A001008 -id:A001008 - cp's OEIS frontend

A308968 Table, read by rows: row n contains the prime factors of A001008(n) (numerator of n-th harmonic number), with multiplicity.

1, 3, 11, 5, 5, 137, 7, 7, 3, 11, 11, 761, 7129, 11, 11, 61, 97, 863, 13, 13, 509, 29, 43, 919, 1049, 1117, 29, 41233, 17, 17, 8431, 37, 1138979, 19, 19, 39541, 37, 7440427, 5, 11167027, 18858053, 3, 23, 23, 53, 227, 761, 583859, 5, 577, 467183, 109, 312408463

Offset: 1

M. F. Hasler, Jul 03 2019

Row 1 is taken to be {1} instead of being empty, by convention.

Length, first = smallest and last = largest term of the rows are given in A308967, A308970 and A308971, respectively. See A308969 for prime divisors without repetition.

			   n | A001008(n) written as product of primes
-----+---------------------------------------------
   1 | 1 (empty product)
   2 | 3
   3 | 11
   4 | 5 * 5
   5 | 137
   6 | 7 * 7
   7 | 3 * 11 * 11
   8 | 761
   9 | 7129
  10 | 11 * 11 * 61
  11 | 97 * 863
  12 | 13 * 13 * 509
  13 | 29 * 43 * 919
  14 | 1049 * 1117
  15 | 29 * 41233
  16 | 17 * 17 * 8431
  17 | 37 * 1138979
  18 | 19 * 19 * 39541
  19 | 37 * 7440427
  20 | 5 * 11167027
etc.

M. F. Hasler, Rows 1 through 170.

Cf. A001008, A308967 (row lengths, for n > 1).

Cf. A308969 (prime divisors without repetition), A308970 (column 1 = first / smallest term of each row), A308971 (last / greatest term in each row).

A308968_row(n)={if(n>1, concat(apply(f->vector(f[2],i,f[1]), Col(factor(A001008(n)))~)),[1])}

A308970 Smallest prime factor of A001008(n), numerator of n-th harmonic number; a(1) = 1.

Original entry on oeis.org

1, 3, 11, 5, 137, 7, 3, 761, 7129, 11, 97, 13, 29, 1049, 29, 17, 37, 19, 37, 5, 18858053, 3, 761, 5, 109, 34395742267, 521, 29, 43, 31, 109, 2917, 269, 3583, 397, 37, 10839223, 199, 737281, 41, 85691034670497533, 7, 140473, 109, 1553, 47, 911, 7, 23982193, 61, 227, 53, 941, 5953

Offset: 1

M. F. Hasler, Jul 03 2019

nonn

Row 1 is taken to be {1} instead of being empty, by convention.

This is the first column of A308968 and A308969, which list the prime factors of A001008.

			   n | A001008(n) written as product of primes
-----+------------------------------------------
   1 | 1 (empty product)
   2 | 3
   3 | 11
   4 | 5 * 5
   5 | 137
   6 | 7 * 7
   7 | 3 * 11 * 11
   8 | 761
   9 | 7129
  10 | 11 * 11 * 61
  11 | 97 * 863
  12 | 13 * 13 * 509
  13 | 29 * 43 * 919
  14 | 1049 * 1117
  15 | 29 * 41233
  16 | 17 * 17 * 8431
  17 | 37 * 1138979
  18 | 19 * 19 * 39541
  19 | 37 * 7440427
  20 | 5 * 11167027
etc., therefore this sequence = 1, 3, 11, 5, 137, 7, 3, 761, ...

Chai Wah Wu, Table of n, a(n) for n = 1..325

Cf. A001008.

Cf. A308967 (number of prime factors), A308968 (table of factorization), A308969 (table of prime divisors), A308971 (greatest prime factor) of A001008(n).

FactorInteger[#][[1,1]]&/@Numerator[HarmonicNumber[Range[60]]] (* Harvey P. Dale, Jul 02 2025 *)

row(n)={if(n>1, factor(A001008(n))[1,1], 1)}

A308971 Largest prime factor of A001008(n), numerator of n-th harmonic number; a(1) = 1.

