A027612 -id:A027612 - cp's OEIS frontend

A354894 a(n) is the numerator of the n-th hyperharmonic number of order n.

1, 5, 47, 319, 1879, 20417, 263111, 261395, 8842385, 33464927, 166770367, 3825136961, 19081066231, 57128792093, 236266661971, 7313175618421, 14606816124167, 102126365345729, 3774664307989373, 3771059091081773, 154479849447926113, 6637417807457499259, 6632660439700528339

Offset: 1

Ilya Gutkovskiy, Jun 10 2022

			1, 5/2, 47/6, 319/12, 1879/20, 20417/60, 263111/210, 261395/56, 8842385/504, ...

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, p. 258.

Eric Weisstein's World of Mathematics, Harmonic Number
Wikipedia, Hyperharmonic number

Cf. A001008, A001700, A027612, A058806, A076174, A124837, A354895 (denominators).

Differs from A049281.

Table[SeriesCoefficient[-Log[1 - x]/(1 - x)^n, {x, 0, n}], {n, 1, 23}] // Numerator
Table[Binomial[2 n - 1, n - 1] (HarmonicNumber[2 n - 1] - HarmonicNumber[n - 1]), {n, 1, 23}] // Numerator

H(n) = sum(i=1, n, 1/i);
a(n) = numerator(binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1))); \\ Michel Marcus, Jun 10 2022

from math import comb
from sympy import harmonic
def A354894(n): return (comb(2*n-1,n-1)*(harmonic(2*n-1)-harmonic(n-1))).p # Chai Wah Wu, Jun 18 2022

a(n) is the numerator of the coefficient of x^n in the expansion of -log(1 - x) / (1 - x)^n.
a(n) is the numerator of binomial(2*n-1,n-1) * (H(2*n-1) - H(n-1)), where H(n) is the n-th harmonic number.
a(n) / A354895(n) ~ log(2) * 2^(2*n-1) / sqrt(Pi * n).

A382812 Numerator of the n-th partial sum of the squares of the harmonic numbers.

Original entry on oeis.org

1, 13, 119, 1577, 3233, 8867, 141563, 2844129, 28119709, 335676251, 3968696491, 55023970333, 758025067309, 799020611041, 1676892996083, 59597395635137, 351844709221043, 2314823924364859, 9114392136427625, 628176680098075, 216039223801697, 5117413095318143, 363066107054194281, 27957386425926920257

Offset: 1

Gary Detlefs, Apr 05 2025

			The squares of the first three harmonic numbers are 1, 9/4, 121/36 which sum to 119/18 so a(3)=119.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A001008, A002805, A382813 (denominators).

Cf. A027611, A027612

H2:= n-> add(harmonic(k)^2, k = 1..n): seq(numer(H2(n)), n=1..25);

Accumulate[HarmonicNumber[Range[30]]^2]//Numerator (* Harvey P. Dale, Aug 10 2025 *)

a(n) = numerator(sum(k=1, n, sum(i=1, k, 1/i)^2)); \\ Michel Marcus, Apr 07 2025

a(n) = numerator((n+1)*H(n)^2-(2*n+1)*H(n) + 2*n), where H(n) is the n-th harmonic number.

a(n) = numerator((S(n)*H(n)^2 + (2*n - 2*S(n) + 1)*H(n)-2*n)/(H(n)-1)), where S(n) is the n-th partial sum of H(n).

A382813 Denominator of the n-th partial sum of the squares of the harmonic numbers.

Original entry on oeis.org

1, 4, 18, 144, 200, 400, 4900, 78400, 635040, 6350400, 64033200, 768398400, 9275666400, 8657288640, 16232416200, 519437318400, 2779951574400, 16679709446400, 60213751101504, 3823095308032, 1216439416192, 26761667156224, 1769615240705312, 127412297330782464, 3062795608913040000

Offset: 1

Gary Detlefs, Apr 05 2025

All terms for n>1 are even.

