A064169 -id:A064169 - cp's OEIS frontend

A074791 Numbers k such that k does not divide the denominator of the k-th harmonic number.

6, 18, 20, 21, 33, 42, 54, 63, 66, 77, 100, 110, 120, 156, 162, 189, 198, 272, 294, 336, 342, 363, 377, 435, 486, 500, 506, 559, 567, 594, 600, 610, 629, 685, 703, 812, 847, 880, 924, 930, 957, 1067, 1166, 1210, 1243, 1247, 1287, 1320, 1332, 1458, 1590, 1640

Offset: 1

Benoit Cloitre, Sep 07 2002

nonn

k such that A064169(k) is different from A027612(k).
Also k such that A096617(k) is different from A001008(k). - Alexander Adamchuk, Jun 26 2006
This sequence contains A036689(k) for all k > 1. - Wouter van Doorn, Nov 06 2024

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A027612, A064169.

Cf. A096617, A001008.

Select[ Range[1700], Mod[ Denominator[ HarmonicNumber[ # ]], # ] != 0 &] (* Robert G. Wilson v, Sep 28 2005 *)
seq = {}; s = 0; Do[s += 1/n; If[! Divisible[Denominator[s], n], AppendTo[seq, n]], {n, 1, 2000}]; seq (* Amiram Eldar, Dec 01 2020 *)

Is a(n) asymptotic to c*n^2 0.5

a(n) < 2*n^2*log(n)^2 for all n > 2. This follows from the fact that for all k > 1 there exists an n such that A036689(k) is equal to A074791(n). - Wouter van Doorn, Nov 06 2024

Better description and more terms from Robert G. Wilson v, Sep 28 2005

A330719 a(n) = denominator(Sum_{k=1..n} (2^(k-1) - 1)/k).

Original entry on oeis.org

1, 2, 2, 4, 4, 12, 12, 24, 8, 40, 40, 120, 120, 840, 840, 1680, 1680, 5040, 5040, 5040, 5040, 55440, 55440, 55440, 11088, 144144, 48048, 48048, 48048, 80080, 80080, 160160, 160160, 2722720, 544544, 4900896, 4900896, 93117024, 93117024, 465585120, 465585120, 465585120

Offset: 1

Amiram Eldar and Thomas Ordowski, Dec 28 2019

Conjecture: if p is an odd prime, then p | A330718(p+1) - a(p+1).
Below 10^6 there is only one pseudoprime, namely 25. Are there others?
Primes p such that p^2 | A330718(p+1) - a(p+1) are 3, 5, 45827, ...

			Denominators of 0, 1/2, 3/2, 13/4, 25/4, 137/12, 245/12, ...

Metin Sariyar, Table of n, a(n) for n = 1..500

Cf. A000265, A002805, A064169, A279683, A290348, A330718.

[Denominator( &+[(2^(k-1)-1)/k: k in [1..n]] ): n in [1..45]]; // G. C. Greubel, Dec 28 2019

seq(denom(add((2^(k-1)-1)/k, k = 1..n)), n = 1..45); # G. C. Greubel, Dec 28 2019

Denominator@Accumulate@Array[(2^(#-1) -1)/# &, 45]
Table[Denominator[-(2^n*LerchPhi[2, 1, n+1] +Pi*I/2 +HarmonicNumber[n])], {n, 45}] (* G. C. Greubel, Dec 28 2019 *)

a(n) = denominator(sum(k=1, n, (2^(k-1)-1)/k)); \\ Michel Marcus, Dec 28 2019

[denominator( sum((2^(k-1)-1)/k for k in (1..n)) ) for n in (1..45)] # G. C. Greubel, Dec 28 2019

a(n) = denominator(-(2^n*LerchPhi(2,1,n+1) + Pi*i/2 + HarmonicNumber(n))). - G. C. Greubel, Dec 28 2019
a(n) = denominator(A279683(n)/n!) for n > 0. - Amiram Eldar and Thomas Ordowski, Jan 15 2020
A000265(a(n)) = A290348(n). - Thomas Ordowski, Mar 29 2025

A001901 Successive numerators of Wallis's approximation to Pi/2 (reduced).

