A119722 -id:A119722 - cp's OEIS frontend

A120290 Numerator of generalized harmonic number H(p-1,2p)= Sum[ 1/k^(2p), {k,1,p-1}] divided by p^2 for prime p>3.

2479157521, 159936660724017234488561, 1119583852472161859174156302552583713828739479026834819554843860744244189

Offset: 3

Alexander Adamchuk, Jul 08 2006

Generalized harmonic number is H(n,m)= Sum[ 1/k^m, {k,1,n} ]. The numerator of generalized harmonic number H(p-1,2p) is divisible by p^2 for prime p>3.

			With prime(3) = 5, a(3) = numerator[ 1 + 1/2^10 + 1/3^10 + 1/4^10 ] / 5^2 = 61978938025 / 25 = 2479157521.

Alexander Adamchuk, First 5 terms.
Eric Weisstein's World of Mathematics, Harmonic Number.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.

Cf. A119722, A099828, A099827, A001008, A007406, A007408, A007410.

Table[Numerator[Sum[1/k^(2*Prime[n]),{k,1,Prime[n]-1}]],{n,3,7}]/Table[Prime[n]^2,{n,3,7}]

a(n) = numerator[ Sum[ 1/k^(2*Prime[n]), {k,1,Prime[n]-1} ]] / Prime[n]^2 for n>2.

A120289 Primes p such that p divides the numerator of Sum_{k=1..n-1} 1/prime(k)^p, where p = prime(n).

Original entry on oeis.org

5, 19, 47, 79, 109

Offset: 1

Alexander Adamchuk, Jul 08 2006

nonn

Next term > 1690. - Michael S. Branicky, Jun 27 2022

			a(1) = 5 because prime 5 divides 275 = numerator(1/2^5 + 1/3^5).
Sum_{k=1..n-1} 1/prime(k)^prime(n) begins:
  n=2: 1/2^3 = 1/8;
  n=3: 1/2^5 + 1/3^5 = 275/7776;
  n=4: 1/2^7 + 1/3^7 + 1/5^7 = 181139311/21870000000;
  n=5: 1/2^11 + 1/3^11 + 1/5^11 + 1/7^11 = 17301861338484245234233/35027750054222100000000000.

Cf. A119722.

from fractions import Fraction
from sympy import isprime, primerange
def ok(p):
    if p < 3 or not isprime(p): return False
    s = sum(Fraction(1, pk**p) for pk in primerange(2, p))
    return s.numerator%p == 0
print([k for k in range(200) if ok(k)]) # Michael S. Branicky, Jun 26 2022

A120347 Numerator of Sum_{k=1..n-1} 1/k^n.

Original entry on oeis.org

1, 9, 1393, 257875, 47463376609, 940908897061, 972213062238348973121, 7727182467755471289426059, 10338014371627802833957102351534201, 26038773205374138944970092886340352227, 205885410277133543091182509665217407908365393153956577

Offset: 2

Alexander Adamchuk, Aug 16 2006, Oct 31 2006

Prime p>2 divides a(p). p^3 divides a(p) for prime p>3. p divides a((p+1)/2) for prime p = {7,11,17,19,23,31,41,43,47,59,67,71,73,79,83,89,97,103,...} = all primes excluding 2 and 3 from A045323[n] Primes congruent to {1, 2, 3, 7} mod 8.

a(n) = Numerator( H(n-1,n) ), where H(k,r) = Sum_{i=1..k} 1/i^r is the generalized harmonic number.

Vincenzo Librandi, Table of n, a(n) for n = 2..49
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A045323, A120289, A120352 (a(prime(n))), A119722 (a(prime(n))/prime(n)^3).

Table[Numerator[Sum[1/k^n,{k,1,n-1}]],{n,2,15}]

a(n) = Numerator(Sum_{k=1..n-1} 1/k^n). a(n) = Numerator[Zeta[n] - Zeta[n,n]].

A130681 Sum[ 1/k^(2p-1), {k,1,p-1}] divided by p^3, for prime p>3.

Original entry on oeis.org

41361119, 126941659254799099843, 201945187495172518712395211386399925751676163316330287629003467281801, 534565103485593943310791656810688803242468895931876288948761507813750601446840308490623197040810555162527973

Offset: 3

Alexander Adamchuk, Jun 29 2007

The generalized harmonic number is H(n,m) = Sum[ 1/k^m, {k,1,n} ]. The numerator of H(p-1,2p-1) is divisible by p^3 for prime p>3. Also the numerator of H(p-1,p) is divisible by p^3 for prime p>3. See A119722(n).

			Prime[3] = 5.
a(3) = numerator[ 1 + 1/2^9 + 1/3^9 + 1/4^9 ] / 5^3 = 5170139875/125 = 41361119.

Alexander Adamchuk, Table of n, a(n) for n = 3..10
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A119722.

