))/n; (conjecture). - Mats Granvik, Mar 04 2013
a(n) = A000035(A000108(n)). - Omar E. Pol, Aug 06 2013
a(n) = 1 iff n=2^k-1 for some k, 0 otherwise. - M. F. Hasler, Jun 20 2014
a(n) = ceiling(log_2(n+2)) - ceiling(log_2(n+1)). - Gionata Neri, Sep 06 2015
From John M. Campbell, Jul 21 2016: (Start)
a(n) = (A000168(n-1) mod 2).
a(n) = (A000531(n+1) mod 2).
a(n) = (A000699(n+1) mod 2).
a(n) = (A000891(n) mod 2).
a(n) = (A000913(n-1) mod 2), for n>1.
a(n) = (A000917(n-1) mod 2), for n>0.
a(n) = (A001142(n) mod 2).
a(n) = (A001246(n) mod 2).
a(n) = (A001246(n) mod 4).
a(n) = (A002057(n-2) mod 2), for n>1.
a(n) = (A002430(n+1) mod 2). (End)
a(n) = 2 - A043529(n). - Antti Karttunen, Nov 19 2017
a(n) = floor(1+log(n+1)/log(2)) - floor(log(2n+1)/log(2)). - Adriano Caroli, Sep 22 2019
This is also the decimal expansion of -Sum_{k>=1} mu(2*k)/(10^k - 1), where mu is the Möbius function (A008683). - Amiram Eldar, Jul 12 2020

Edited by M. F. Hasler, Jun 20 2014

A086463 Decimal expansion of Pi^2/18.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 21 2003

The sequence of repeating coefficients [1,-1,-2,-1,1,2] in the sum in the formula section, is equal to the 6th column in A191898. - Mats Granvik, Mar 19 2012

			0.548311355616075478824138388882008396406316633735...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.1, p. 20.
A. Holroyd, Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation, Prob. Th. and Related Fields 125, 195-224, 2003.

J. M. Borwein and R. Girgensohn, Evaluations of binomial series, Aequat. Math. 70 (2005) 25-36.
A. Holroyd, Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation, arXiv:math/0206132 [math.PR], 2002.
Ji-Cai Liu, On two congruences involving Franel numbers, arXiv:2002.03650 [math.NT], 2020.
Courtney Moen, Infinite series with binomial coefficients, Math. Mag. 64 (1) (1991) 53-55.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27.
Eric Weisstein's World of Mathematics, Bootstrap Percolation.
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Index entries for transcendental numbers.

Cf. A007814, A073010, A073016, A086464, A112093, A112094, A130549, A130550.

RealDigits[Pi^2/18,10,120][[1]] (* Harvey P. Dale, Aug 14 2011 *)

Pi^2/18 \\ Charles R Greathouse IV, Mar 20 2012

Sum[1/n^2/Binomial[2n,n], {n,Infinity}].
Pi^2/18 = A013661/3 = Sum[1/(i+0)^2 - 1/(i+1)^2 - 2/(i+2)^2 - 1/(i+3)^2 + 1/(i+4)^2 + 2/(i+5)^2, {i =1, 7, 13, 19, 25,.. infinity, stride of 6}]. - Mats Granvik, Mar 19 2012
Equals Sum_{k>=1} (H(k) - 2*H(2k))/((-3^k)*k). See Liu. - Michel Marcus, Feb 11 2020
Equals Sum_{k>=1} A007814(k)/k^2. - Amiram Eldar, Jul 13 2020
Equals (2/9) * Sum_{k>=0} (-1)^k*(7*k+5)*k!^3/((2*k+1)*(3*k+2)!) [Gosper 1974] - R. J. Mathar, Feb 07 2024
Continued fraction expansion: 1/(2 - 2/(13 - 48/(34 - 270/(65 - ... - 2*(2*n - 1)*n^3/(5*n^2 + 6*n + 2 - ... ))))). See A130549. - Peter Bala, Feb 16 2024