Original entry on oeis.org

1, 2, 4, 16, 64, 128, 256, 2048, 16384, 32768, 65536, 262144, 1048576, 2097152, 4194304, 67108864, 1073741824, 2147483648, 4294967296, 17179869184, 68719476736, 137438953472, 274877906944, 2199023255552

Offset: 0

N. J. A. Sloane

If p is prime, then a(p-2) == - A001902(p-2) (mod p). Cf. A064169 (third comment) and my formula here. Such pseudoprimes are 1467, 7831, ... Primes p such that a(p-2) == - A001902(p-2) (mod p^2) are 5, 45827, ... Cf. A355959, see also A330719 (third comment). - Thomas Ordowski, Oct 19 2024

			From _Wolfdieter Lang_, Dec 07 2017: (Start)
The Wallis numerators (N) and denominators (D) with partial products A(n) = A001900(n) and B(n) = A000246(n+1) in unreduced form, and a(n) and b(n) = A001902(n) in reduced form.
n, k:     0  1  2  3  4   5    6     7      8        9       10 ...
N(k):     1  2  2  4  4   6    6     8      8       10       10 ...
D(k):     1  1  3  3  5   5    7     7      9        9        9 ...
A(n):     1  2  4 16 64 384 2304 18432 147456  1474560 14745600 ...
B(n):     1  1  3  9 45 225 1575 11025  99225   893025  9823275 ...
a(n):     1  2  4 16 64 128  256  2048  16384    32768    65536 ...
b(n):     1  1  3  9 45  75  175  1225  11025    19845    43659 ...
n = 5: numerator(1*2*2*4*4*6/(1*1*3*3*5*5)) = numerator(384/225) = numerator(128/75) = 128. (End)

H.-D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146.

Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004.
Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), Amer. Math. Monthly 112 (2005), 729-734 and 113 (2006), 670.
Eric Weisstein's World of Mathematics, Pi
Eric Weisstein's World of Mathematics, Pi Continued Fraction
Index to divisibility sequences

Denominators are A001902. Subsequence of A000079.

a[n_?EvenQ] := n!!^2/((n - 1)!!^2*(n + 1)); a[n_?OddQ] := ((n - 1)!!^2*(n + 1))/n!!^2; Table[a[n] // Numerator, {n, 0, 23}] (* Jean-François Alcover, Jun 19 2013 *)

(2*2*4*4*6*6*8*8*...*2n*2n*...)/(1*3*3*5*5*7*7*9*...*(2n-1)*(2n+1)*...) for n >= 1.
From Wolfdieter Lang, Dec 07 2017: (Start)
1/1 * 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ...; partial products (reduced). Here the numerators with offset 0.
a(n) = numerator(W(n)), for n >= 0, with W(n) = Product_{k=0..n} N(k)/D(k) (reduced), with N(k) = 2*floor((k+1)/2) for k >= 1 and N(0) = 1, and D(k) = 2*floor(k/2) + 1, for k >= 0. (End)
a(n) is the numerator of the continued fraction [1;1,1/2,1/3,...,1/n]. - Thomas Ordowski, Oct 19 2024

A119947 Triangle of numerators in the square of the matrix A[i,j] = 1/i for j <= i, 0 otherwise.

Original entry on oeis.org

1, 3, 1, 11, 5, 1, 25, 13, 7, 1, 137, 77, 47, 9, 1, 49, 29, 19, 37, 11, 1, 363, 223, 153, 319, 107, 13, 1, 761, 481, 341, 743, 533, 73, 15, 1, 7129, 4609, 3349, 2509, 1879, 275, 191, 17, 1, 7381, 4861, 3601, 2761, 2131, 1627, 1207, 121, 19, 1, 83711, 55991, 42131, 32891, 25961

Offset: 1

Wolfdieter Lang, Jul 20 2006

The triangle of the corresponding denominators is A119948. The rationals appear in lowest terms (while in A027446 they are row-wise on the least common denominator).
The triangle with row number i multiplied with the least common multiple (LCM) of its denominators yields A027446.
First column is A001008. - Tilman Neumann, Oct 01 2008
Column 2 is A064169. - Clark Kimberling, Aug 13 2012
Third diagonal (11, 13, 47, ...) is A188386. - Clark Kimberling, Aug 13 2012

