A027611 -id:A027611 - cp's OEIS frontend

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

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).

A368810 a(n) = numerator of Sum_{i=1..n} Sum_{j=1..n} (1/i + 1/j).

Original entry on oeis.org

2, 6, 11, 50, 137, 147, 363, 1522, 7129, 7381, 83711, 86021, 1145993, 1171733, 1195757, 4873118, 42142223, 42822903, 275295799, 279175675, 56574159, 19093197, 444316699, 1347822955, 34052522467, 34395742267, 312536252003, 315404588903, 9227046511387

Offset: 1

Mats Granvik, Jan 06 2024

Cf. A027611, A096620 (denominators), A193758.

Numerator[Table[Sum[Sum[1/i + 1/j, {i, 1, n}], {j, 1, n}], {n, 1, 29}]]

from sympy import harmonic
def A368810(n): return ((n<<1)*harmonic(n)).p # Chai Wah Wu, Feb 04 2024

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

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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A368810 a(n) = numerator of Sum_{i=1..n} Sum_{j=1..n} (1/i + 1/j).

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

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Author

Keywords

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Crossrefs

Programs

Maple

Mathematica

Formula