Table[ Numerator[ Sum[ 1/k^(2*Prime[n]-1), {k,1,Prime[n]-1} ] ] / Prime[n]^3, {n,3,10} ]

a(n)=p=prime(n);numerator(sum(i=1,p-1,1/i^(2*p-1)))/p^3 \\ Ralf Stephan, Nov 10 2013

a(n) = Numerator[ Sum[ 1/k^(2*Prime[n]-1), {k,1,Prime[n]-1} ] ] / Prime[n]^3 for n>2.

a(n) = A228426(A000040(n))/A000040(n)^3.

Edited by Ralf Stephan, Nov 10 2013

A120352 Numerator of Sum[ 1/k^p, {k,1,p-1} ], where p = Prime[n].

Original entry on oeis.org

1, 9, 257875, 940908897061, 26038773205374138944970092886340352227, 5706439637514064062030256049808675747470805004854626598761, 3819751175863358416058062379293843331497647520922258560223903226691067255782388923965399403291707829

Offset: 1

Alexander Adamchuk, Aug 16 2006, Oct 31 2006

p^3 divides a(n) for n>2. A119722[n] = a(n)/p^3, p=Prime[n].

Numerators of Sum[ 1/k^n, {k,1,n-1} ] are listed in A120347(n) = {1, 9, 1393, 257875, 47463376609, 940908897061, ...}.

Cf. A119722.

Cf. A120347.

Table[Numerator[Sum[1/k^Prime[n],{k,1,Prime[n]-1}]],{n,1,8}]

a(n) = Numerator[ Sum[ 1/k^Prime[n], {k,1,Prime[n]-1} ]]. a(n) = Numerator[ Zeta[p] - Zeta[p,p] ], for p = Prime[n].

a(n) = A120347[ Prime[n] ].

A130682 Numerator of generalized harmonic number H(p-1,p^2) = Sum_{k=1..p-1} 1/k^(p^2) divided by p^4 for prime p>3.

Original entry on oeis.org

1526339511795367850762323, 187024220802620550798074497168768775337833066860651232788557036897081398718783708709

Offset: 3

Alexander Adamchuk, Jun 29 2007

The generalized harmonic number is H(n,m) = Sum_{k=1..n} 1/k^m. The numerator of the generalized harmonic number H(p-1,p) is divisible by p^3 for prime p>3. See A119722(n). The numerator of the generalized harmonic number H(p-1,p^2) is divisible by p^4 for prime p>3. In general, the numerator of the generalized harmonic number H(p-1,p^k) is divisible by p^(k+2) for prime p>3.

			Prime[3] = 5.
a(3) = numerator[ 1 + 1/2^25 + 1/3^25 + 1/4^25 ] / 5^4 = 953962194872104906726451875/625 = 1526339511795367850762323.

Alexander Adamchuk, Jun 29 2007, Table of n, a(n) for n = 3..6
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A119722 = Numerator of generalized harmonic number H(p-1, p)= Sum[ 1/k^p, {k, 1, p-1}] divided by p^3 for prime p>3.

Table[ Numerator[ Sum[ 1/k^(Prime[n]^2), {k,1,Prime[n]-1} ] ] / Prime[n]^4, {n,3,10} ]

a(n) = Numerator[ Sum[ 1/k^(Prime[n]^2), {k,1,Prime[n]-1} ] ] / Prime[n]^4 for n>2.

A116184 Numbers n such that 37^3 divides the numerator of generalized harmonic number H(36,n) = Sum[ 1/k^n, {k,1,36} ].

Original entry on oeis.org

3, 37, 39, 73, 75, 111, 147, 148, 183, 185, 219, 221, 255, 259, 291, 295, 327, 333, 363, 369, 399, 407, 435, 443, 471, 481, 507, 517, 543, 555, 579, 591, 615, 629, 651, 665, 687, 703, 723, 739, 759, 777, 795, 813, 831, 851, 867, 887, 903, 925, 939, 961, 975

Offset: 1

Alexander Adamchuk, Apr 08 2007

Note the pattern in the first differences of a(n): {34,2,34,2,36,36,1,35,2,34,2,34,4,32,4,32,6,30,6,30,8,28,8,28,10,26,10,26,12,24,12,24,14,22,14,22,16,20,16,20,18,18,18,18,20,16,20,16,22,14,22,14,24,...}. Conjecture: All terms of the arithmetic progression 3+36k belong to a(n). Prime terms in a(n) are {3, 37, 73, 443, 739, 887, 1109, ...}. It appears that all primes in a(n) that are greater than 37 are of the form 37k-1. For example, 73 = 37*2-1, 443 = 37*12-1, 739 = 37*20-1, 887 = 37*24-1, 1109 = 37*30-1. Many terms in a(n) are the multiples of 37. There are terms of the form 37*m with m = {1,3,4,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,37,39,41,...}. Note that 37^4 divides the numerator of generalized harmonic number H(36,n) for n = {111, 147, 1047, 1369, 1443, 1479, ...} = {3*37, 3+4*36, 3+29*36, 37^2, 3+40*36, 3+41*36, ...}.