			The squares of the first three harmonic numbers are 1, 9/4, 121/36 which sum to 119/18 so a(3) = 18.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A001008, A002805, A382812 (numerators).

Cf. A027611, A027612.

H2:= n-> add(harmonic(k)^2, k = 1..n): seq(denom(H2(n)), n=1..25);

a(n) = denominator(sum(k=1, n, sum(i=1, k, 1/i)^2)); \\ Michel Marcus, Apr 07 2025

a(n) = denominator((n+1)*H(n)^2-(2*n+1)*H(n)+2*n), where H(n) is the n-th harmonic number.

a(n) = denominator((S(n)*H(n)^2+(2*n-2*S(n)+1)*H(n) - 2*n)/(H(n) - 1)), where S(n) = the n-th partial sum of H(n).

A120487 Denominator of 1^n/n + 2^n/(n-1) + 3^n/(n-2) + ... + (n-1)^n/2 + n^n/1.

Original entry on oeis.org

1, 2, 3, 12, 5, 20, 35, 280, 63, 2520, 385, 27720, 6435, 8008, 45045, 720720, 85085, 4084080, 969969, 739024, 29393, 5173168, 7436429, 356948592, 42902475, 2974571600, 717084225, 80313433200, 215656441, 2329089562800, 4512611027925

Offset: 1

Alexander Adamchuk, Jul 22 2006

Numerator is A115071(n).

Also a(n) is denominator of (n+1)^(n+1) * (H(n+1) - 1), where H(k) is harmonic number, H(k) = Sum_{i=1..k} 1/i = A001008(k)/A002805(k). - Alexander Adamchuk, Jan 02 2007

Cf. A115071, A027612, A031971, A002805.

Cf. A001008, A002805.

Denominator[Table[Sum[k^n/(n-k+1),{k,1,n}],{n,1,50}]]
Table[ Denominator[ (n+1)^(n+1) * Sum[ 1/i,{i,2,n+1} ] ], {n,1,40} ] (* Alexander Adamchuk, Jan 02 2007 *)

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

a(n) = denominator((n+1)^(n+1) * Sum_{i=2..n+1} 1/i). - Alexander Adamchuk, Jan 02 2007

A123369 Number of prime divisors of n-th Conway and Guy second-order harmonic number (counted with multiplicity).

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 2, 1, 3, 2, 2, 2, 3, 2, 4, 3, 1, 2, 5, 3, 3, 2, 2, 1, 3, 3, 3, 1, 1, 2, 2, 2, 5, 2, 2, 2, 5, 1, 3, 4, 4, 3, 3, 3, 5, 4, 3, 3, 3, 2, 2, 6, 2, 3, 4, 2, 4, 2, 3, 3, 2, 4, 4, 4, 3, 3, 3, 3, 4, 2, 3, 4, 2, 2, 5, 3, 2, 2, 4, 4, 2, 2, 1, 6, 4, 2, 5, 3, 5, 1, 2, 2, 3, 4, 2, 3, 3, 3, 5

Offset: 1

Jonathan Vos Post, Nov 09 2006

We must include multiplicity in the definition due to terms such as a(16) = 29889983 = 19 * 31^2 * 1637. The primes are those n for which a(n) = Omega(A027612(n))= 1, namely a(2) = 5, a(3) = 13, a(6) = 223, a(9) = 4861, a(18) = 197698279, a(25) = 25472027467. The semiprimes are those for which a(n) = 2, such as when n = 4, 5, 7, 8, 11, 12, 13, 15, 19, 23, 24. The 3-almost primes are those for which a(n) = 3, as with the "3-brilliant" a(10) = 55991 = 13 * 59 * 73, a(14), a(17), a(21), a(22), a(26).

			a(20) = 5 because A027612(20) = 41054655 = 3 * 5 * 23 * 127 * 937 has 5 prime factors.

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, pp. 143 and 258-259.

