A076637 -id:A076637 - cp's OEIS frontend

A061002 As p runs through the primes >= 5, sequence gives { numerator of Sum_{k=1..p-1} 1/k } / p^2.

1, 1, 61, 509, 8431, 39541, 36093, 375035183, 9682292227, 40030624861, 1236275063173, 6657281227331, 2690511212793403, 5006621632408586951, 73077117446662772669, 4062642402613316532391, 46571842059597941563297, 8437878094593961096374353

Offset: 3

N. J. A. Sloane, May 15 2001

This is an integer by a theorem of Waring and Wolstenholme.
Conjecture: If p is the n-th prime and H(n) is the n-th harmonic number, then denominator(H(p)/H(p-1))/numerator(H(p-1)/p^2) = p^3. A193758(p)/a(n) = p^3, p > 3. - Gary Detlefs, Feb 20 2013
The sequence which gives the numerators of H_{p-1} = Sum_{k=1..p-1} 1/k for p prime >= 5 is A076637. - Bernard Schott, Dec 02 2018

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, p. 388 Problem 5.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 115.

Muniru A Asiru, Table of n, a(n) for n = 3..340
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Cf. A000040, A001008, A001248, A120285, A185399, A193758.

List(List(Filtered([5..80],p->IsPrime(p)),i->Sum([1..i-1],k->1/k)/i^2),NumeratorRat); # Muniru A Asiru, Dec 02 2018

A061002:=proc(n) local p;
  p:=ithprime(n);
  (1/p^2)*numer(add(1/i,i=1..p-1));
end proc;
[seq(A061002(n),n=3..20)];

Table[Function[p, Numerator[Sum[1/k, {k, p - 1}]/p^2]]@ Prime@ n, {n, 3, 20}] (* Michael De Vlieger, Feb 04 2017 *)

a(n) = my(p=prime(n)); numerator(sum(k=1, p-1, 1/k))/p^2; \\ Michel Marcus, Dec 03 2018

a(n) = A001008(p-1)/p^2, p=A000040(n). - R. J. Mathar, Jan 09 2017

a(n) = A120285(n)/A001248(n). - R. J. Mathar, Jan 09 2017

A322434 Numbers k that are the numerator of a harmonic number such that k is divisible by the square of a prime >= 5.

Original entry on oeis.org

25, 49, 363, 7381, 86021, 2436559, 14274301, 19093197, 315404588903, 9227046511387, 9304682830147, 54801925434709, 2078178381193813, 12309312989335019, 5943339269060627227, 14063600165435720745359, 254381445831833111660789, 15117092380124150817026911

Offset: 1

Bernard Schott, Dec 08 2018

From Bernard Schott, Dec 09 2018: (Start)
In the data, there are two families of numerators.
1) Numerators of harmonic numbers H_{p-1} which are divisible by p^2 for p >= 5, by Wolstenholme's theorem; these numerators are in A076637.
2) Numerators of harmonic numbers which are also divisible by squares of primes, but not as a result of Wolstenholme's theorem. E.g., the numbers 363, numerator of H_7, and 9227046511387, numerator of H_{29}, are respectively divisible by 11^2 and 43^2. Up through H_{60}, only the two numerators of H_7 and H_{29} belong to this second family.
(End)
The third term in the second family is the numerator of H_{297} ~ 1.153... * 10^129 which is divisible by 1153^2, and the fourth numerator of H_k has k > 335. - Amiram Eldar, Dec 09 2018
Numbers n <= 50000 such that 11^2 divides the numerator of the n-th harmonic number: 7, 10 (by Wolstenholme's theorem), 848, 9328, 9338, 10583, 10591, 102718, 102721, 102728, 116413, 116423. - Jon E. Schoenfield, Dec 09 2018

			49 is a term because the numerator of the harmonic number H_6 = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 = 49/20 is divisible by the square of 7 (by Wolstenholme's Theorem).
363 is a term because the numerator of the harmonic number H_7 = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 = 363/140 is divisible by the square of 11.

Hisanori Mishima, Wolstenholme Numbers, WIFC (World Integer Factorization Center).

Cf. A001008, A076637.

Select[Numerator@ HarmonicNumber@ Range@ 60, AnyTrue[FactorInteger[#], And[First[#] > 2, Last[#] > 1] &] &] (* Michael De Vlieger, Dec 08 2018 *)

isok(nh) = {my(f = factor(nh)); for (i=1, #f~, if ((f[i, 1] >= 5) && (f[i, 2] >= 2), return (1)); ); return (0); }
lista(nn) = {my(h = 0, nh); for (n=1, nn, h += 1/n; nh = numerator(h); if (isok(nh), print1(nh, ", "); ); ); } \\ _Michel Marcus, Dec 08 2018

A076638 Denominators of harmonic numbers when the numerators are divisible by squares of primes >= 5 in the case of Wolstenholme's Theorem.