Eric Weisstein, The World of Mathematics: Wolstenholme's Theorem.
Eric Weisstein, The World of Mathematics: Harmonic Number.

Cf. A007408 = Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^3. Cf. A119722, A017533.

Do[ f=Numerator[ Sum[ 1/k^n, {k,1,36} ] ]; If[ IntegerQ[ f/37^3 ], Print[n] ], {n,1,1000}]

A125194 Numerator of generalized harmonic number H((p-1)/2,2p)= Sum[ 1/k^(2p), {k,1,(p-1)/2}] divided by p^2 for prime p>3.

Original entry on oeis.org

41, 1599366601, 10877829357646990581304675244472669289, 100935935338172297894217692920950359818733561, 9217176064595104612826996436899733706027947436610177335077693637792069056822883934927465549747441

Offset: 3

Alexander Adamchuk, Jan 13 2007

Generalized harmonic number is H(n,m)= Sum[ 1/k^m, {k,1,n} ]. The numerator of generalized harmonic number H(p-1,2p) is divisible by p^2 for prime p>3 (see A120290(n)). The numerator of generalized harmonic number H((p-1)/2,2p) is divisible by p^2 for prime p>3.

			Prime[3] = 5.
a(3) = Numerator[ 1 + 1/2^10 ] / 5^2 = 1025 / 25 = 41.

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

Cf. A120290, A119722, A001008, A007406, A007408, A007410.

Do[p=Prime[k];f=0;Do[f=f+1/n^(2p);g=Numerator[f];If[IntegerQ[g/(p)^2],Print[{p,g/p^2}]],{n,1,(p-1)/2}],{k,1,100}]

a(n) = Numerator[ Sum[ 1/k^(2*Prime[n]), {k,1,(Prime[n]-1)/2} ]] / Prime[n]^2 for n>2.

A128820 Numerator of alternating generalized harmonic number H'(p-1,2p) = Sum_{k=1..p-1} (-1)^(k+1)/k^(2*p) divided by p^2 for prime p > 2.

Original entry on oeis.org

7, 2474315503, 53305712401979540402437, 5597916593064896381208777124641713285719656398067086247546781015747740847, 192635872080422175485338764164035657976855166649911323825254242037669356649787653784405726270977624462974729613783

Offset: 2

Alexander Adamchuk, Apr 10 2007

nonn

Alternating generalized harmonic number is H'(n,m) = Sum_{k=1..n} (-1)^(k+1)*1/k^m. Numerator of H'(p-1,2n) is divisible by p for all integers n > 0 and primes p > 2. Numerator of H'(p-1,2p) is divisible by p^2 for prime p > 2.

			prime(2) = 3; a(2) = numerator(1 - 1/2^6) / 3^2 = 63/9 = 7.
prime(3) = 5; a(3) = numerator(1 - 1/2^10 + 1/3^10 - 1/4^10) / 5^2 = 61857887575/25 = 2474315503.

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

Cf. A119722 (numerator of generalized harmonic number H(p-1, p) = Sum_{k=1..p-1} 1/k^p divided by p^3 for prime p>3).

Cf. A001008, A119682, A120296.

Table[ Numerator[ Sum[(-1)^(k+1)*1/k^(2*Prime[n]), {k,1,Prime[n]-1} ] ] / Prime[n]^2, {n,2,10} ]

a(n) = numerator(Sum_{k=1..prime(n)-1} (-1)^(k+1)/k^(2*prime(n))) / prime(n)^2 for n > 1.

cp's OEIS Frontend

A120290 Numerator of generalized harmonic number H(p-1,2p)= Sum[ 1/k^(2p), {k,1,p-1}] divided by p^2 for prime p>3.

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A120289 Primes p such that p divides the numerator of Sum_{k=1..n-1} 1/prime(k)^p, where p = prime(n).

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A120347 Numerator of Sum_{k=1..n-1} 1/k^n.

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A130681 Sum[ 1/k^(2p-1), {k,1,p-1}] divided by p^3, for prime p>3.

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A120352 Numerator of Sum[ 1/k^p, {k,1,p-1} ], where p = Prime[n].

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A130682 Numerator of generalized harmonic number H(p-1,p^2) = Sum_{k=1..p-1} 1/k^(p^2) divided by p^4 for prime p>3.

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A116184 Numbers n such that 37^3 divides the numerator of generalized harmonic number H(36,n) = Sum[ 1/k^n, {k,1,36} ].

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A125194 Numerator of generalized harmonic number H((p-1)/2,2p)= Sum[ 1/k^(2p), {k,1,(p-1)/2}] divided by p^2 for prime p>3.

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