			The rationals are [1]; [3/4, 1/4]; [11/18, 5/18, 1/9]; [25/48, 13/48, 7/48, 1/16]; ... See the W. Lang link for more.
From _Clark Kimberling_, Aug 13 2012: (Start)
As a triangle given by f(n,m) = Sum_{h=m..n} 1/h, the first six rows are:
    1
    3    1
   11    5    1
   25   13    7    1
  137   77   47    9    1
   49   29   19   37   11    1
  363  223  153  319  107   13    1
(End)

Wolfdieter Lang, First ten rows and rationals.

Cf. A002024: i appears i times (denominators in row i of the matrix A).
Row sums give A119949. Row sums of the triangle of rationals always give 1.
For the cube of this matrix see the rational triangle A119935/A119932 and A027447; see A027448 for the fourth power.
Cf. A001008, A027446, A064169, A119948, A188386.

f[n_, m_] := Numerator[Sum[1/k, {k, m, n}]]
Flatten[Table[f[n, m], {n, 1, 10}, {m, 1, n}]]
TableForm[Table[f[n, m], {n, 1, 10}, {m, 1, n}]] (* Clark Kimberling, Aug 13 2012 *)

A119947_upto(n)={my(M=matrix(n,n,i,j,(j<=i)/i)^2);vector(n,r,apply(numerator,M[r,1..r]))} \\ M. F. Hasler, Nov 05 2019

a(i,j) = numerator(r(i,j)) with r(i,j):=(A^2)[i,j], where the matrix A has elements a[i,j] = 1/i if j<=i, 0 if j>i, (lower triangular).

Edited by M. F. Hasler, Nov 05 2019

A064168 Sum of numerator and denominator in n-th harmonic number, 1 + 1/2 + 1/3 +...+ 1/n.

Original entry on oeis.org

2, 5, 17, 37, 197, 69, 503, 1041, 9649, 9901, 111431, 113741, 1506353, 1532093, 1556117, 3157279, 54394463, 18358381, 352893319, 71354639, 24031221, 24266365, 563299563, 1704771547, 42976237267, 43319457067, 392849685203, 395718022103, 11556136074187

Offset: 1

Leroy Quet, Sep 19 2001

Numerator and denominator in definition have no common factors >1.

			The 3rd harmonic number is 11/6. So a(3) = 11 + 6 = 17.

Brian Hayes, A Tantonalizing Problem

Cf. A001008, A002805, A064167, A064169.

h:= n-> numer(sum(1/k,k=1..n))+denom(sum(1/k,k=1..n)): seq(h(n),n=1..30);  # Emeric Deutsch, Nov 18 2004

Numerator[#]+Denominator[#]&/@HarmonicNumber[Range[30]] (* Harvey P. Dale, Jul 04 2017 *)

a(n) = my(h=sum(k=1, n, 1/k)); numerator(h) + denominator(h); \\ Michel Marcus, Sep 07 2019

More terms from Emeric Deutsch, Nov 18 2004

A064167 Product of numerator and denominator of the n-th harmonic number, 1 + 1/2 + 1/3 +...+ 1/n.

Original entry on oeis.org

1, 6, 66, 300, 8220, 980, 50820, 213080, 17965080, 18600120, 2320468920, 2384502120, 412970037480, 422245703880, 430902992520, 1756076802480, 516336630329520, 58297387228080, 21362271268818480, 866533600973040, 97555876321904, 98772315738096, 52866073370045936, 481103506052529360

Offset: 1

Leroy Quet, Sep 19 2001

nonn

Numerator and denominator in definition have no common divisors >1.

			The 3rd harmonic number is 11/6. So a(3) = 11 * 6 = 66.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001008, A002805, A064168, A064169.

Numerator[#]Denominator[#]&/@HarmonicNumber[Range[30]] (* Harvey P. Dale, May 01 2022 *)

a(n) = my(h=sum(k=1, n, 1/k)); numerator(h) * denominator(h); \\ Michel Marcus, Sep 07 2019

More terms from Michel Marcus, Sep 07 2019

A104174 Numerator of the fractional part of a harmonic number.