Original entry on oeis.org

12, 20, 2520, 27720, 720720, 4084080, 5173168, 80313433200, 2329089562800, 13127595717600, 485721041551200, 2844937529085600, 1345655451257488800, 3099044504245996706400, 54749786241679275146400, 3230237388259077233637600

Offset: 1

Michael Gilleland (megilleland(AT)yahoo.com), Oct 23 2002

From Bernard Schott, Dec 28 2018: (Start)
By Wolstenholme's Theorem, if p prime >= 5, the numerator of the harmonic number H_{p-1} is always divisible by p^2. The obtained quotients are in A061002.
The numerators of H_7 and H_{29} are also divisible by prime squares, respectively by 11^2 and 43^2, but not in the case of Wolstenholme's theorem, so the denominators of H_7 and H_{29} are not in this sequence here. (End)

			a(1)=12 because the numerator of H_4 = 25/12 is divisible by the square of 5;
a(2)=20 because the numerator of H_6 = 49/20 is divisible by the square of 7.

Eric Weisstein's World of Mathematics, Wolstenholme's Theorem

Cf. A076637, A185399.

a[p_] := Denominator[HarmonicNumber[p - 1]]; a /@ Prime@Range[3, 20] (* Amiram Eldar, Dec 28 2018 *)

More terms added by Amiram Eldar, Dec 04 2018

A326227 Indices of nonsquarefree numerators (A001008) of harmonic numbers H(n) = Sum_{k=1..n} 1/k.

Original entry on oeis.org

4, 6, 7, 10, 12, 16, 18, 22, 28, 29, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270, 276

Offset: 1

M. F. Hasler, Jul 03 2019

It appears that the first term of A001008 having a cubic factor is A001008(848) = 11^3 * 1871 * C359.

By Wolstenholme's Theorem, p^2 divides A001008(p-1) whenever p >= 5 is prime (cf. A076637); see A308968 for illustration. Therefore, A006093 \ {1, 2} (primes - 1) is a subsequence. (Thanks to Bernard Schott.)

Cf. A001008, A076637, A322434.

Cf. A308967 (number of prime factors), A308968 (table of factorization), A308969 (table of prime divisors), A308970 & A308971 (smallest & largest prime factor) of A001008(n).

is_A326227(n)={n>3&&vecmax(factor(A001008(n))[,2])>1} \\ Add ,0 in factor() for much faster but possibly incorrect results [false negative].
for(n=1,oo, is_A326227(n) && print1(n","))

A330014 When prime(n) is an odd prime (n >= 2) and N(n) / D(n) = Sum_{k=1..prime(n)-1} 1/k^3, then prime(n) divides N(n) and a(n) = N(n) / prime(n).

Original entry on oeis.org

3, 407, 4081, 1742192177, 1964289620189, 26430927136768997, 12913609418092462447, 14639800647032731764901, 21461951639001843544904995612963, 489697309796854053100609288112563213, 97796057728171000155497946604711651753457

Offset: 2

Bernard Schott, Nov 27 2019

nonn

The idea of this sequence comes from the 1st exercise of "sélection de la délégation française" in 2005 for IMO 2006 where it was asked to prove that prime(n) divides N(n) [See reference].

The first fractions N(n)/D(n) are 9/8, 2035/1728, 28567/24000, 19164113947/16003008000, 25535765062457/21300003648000, ...

			For prime(4) = 7 then 1 + 1/2^3 + 1/3^3 + 1/4^3 + 1/5^3 + 1/6^3 = 28567/24000 and 28567/7 = 4081, a(4) = 4081.

Guy Alarcon and Yves Duval, TS: Préparation au Concours Général, RMS, Collection Excellence, Paris, 2010, chapitre 10, Exercices de sélection de la délégation française en Octobre 2005 pour OIM 2006, Exercice 1, p. 169, p. 179.

Index to sequences related to Olympiads.

Cf. A076637, A061002, A076637 (Wolstenholme's Theorem).

[(Numerator(&+ [1/(k-1)^3:k in [2..NthPrime(n)]])) / NthPrime(n):n in [2..12]]; // Marius A. Burtea, Nov 27 2019

a[n_] := Numerator[Sum[1/(i- 1)^3, {i, 2,(p = Prime[n])}]]/p; Array[a, 11, 2] (* Amiram Eldar, Nov 27 2019 *)

cp's OEIS Frontend

A061002 As p runs through the primes >= 5, sequence gives { numerator of Sum_{k=1..p-1} 1/k } / p^2.

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A322434 Numbers k that are the numerator of a harmonic number such that k is divisible by the square of a prime >= 5.

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A076638 Denominators of harmonic numbers when the numerators are divisible by squares of primes >= 5 in the case of Wolstenholme's Theorem.

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A326227 Indices of nonsquarefree numerators (A001008) of harmonic numbers H(n) = Sum_{k=1..n} 1/k.

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A330014 When prime(n) is an odd prime (n >= 2) and N(n) / D(n) = Sum_{k=1..prime(n)-1} 1/k^3, then prime(n) divides N(n) and a(n) = N(n) / prime(n).

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