G. C. Greubel, Table of n, a(n) for n = 1..150
Eric Weisstein's World of Mathematics, Harmonic Number, MathWorld, see discussion of Conway and Guy (1996) definition of the second-order harmonic number.

Cf. A001222 Number of prime divisors of n (counted with multiplicity), A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) +...+ (n-1)/2 + n, A027611, A001008, A002805, A001705, A006675, A093418.

PrimeOmega[Numerator[Table[Sum[k/(n - k + 1), {k, 1, n}], {n, 1, 50}]]] (* G. C. Greubel, Jan 22 2017 *)

a(n) = A001222(A027612(n)) = Omega(Numerator of 1/n + 2/(n-1) + 3/(n-2) +...+ (n-1)/2 + n).

A316297 a(n) = n! times the denominator of the n-th harmonic number H(n).

Original entry on oeis.org

1, 4, 36, 288, 7200, 14400, 705600, 11289600, 914457600, 9144576000, 1106493696000, 13277924352000, 2243969215488000, 31415569016832000, 471233535252480000, 15079473128079360000, 4357967734014935040000, 26147806404089610240000, 9439358111876349296640000

Offset: 1

Matthew Campbell, Jun 29 2018

nonn

			a(4) = 4! * A002805(4) = 24 * 12 = 288.

Cf. A000142, A001008, A002805, A027611, A027612, A124837, A124838.

H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
a:= n-> denom(H(n))*n!:
seq(a(n), n=1..20);  # Alois P. Heinz, Jul 21 2018

a[n_] := n! Denominator@HarmonicNumber@n; Array[a, 18] (* Robert G. Wilson v, Jun 30 2018 *)

a(n) = n! * denominator(sum(k=1, n, 1/k)); \\ Michel Marcus, Aug 12 2018

a(n) = A000142(n) * A002805(n).

A119389 Numerator of (1^2/n + 2^2/(n-1) + ... + k^2/(n-k+1) + ... + (n-1)^2/2 + n^2/1).

Original entry on oeis.org

1, 9, 34, 265, 186, 1141, 2868, 31401, 18635, 477301, 91192, 8051069, 4508441, 3336145, 22048024, 410111791, 223063947, 3057889621, 823596665, 706952715, 125961187, 6173866701, 9838037952, 521135614075, 275363139571

Offset: 1

Alexander Adamchuk, Jul 26 2006

p divides a(p-1) for prime p>2. p divides a(2p-1) for all prime p. p divides a(3p-1) for all prime p. p divides a(4p-1) for all prime p except p=3. p divides a(5p-1) for prime p>3. p divides a(6p-1) for all prime except p=5. . p^2 divides a(p^2-1) for prime p>2. p^2 divides a(2p^2-1) for all prime p. p^2 divides a(3p^2-1) for all prime p. . p^3 divides a(p^3-1) for prime p>2. . p^k divides a(p^k-1) for prime p>2 and integer k>1. p^k divides a(m*p^k-1) for all prime p and integer m,k>1.

Cf. A027612, A001008, A002805, A000217.

Numerator[Table[Sum[k^2/(n-k+1),{k,1,n}],{n,1,50}]]

a(n) = Numerator[Sum[k^2/(n-k+1),{k,1,n}]]. a(n) = Numerator[HarmonicNumber[n]*(n+1)^2 - 3*n(n+1)/2]. a(n) = Numerator[A001008[n]/A002805[n]*(n+1)^2 - 3*A000217[n]].

A119782 Numerator of Sum[ i^3/(n-i+1)^2, {i,1,n}].

Original entry on oeis.org

1, 33, 262, 10325, 7227, 923027, 4588732, 49631229, 29087285, 1618749187, 152762993, 69307256941, 191848206263, 702044812445, 9181629908344, 2873944084323677, 1547933530301469, 1198197057199579439

Offset: 1

Alexander Adamchuk, Jun 25 2006

Prime p divides a(2p-1). Prime p divides a(3p-1), p>2. Prime p divides a(p-1), p>2. p^2 divides a(p^2-1) for prime p>2. p^3 divides a(p^3-1) for prime p>2. p^4 divides a(p^4-1) for prime p>2. p^5 divides a(p^5-1) for prime p>2.