Original entry on oeis.org

1, 3, 11, 5, 137, 7, 11, 761, 7129, 61, 863, 509, 919, 1117, 41233, 8431, 1138979, 39541, 7440427, 11167027, 18858053, 227, 583859, 467183, 312408463, 34395742267, 215087, 375035183, 4990290163, 17783, 2667653736673, 535919, 199539368321, 15088528003, 137121586897

Offset: 1

M. F. Hasler, Jul 03 2019

nonn

Initial terms coincide with A120299 = greatest prime factor of Stirling numbers of first kind A000254. They differ when the unreduced denominator of H(n), equal to n!, is divisible by this factor, i.e., A120299(n) <= n. Can this ever happen?

			   n | A001008(n) written as product of primes
-----+------------------------------------------
   1 | 1 (empty product)
   2 | 3
   3 | 11
   4 | 5 * 5
   5 | 137
   6 | 7 * 7
   7 | 3 * 11 * 11
   8 | 761
   9 | 7129
  10 | 11 * 11 * 61
  11 | 97 * 863
  12 | 13 * 13 * 509
  13 | 29 * 43 * 919
  14 | 1049 * 1117
  15 | 29 * 41233
  16 | 17 * 17 * 8431
  17 | 37 * 1138979
  18 | 19 * 19 * 39541
  19 | 37 * 7440427
  20 | 5 * 11167027
etc., therefore this sequence = 1, 3, 11, 5, 137, 7, 11, 761, 7129, 61, ...

Amiram Eldar, Table of n, a(n) for n = 1..325

Cf. A001008, A006530.

Cf. A308967 (number of prime factors), A308968 (table of factorization), A308969 (table of prime divisors), A308970 (smallest prime factor) of A001008(n).

Array[FactorInteger[Numerator@HarmonicNumber[#]][[-1, 1]] &, 35] (* Michael De Vlieger, Jul 04 2019 *)

a(n)={if(n>1, factor(A001008(n))[1,1], 1)}

a(n) = A006530(A001008(n)). - Amiram Eldar, Feb 24 2020

A079353 Numbers n such that the best rational approximation to H(n) with denominator <=n is an integer, where H(n) denotes the n-th harmonic number (A001008/A002805).

Original entry on oeis.org

1, 3, 4, 10, 11, 30, 31, 82, 83, 226, 227, 615, 616, 1673, 1674, 4549, 4550, 12366, 12367, 33616, 33617

Offset: 1

Benoit Cloitre, Feb 14 2003

From Robert Israel, May 19 2014: The definition is unclear. For example, how does 10 fit in? H(10) = 7381/2520, and the best approximation with denominator <= 10 is 29/10, which is not an integer. Similarly, I don't see how 31, 82, 227, 616, or 1674 fit the definition, as according to my computations the best approximations in these cases are 125/31, 409/82, 1363/227, 4313/616, 13393/1674.
Response from David Applegate, May 20 2014: I suspect, without deep investigation, that what was meant by "best rational approximation to" is "continued fraction convergent". The continued fraction convergents to H(10)=7381/2520 are 2, 3, 41/14, 495/169, ... The continued fraction convergents to H(31) are 4, 145/36, 149/37, 443/110, ... The continued fraction convergents to H(82) are 4, 5, 499/100, 2001/401, ... I haven't verified that the rest of the terms match this definition.
Response from Ray Chandler, May 20 2014: I confirm that definition matches the listed terms and continues with 4549, 4550 and no others less than 10000.
Added by Ray Chandler, May 29 2014: Except for the beginning terms A079353 appears to be the union of A115515 and A002387 (compare A242654).

			H(11)=83711/27720 and the best approximation to H(11) among the fractions of form k/11, k>=0, is 33/11=3, an integer. Hence 11 is in the sequence.

Cf. A001008/A002805, A002387, A004080, A115515.

See A242654 for the most likely continuation.

okQ[n_] := Select[Convergents[N[HarmonicNumber[n], 30], 10], Denominator[#] <= n &][[-1]] // IntegerQ;
Reap[For[n = 1, n <= 40000, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] // Quiet (* Jean-François Alcover, Apr 10 2019 *)

a(16)-a(17) from Ray Chandler, May 20 2014
Edited by N. J. A. Sloane, May 29 2014
a(18)-a(21) from Jean-François Alcover, Apr 10 2019

A092194 Primes p that divide A001008(k), the numerator of the k-th harmonic number H(k), for some k < p-1.

Original entry on oeis.org

11, 29, 37, 43, 53, 61, 97, 109, 137, 173, 199, 227, 257, 269, 271, 313, 347, 353, 379, 397, 401, 409, 421, 433, 439, 509, 521, 577, 599, 601, 617, 641, 643, 647, 659, 677, 733, 761, 773, 809, 821, 827, 839, 863, 911, 919, 929, 937, 941, 947, 953, 971, 1009

Offset: 1

T. D. Noe, Feb 24 2004

nonn

These primes are a subset of the non-harmonic primes A092102. Because these primes are analogous to the irregular primes A000928 that divide the numerators of Bernoulli numbers, they might be called H-irregular primes. The density of these primes is about 0.4 -- very close to the density of irregular primes.