A171462 Number of hands a bartender needs to have in order to win at the blind bartender's problem with n glasses in a cycle.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 6, 4, 6, 8, 10, 8, 12, 12, 12, 8, 16, 12, 18, 16, 18, 20, 22, 16, 20, 24, 18, 24, 28, 24, 30, 16, 30, 32, 30, 24, 36, 36, 36, 32, 40, 36, 42, 40, 36, 44, 46, 32, 42, 40, 48, 48, 52, 36, 50, 48, 54, 56, 58, 48, 60, 60, 54, 32, 60, 60, 66, 64, 66, 60, 70, 48, 72

Offset: 1

Richard Ehrenborg, Dec 09 2009

For n greater than 1, the n-th entry is given by n*(1-1/p) where p is largest prime dividing n.

			a(4) = 2 since in the classical problem with 4 glasses on a tray, the blind bartender needs 2 hands.

W. T. Laaser and L. Ramshaw, Probing the Rotating Table, The Mathematical Gardner (edited by David A. Klarner), Prindle, Weber & Schmidt, Boston, Massachusetts, 1981, pages 285-307.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. Ehrenborg and C. Skinner, The blind bartender's problem, Journal of Combinatorial Theory, Series A 70 (1995), 249-266.
T. Lewis and S. Williard, The rotating table, Mathematics Magazine, vol. 53, no. 3 (May 1980) pages 174-179.

Cf. A006530, A052126, A060681.

a171462 n = div n p * (p - 1) where p = a006530 n
-- Reinhard Zumkeller, Apr 06 2015

{0}~Join~Array[# (1 - 1/FactorInteger[#][[-1, 1]]) &, 72, 2] (* Michael De Vlieger, Jul 08 2020 *)

a(n) = {if (n == 1, return (0)); f = factor(n); p = f[#f~,1]; return (n * (p - 1)/p);} \\ Michel Marcus, Jun 09 2013

from sympy import primefactors
def a(n): return 0 if n == 1 else n - n//(primefactors(n)[-1])
print([a(n) for n in range(1, 74)]) # Michael S. Branicky, Apr 19 2021

Conjecture: n > 1: k=1..n: a(n) = -n*min(A191898(n, k)/k). Verified up to n=10000. - Mats Granvik, Apr 19 2021

a(n) = n - A052126(n) = n - n/A006530(n). - Antti Karttunen, Jan 03 2024

A182448 Decimal expansion of Pi^2/15.

Original entry on oeis.org

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

Offset: 0

Mats Granvik, Apr 29 2012

			0.65797362673929...

Thomas L. Curtright, Bernoulli Partitions, arXiv:2502.09633 [math.CO], 2025. See p. 3. Mentions this sequence.
László Tóth, Linear combinations of Dirichlet series associated with the Thue-Morse sequence, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 22 (2022), #A98; arXiv preprint, arXiv:2211.13570 [math.NT], 2022.
Index entries for transcendental numbers.

Cf. A000796, A008836, A010060, A013661, A013662, A086463, A191898, A249103.

Cf. A347328, A347329, A347330, A347331.

RealDigits[N[Sum[1/(n + 0)^2 - 1/(n + 1)^2 + 1/(n + 2)^2 - 1/(n + 3)^2 - 4/(n + 4)^2 - 1/(n + 5)^2 + 1/(n + 6)^2 - 1/(n + 7)^2 + 1/(n + 8)^2 + 4/(n + 9)^2, {n, 1, Infinity, 10}], 90]][[1]]
RealDigits[N[Sum[LiouvilleLambda[n]/n^2, {n, 1, Infinity}], 90]][[1]]
RealDigits[Pi^2/15,10,120][[1]] (* Harvey P. Dale, May 28 2017 *)