Cf. A007406, A001008, A027612.

Numerator[Table[Sum[i^3/(n-i+1)^2,{i,1,n}],{n,1,50}]]

a(n) = numerator[ Sum[i^3/(n-i+1)^2, {i,1,n}]].

A119783 Numerator of Sum_{i=1..n} i^3/(n-i+1).

Original entry on oeis.org

1, 17, 94, 965, 841, 6167, 18044, 225489, 75220, 4280111, 899494, 86645897, 26288822, 41914055, 296912584, 5893703327, 1703822073, 49486578079, 14076109870, 12725008135, 1190932611, 122366761563, 203957791852

Offset: 1

Alexander Adamchuk, Jun 25 2006

Prime p divides a(2p-1), p>3. Prime p divides a(3p-1), p>2. Prime p divides a(p-1), p>3. p^2 divides a(p^2-1) for prime p>3. p^3 divides a(p^3-1) for prime p>3. p^4 divides a(p^4-1) for prime p>3. p^5 divides a(p^5-1) for p>3.

Cf. A007406, A001008, A027612.

Numerator[Table[Sum[i^3/(n-i+1),{i,1,n}],{n,1,50}]]

a(n) = numerator( Sum_{i=1..n} i^3/(n-i+1) ).

A300910 Expansion of e.g.f. 1/(1 - x)^(x/(1 - x)^2).

Original entry on oeis.org

1, 0, 2, 15, 116, 1070, 11754, 149436, 2145296, 34193736, 598061160, 11377384920, 233732130312, 5153974126704, 121354505626704, 3037419444974040, 80497938647953920, 2251124265581428800, 66225476356207660224, 2044005966844402035456, 66025689709572751040640, 2227221130525199246067840, 78301158190416233445985920

Offset: 0

Ilya Gutkovskiy, Mar 15 2018

nonn

Exponential transform of A006675.

			1/(1 - x)^(x/(1 - x)^2) = 1 + 2*x^2/2! + 15*x^3/3! + 116*x^4/4! + 1070*x^5/5! + 11754*x^6/6! + 149436*x^7/7! + ...

N. J. A. Sloane, Transforms

Cf. A000254, A001705, A006675, A027611, A027612, A087761, A300491.

a:=series(1/(1-x)^(x/(1-x)^2),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[1/(1 - x)^(x/(1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[k k! (HarmonicNumber[k] - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

E.g.f.: A(x) = exp(B(x)*C(x)), where B(x) is the g.f. of the sequence {0, 1, 2, 3, 4, 5, ...} and C(x) is the g.f. of the sequence {0, 1, 1/2, 1/3, 1/4, 1/5, ...}.

a(0) = 1; a(n) = Sum_{k=1..n} k*k!*(H(k)-1)*binomial(n-1,k-1)*a(n-k), where H(k) is the k-th harmonic number.

cp's OEIS Frontend

A354894 a(n) is the numerator of the n-th hyperharmonic number of order n.

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A382812 Numerator of the n-th partial sum of the squares of the harmonic numbers.

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A382813 Denominator of the n-th partial sum of the squares of the harmonic numbers.

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A120487 Denominator of 1^n/n + 2^n/(n-1) + 3^n/(n-2) + ... + (n-1)^n/2 + n^n/1.

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A123369 Number of prime divisors of n-th Conway and Guy second-order harmonic number (counted with multiplicity).

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A316297 a(n) = n! times the denominator of the n-th harmonic number H(n).

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A119389 Numerator of (1^2/n + 2^2/(n-1) + ... + k^2/(n-k+1) + ... + (n-1)^2/2 + n^2/1).

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A119782 Numerator of Sum[ i^3/(n-i+1)^2, {i,1,n}].