These primes are called Harmonic irregular primes in the Wikipedia entry for "Regular prime" (see links). It may be noted that if p is known to be of this type and H(k) is the smallest Harmonic number divisible by p, then not only does k < p-1 hold, but k <= (p-1)/2. This is because, by symmetry, H(p-1-n) == H(n) (mod p), so that any eligible k lying between (p+1)/2 and p-1 would have a counterpart in the range between 1 and (p-1)/2. Furthermore, the minimal k cannot be exactly equal to (p-1)/2, because then p would be a Wieferich prime (A001220) and would also divide H(Int(p/4)). Thus k <= (p-3)/2, and this inequality is sharp because exact equality holds for p = 29, 37, 3373 (see A072984). - John Blythe Dobson, Apr 09 2015

Robert Israel, Table of n, a(n) for n = 1..2500
Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Irregular Prime
Wikipedia, Regular prime

Cf. A072984 (least k such that prime(n) divides A001008(k)).

A1008:= 1: h:= 1: for n from 2 to 10000 do h:= h+1/n; A1008:= A1008, numer(h) od:
A1008:= [A1008]:
filter:= proc(p) ormap(i -> A1008[i] mod p = 0, [$1..p-2]) end proc:
select(filter, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Jan 20 2025

n=2; Table[cnt=0; While[cnt==0, p=Prime[n]; k=1; h=0; While[cnt==0 && k<=(p-1)/2, h=h+1/k; If[Mod[Numerator[h], p]==0, cnt++ ]; k++ ]; n++ ]; p, {100}]

A177734 Largest k such that prime(n) divides the numerator of the k-th harmonic number (=A001008(k)).

Original entry on oeis.org

22, 24, 102728, 1011849771855214912968404217247, 168, 288, 848874360, 528, 695552, 886725671, 50641, 1680, 2359785, 10776888210, 414839198, 42176361744, 226972, 4488, 9094138358932, 5328, 6240

Offset: 2

Max Alekseyev, May 12 2010

For p = prime(n), Boyd defines J_p to be the set of numbers k such that p divides A001008(k). This sequence gives the largest element of J_p. The smallest element of J_p is given by A072984. The size of J_p is given by A092103.

Term a(23) is too large to include, see b-file. - Max Alekseyev, Apr 04 2025

Max Alekseyev, Table of n, a(n) for n = 2..220
David W. Boyd, A p-adic study of the partial sums of the harmonic series, Experimental Math., Vol. 3 (1994), No. 4, 287-302. [WARNING: Table 2 contains miscalculations for p=19, 47, 59, ... - _Max Alekseyev_, Feb 10 2016]
Leonardo Carofiglio, Giacomo Cherubini, and Alessandro Gambini, On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers, arXiv:2503.15714 [math.NT], 2025.

Cf. A072984, A092103, A092193.

For p = prime(n) in A092101, a(n) = p^2 - 1.

a(5) computed by Boyd.

a(8)-a(22) from Max Alekseyev, Oct 23 2012

A256103 a(n) = gcd(A001008(m(n)), m(n)), with m(n) = A256102(n), n >= 1.

Original entry on oeis.org

5, 7, 11, 11, 13, 17, 19, 23, 29, 31, 43, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 11, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223

Offset: 1

Wolfdieter Lang, Apr 16 2015

See A256102. The entry a(n) gives the quotient of the numerator of the harmonic sum of the first A256102(n) positive integers and the denominator of the harmonic mean of the same numbers. For each positive integer values m not from A256102 this quotient is 1.

			n = 1: gcd(A001008(20), 20) = gcd(55835135, 20) = 5. A001008(20)/A175441(20) = 55835135/11167027 = 5.
Because 19 is not from A256102 one has A001008(19) = A175441(19) = 275295799.

Cf. A256102, A002805, A001008, A175441.

a(n) = gcd(A001008(m(n)), m(n)), with m(n) = A256102(n), n >= 1.

a(n) = A001008(m(n))/A175441(m(n)), with m(n) = A256103(n), n >= 1.

A308969 Table, read by rows: row n contains the prime divisors of A001008 (numerator of n-th harmonic number), without repetitions.

Original entry on oeis.org

1, 3, 11, 5, 137, 7, 3, 11, 761, 7129, 11, 61, 97, 863, 13, 509, 29, 43, 919, 1049, 1117, 29, 41233, 17, 8431, 37, 1138979, 19, 39541, 37, 7440427, 5, 11167027, 18858053, 3, 23, 53, 227, 761, 583859, 5, 577, 467183, 109, 312408463

Offset: 1

M. F. Hasler, Jul 03 2019

Row 1 is taken to be {1} instead of being empty, by convention.

			   n | A001008(n) written as product of primes
-----+---------------------------------------------
   1 | 1 (empty product)
   2 | 3
   3 | 11
   4 | 5 * 5   (So 5 is the only prime divisor, and row(4) = {5}.)
   5 | 137
   6 | 7 * 7
   7 | 3 * 11 * 11       whence row(7) = {3, 11}.
   8 | 761
   9 | 7129
  10 | 11 * 11 * 61      whence row(10) = {11, 61}.
  11 | 97 * 863
  12 | 13 * 13 * 509     whence row(16) = {13, 509}.
  13 | 29 * 43 * 919     whence row(13) = {29, 43, 919}.
  14 | 1049 * 1117
  15 | 29 * 41233
  16 | 17 * 17 * 8431    whence row(16) = {17, 8431}.
  17 | 37 * 1138979
  18 | 19 * 19 * 39541   whence row(18) = {19, 39541}.
  19 | 37 * 7440427
  20 | 5 * 11167027
etc.