PARI
```
Pi^2/15 \\ Michel Marcus, Oct 21 2014
```

See Mathematica code.
Equals Gamma(4)*zeta(4)/Pi^2 = zeta(4)/zeta(2) = A013662/A013661 = Product_{p prime} (p^2/(p^2+1)). - Stanislav Sykora, Oct 21 2014
Equals (1/10) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/3)^2 - 1/(n + 2/3)^2 ). - Peter Bala, Oct 31 2019
Equals Sum_{k>=1} A008836(k)/k^2. - Amiram Eldar, Jun 23 2020
Equals (1/10) * Sum_{k>=1} (5*t(k-1) + 3*t(k))/k^2, where t(k) = A010060(k) (Tóth, 2022). - Amiram Eldar, Feb 04 2024
Equals 3/5 + (1/5) * Sum_{n>=1} 1/(n^2*(n+1)^2). - Davide Rotondo, May 28 2025
Equals 1/A082020 = A164102/30 = A195055/5. - Hugo Pfoertner, May 28 2025

Offset corrected and more terms added by Rick L. Shepherd, Jan 08 2014

A188934 Decimal expansion of (1+sqrt(17))/4.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 13 2011

Decimal expansion of the length/width ratio of a (1/2)-extension rectangle.  See A188640 for definitions of shape and r-extension rectangle.
A (1/2)-extension rectangle matches the continued fraction [1,3,1,1,3,1,1,3,1,1,3,...] for the shape L/W=(1+sqrt(17))/4.  This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...].  Specifically, for the (1/2)-extension rectangle, 1 square is removed first, then 3 squares, then 1 square, then 1 square,..., so that the original rectangle of shape (1+sqrt(17))/4 is partitioned into an infinite collection of squares.
Conjecture: This number is an eigenvalue to infinitely many n*n submatrices of A191898, starting in the upper left corner, divided by the row index. For the first few characteristic polynomials see A260237 and A260238. - Mats Granvik, May 12 2016.

			1.2807764064044151374553524639935192562...

Cf. A188640, A188935.

Essentially the same as A188485.

r = 1/2; t = (r + (4 + r^2)^(1/2))/2; RealDigits[ N[ FullSimplify@ t, 111]][[1]]
(* for the continued fraction *) ContinuedFraction[t, 120]
RealDigits[(1 + Sqrt@ 17)/4, 10, 111][[1]] (* Or *)
RealDigits[Exp@ ArcSinh[1/4], 10, 111][[1]] (* Robert G. Wilson v, Aug 17 2011 *)

(sqrt(17)+1)/4 \\ Charles R Greathouse IV, May 12 2016

A199514 Numerators of zeros to a symmetric polynomial. Numerators of mu(n)^2*(n/(n - phi(n))).

Original entry on oeis.org

2, 3, 0, 5, 3, 7, 0, 0, 5, 11, 0, 13, 7, 15, 0, 17, 0, 19, 0, 7, 11, 23, 0, 0, 13, 0, 0, 29, 15, 31, 0, 33, 17, 35, 0, 37, 19, 13, 0, 41, 7, 43, 0, 0, 23, 47, 0, 0, 0, 51, 0, 53, 0, 11, 0, 19, 29, 59, 0, 61, 31, 0, 0, 65, 33, 67, 0, 69, 35, 71, 0, 73, 37, 0, 0, 77, 13, 79, 0, 0, 41, 83, 0, 85, 43, 87, 0, 89, 0

Offset: 2

Mats Granvik, Nov 07 2011

The polynomials are defined as the determinant of a symmetric matrix with the following definition:
T(n, 1) = 1, T(1, k) = 1, T(n, k) = If n < k, x - Sum_(i = 1)^(i = n - 1) of T(k - i, n), otherwise x - Sum_(i = 1)^(i = k - 1) of T(k - i, n).
Eric Naslund on Mathematics Stack Exchange kindly gave the description in terms of arithmetic functions. The sequence of fractions A199514/A199515 is an integer only for prime numbers. As the matrix gets bigger there are fractions as zeros that are greater than small prime numbers.