Original entry on oeis.org

0, 1, 5, 1, 17, 9, 83, 201, 2089, 2341, 551, 2861, 64913, 90653, 114677, 274399, 5385503, 2022061, 42503239, 9276623, 3338549, 3573693, 87368107, 276977179, 7281378067, 7624597867, 71595952403, 74464289303, 2239777822987

Offset: 1

Georg Haass (geha5001(AT)stud.uni-saarland.de), Mar 10 2005

nonn

Clark Kimberling, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Book Stacking Problem.

Cf. A064169, A075135.

h[n_] := Sum[1/k, {k, 1, n}]
Table[Numerator[FractionalPart[h[n]]], {n, 1, 30}]
(* Clark Kimberling, Aug 13 2012 *)
FractionalPart[HarmonicNumber[Range[30]]]//Numerator (* Harvey P. Dale, Jul 28 2019 *)

a(n) = numerator(frac(sum(k=1, n, 1/k))); \\ Michel Marcus, Sep 27 2021

from sympy import harmonic
def A104174(n): return (lambda x: x.p % x.q)(harmonic(n)) # Chai Wah Wu, Sep 26 2021

a(n) = numerator(frac(Sum_{k=1..n} 1/k)). [edited by Michel Marcus, Sep 27 2021]

A358464 a(n) is the greatest m such that Sum_{k = 1..m} 1/(1 + n*k) <= 1.

Original entry on oeis.org

2, 6, 16, 42, 110, 288, 761, 2020, 5388, 14417, 38681, 103994, 280032, 755031, 2037848, 5504884, 14880978, 40250609, 108926101, 294902398, 798703663, 2163878141

Offset: 1

Thomas Scheuerle, Nov 18 2022

This sequence coincidences with 2*Fibonacci(2*n) (A025169) for the first 6 terms.

			a(2) = 6 because Sum_{m = 1..a(2)} 1/(1+2*m) = 43024/45045 < 1, but Sum_{m = 1..a(2)+1} 1/(1+2*m) = 46027/45045 > 1.

David Borwein and Jonathan M. Borwein, Some Remarkable Properties of Sinc and Related Integrals, The Ramanujan Journal, 5 (2001) 73 - 89.
Hanspeter Schmid, Two curious integrals and a graphic proof, Elemente der Mathematik, 69 (2014) 11 - 17.

Cf. A002387, A025169, A064169.

a(n) = {my(k = 2*fibonacci(2*n)-1);my(b = (psi(k+(1/n))-psi(1+(1/n)))/n); until(b > 1, b = b+(1/(1+n*k)); k=k+1 );k-2}

ceiling(digamma(a(n)+(1/n)+1) - digamma((1/n)+1)) = n.
Integral_{x=0..oo} Product_{k=0..m} sinc(x/(n*k+1)) dx = Pi for 0 <= m <= a(n). See links Schmid and Borwein.
ceiling(Sum_{m = 0..oo} ( 1/(m+1) * Sum_{k = 0..m} (-1)^k*binomial(m, k)*log( (a(n)+(1/n)+1+k) / ((1/n)+1+k) ) )) = n.
a(n) ~ floor(exp(n + digamma(1+(1/n))) - (1/2) - (1/n)). This appears to be accurate for at least n < 22.

Showing 1-10 of 11 results. Next

cp's OEIS Frontend

A309391 a(n) = gcd(n, A064169(n-2)) for n > 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A064404 When the numerator - denominator (A064169) in n-th harmonic number is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A074791 Numbers k such that k does not divide the denominator of the k-th harmonic number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A330719 a(n) = denominator(Sum_{k=1..n} (2^(k-1) - 1)/k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A001901 Successive numerators of Wallis's approximation to Pi/2 (reduced).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A119947 Triangle of numerators in the square of the matrix A[i,j] = 1/i for j <= i, 0 otherwise.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A064168 Sum of numerator and denominator in n-th harmonic number, 1 + 1/2 + 1/3 +...+ 1/n.

Views

Author

Keywords

Comments