Cf. A001008.

Cf. A308967 (number of prime factors), A308968 (table of factorization), A308970 & A308971 (smallest & greatest prime factor) of A001008(n).

Table[FactorInteger[Numerator[HarmonicNumber[n]]][[All,1]],{n,30}]// Flatten (* Harvey P. Dale, Sep 14 2020 *)

row(n)={if(n>1, factor(A001008(n))[,1]~, [1])}

A349850 Decimal expansion of Sum_{k>=1} H(k)*F(k)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 02 2021

			3.96874800690391485217106364061998568869842436396224...

Hideyuki Ohtsuka, Problem B-1200, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 54, No. 4 (2016), p. 367; Harmonic and Fiboancci [sic]/Lucas Numbers, Solution to Problem B-1200 by Kenny B. Davenport, ibid., Vol. 55, No. 4 (2017), pp. 372-373.

Cf. A000045, A001008, A001622, A002162, A002163, A002390, A002805, A349851.

RealDigits[2*Log[2] + 12*Log[GoldenRatio]/Sqrt[5], 10, 100][[1]]

Equals log(4*phi^(12/sqrt(5))) = 2*log(2) + 12*log(phi)/sqrt(5), where phi is the golden ratio (A001622).

A035048 Numerators of alternating sum transform (PSumSIGN) of Harmonic numbers H(n) = A001008/A002805.

Original entry on oeis.org

1, 1, 4, 3, 23, 11, 176, 25, 563, 137, 6508, 49, 88069, 363, 91072, 761, 1593269, 7129, 31037876, 7381, 31730711, 83711, 744355888, 86021, 3788707301, 1145993, 11552032628, 1171733, 340028535787, 1195757

Offset: 1

N. J. A. Sloane

p^2 divides a(2p-2) for prime p>3. a(2p-2)/p^2 = A061002(n) = A001008(p-1)/p^2 for prime p>2. - Alexander Adamchuk, Jul 07 2006

Robert Israel, Table of n, a(n) for n = 1..2000
N. J. A. Sloane, Transforms

Cf. A035047, A002428, A001008, A058313, A061002.

S:= series(log(1-x)/(x^2-1), x, 101):
seq(numer(coeff(S,x,j)), j=1..100); # Robert Israel, Jun 02 2015

Numerator[Table[Sum[(-1)^(k+1)*Sum[(-1)^(i+1)*1/i,{i,1,k}],{k,1,n}],{n,1,50}]] (* Alexander Adamchuk, Jul 07 2006 *)

a(n)=numerator(polcoeff(log(1-x)/(x^2-1)+O(x^(n+1)),n))

G.f. for A035048(n)/A035047(n) : log(1-x)/(x^2-1). - Benoit Cloitre, Jun 15 2003
a(n) = Numerator[Sum[(-1)^(k+1)*Sum[(-1)^(i+1)*1/i,{i,1,k}],{k,1,n}]]. - Alexander Adamchuk, Jul 07 2006
a(n) = numerator((-1)^(n+1)*1/2*(log(2)+(-1)^(n+1)*(gamma+1/2*(psi(1+n/2)-psi(3/2+n/2))+psi(2+n)))), with gamma the Euler-Mascheroni constant. - - Gerry Martens, Apr 28 2011

cp's OEIS Frontend

A308968 Table, read by rows: row n contains the prime factors of A001008(n) (numerator of n-th harmonic number), with multiplicity.

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A308970 Smallest prime factor of A001008(n), numerator of n-th harmonic number; a(1) = 1.

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A308971 Largest prime factor of A001008(n), numerator of n-th harmonic number; a(1) = 1.

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A079353 Numbers n such that the best rational approximation to H(n) with denominator <=n is an integer, where H(n) denotes the n-th harmonic number (A001008/A002805).

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A092194 Primes p that divide A001008(k), the numerator of the k-th harmonic number H(k), for some k < p-1.

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A177734 Largest k such that prime(n) divides the numerator of the k-th harmonic number (=A001008(k)).

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A256103 a(n) = gcd(A001008(m(n)), m(n)), with m(n) = A256102(n), n >= 1.

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A308969 Table, read by rows: row n contains the prime divisors of A001008 (numerator of n-th harmonic number), without repetitions.

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