			The 7 X 7 symmetric matrix is:
  1......1......1......1......1......1......1
  1...-1+x......1...-1+x......1...-1+x......1
  1......1...-2+x......1......1...-2+x......1
  1...-1+x......1.....-1......1...-1+x......1
  1......1......1......1...-4+x......1......1
  1...-1+x...-2+x...-1+x......1...2-2x......1
  1......1......1......1......1......1...-6+x
Taking the determinant of the matrix above gives the polynomial: -1260 x + 2322 x^2 - 1594 x^3 + 506 x^4 - 74 x^5 + 4 x^6
The polynomials for the first seven matrices are:
  1,
  -2 + x,
  6 - 5 x + x^2,
  -6 x + 5 x^2 - x^3,
  30 x - 31 x^2 + 10 x^3 - x^4,
  180 x - 306 x^2 + 184 x^3 - 46 x^4 + 4 x^5,
  -1260 x + 2322 x^2 - 1594 x^3 + 506 x^4 - 74 x^5 + 4 x^6,
  ...
and their zeros respectively are:
  {}
  2
  2, 3
  2, 3, 0
  2, 3, 0, 5
  2, 3, 0, 5, 3/2
  2, 3, 0, 5, 3/2, 7
  ...

Antti Karttunen, Table of n, a(n) for n = 2..65537
Mats Granvik, Are the primes found as a subset in this sequence?, Mathematics Stack Exchange.

Cf. A000010, A008683, A008966, A051953, A191898. Denominators: A199515.

Table[Numerator[MoebiusMu[n]^2*(n/(n - EulerPhi[n]))], {n, 2, 90}]
(* or *)
Clear[nn, t, n, k, M, x];
nn = 90;
a = Range[nn]*0;
Do[
t[n_, 1] = 1;
t[1, k_] = 1;
t[n_, k_] :=
  t[n, k] =
   If[n < k,
    If[And[n > 1, k > 1], x - Sum[t[k - i, n], {i, 1, n - 1}], 0],
    If[And[n > 1, k > 1], x - Sum[t[n - i, k], {i, 1, k - 1}], 0]];
M = Table[Table[t[n, k], {k, 1, i}], {n, 1, i}];
a[[i]] = x /. Solve[Det[M] == 0, x], {i, 1, nn}];
a[[1]] = {};
b = Differences[Table[Total[a[[i]]], {i, 1, nn}]];
Numerator[b]

A199514(n) = numerator(issquarefree(n)*(n/(n-eulerphi(n)))); \\ Antti Karttunen, Sep 07 2018

a(n)/A199515(n) = A008683(n)^2*(n/(n - A000010(n))), n > 1.

a(n) = numerator of A008966(n)*(n/A051953(n)). - Antti Karttunen, Sep 07 2018

A230284 Denominators to Dirichlet inverse of Euler totient based version of series expansion for x/LambertW(x).

Original entry on oeis.org

1, 1, 2, 3, 8, 15, 144, 35, 5760, 315, 5600, 693, 43545600, 1001, 6706022400, 6435, 14014, 109395, 376610217984000, 46189, 128047474114560000, 323323, 2540395, 2028117, 26976017466662584320000, 96577, 3241475864250624, 35102025, 2126818694000, 5386025

Offset: 1

Mats Granvik, Oct 15 2013

G. C. Greubel, Table of n, a(n) for n = 1..385

Cf. A191898, A177885, A230283 (numerators).

Similar to but strictly different from A264235.

Clear[nn, n, k, s, x]; nn = 22; Denominator[CoefficientList[1 + Integrate[1 + Expand[Sum[Exp[Limit[Zeta[s]*Sum[(If[n == 1, 0, Table[DivisorSum[m, # MoebiusMu[#] &], {m, nn}][[GCD[n, k]]]])/(k)^(s - 1), {k, 1, n}], s -> 1]]*(-x)^n, {n, 1, nn}]], x], x]]

A230283 Numerators to Dirichlet inverse of Euler totient based version of series expansion for x/LambertW(x).

Original entry on oeis.org

1, 1, -1, 2, -9, 8, -625, 2, -117649, 128, -6561, 8, -25937424601, 18, -23298085122481, 16, -9, 32768, -48661191875666868481, 400, -104127350297911241532841, 648, -81, 256, -907846434775996175406740561329, 490, -59604644775390625, 1024, -2541865828329, 1296

Offset: 1

Mats Granvik, Oct 15 2013

The coefficients of the series expansion of x/Lambert(x) expanded at 0 can be seen as exponentiated numerators in convergents of zeta function limits of truncated Dirichlet series for logarithms. Those numerators are defined by simple recurrences. Letting those recurrences run in cross directions to each other, one get the Dirichlet inverse of the Euler totient in a greatest common divisor matrix, and the von Mangoldt function as convergents of Dirichlet series. Since x/LambertW(x) is good at approximately describing the nontrivial Riemann zeta zeros and since the Riemann zeta zeros are the frequencies that build up the von Mangoldt function, this prime number or von Mangoldt function version of the x/LambertW(x) is motivated.

G. C. Greubel, Table of n, a(n) for n = 1..385

Cf. A191898, A177885, A230284 (denominators).

Clear[nn, n, k, s, x]; nn = 22; Numerator[CoefficientList[1 + Integrate[1 + Expand[Sum[Exp[Limit[Zeta[s]*Sum[(If[n == 1, 0, Table[DivisorSum[m, # MoebiusMu[#] &], {m, nn}][[GCD[n, k]]]])/(k)^(s - 1), {k, 1, n}], s -> 1]]*(-x)^n, {n, 1, nn}]], x], x]]

A260237 Numerators of the characteristic polynomials of the von Mangoldt function matrix.

Original entry on oeis.org

0, 1, -1, -1, -1, 1, 1, 11, -1, -1, 0, -3, -9, 5, 1, 0, 3, 81, 7, -73, -1, 0, 3, 73, -1261, -1183, 53, 1, 0, -3, -1231, 5251, 8989, 1451, -731, -1, 0, 0, 7, 397, -12491, -19877, -15047, 1567, 1, 0, 0, 0, -7, -1483, 50111, 69761, 45959, -5261, -1

Offset: 1

Mats Granvik, Jul 20 2015

The von Mangoldt function matrix is the symmetric matrix A191898 divided by either the row index or the column index.

Every eigenvalue of a smaller von Mangoldt function matrix appears to be common to infinitely many larger von Mangoldt matrices. The eigenvalues of smaller von Mangoldt function matrices also repeat within larger von Mangoldt function matrices.

			The first term set to zero is not part of the characteristic polynomials. It is only there for the formatting of the table.
{
{0},
{1, -1},
{-1, -1, 1},
{1, 11, -1, -1},
{0, -3, -9, 5, 1},
{0, 3, 81, 7, -73, -1},
{0, 3, 73, -1261, -1183, 53, 1},
{0, -3, -1231, 5251, 8989, 1451, -731, -1},
{0, 0, 7, 397, -12491, -19877, -15047, 1567, 1},
{0, 0, 0, -7, -1483, 50111, 69761, 45959, -5261,-1}
}

Denominators in A260238.

Clear[nnn, nn, T, n, k, x]; nnn = 9; T[n_, k_] := T[n, k] = Which[n < 1 || k < 1, 0, n == 1 || k == 1, 1, k > n, T[k, n], n > k, T[k, Mod[n, k, 1]], True, -Sum[T[n, i], {i, n - 1}]]; b = Table[CoefficientList[CharacteristicPolynomial[Table[Table[T[n, k]/n, {k, 1, nn}], {n, 1, nn}], x], x], {nn, 1, nnn}]; Flatten[{0,Numerator[b]}]

cp's OEIS Frontend

A007917 Version 1 of the "previous prime" function: largest prime <= n.

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A036987 Fredholm-Rueppel sequence.

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A086463 Decimal expansion of Pi^2/18.

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A171462 Number of hands a bartender needs to have in order to win at the blind bartender's problem with n glasses in a cycle.

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A182448 Decimal expansion of Pi^2/15.

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A188934 Decimal expansion of (1+sqrt(17))/4.

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