A061201 -id:A061201 - cp's OEIS frontend

A212058 Number of (w,x,y,z) with all terms in {1,...,n} and w>=xyz.

0, 1, 5, 12, 25, 41, 66, 94, 132, 176, 229, 285, 359, 436, 522, 617, 727, 840, 971, 1105, 1257, 1418, 1588, 1761, 1964, 2173, 2391, 2619, 2865, 3114, 3390, 3669, 3969, 4278, 4596, 4923, 5286, 5652, 6027, 6411, 6825, 7242, 7686, 8133, 8598

Offset: 0

Clark Kimberling, Apr 30 2012

nonn

a(n)+A212057(n)=n^4. For a guide to related sequences, see A211795.

Cf. A211795, A061201.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w >= x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212058 *)
(* Peter J. C. Moses, Apr 13 2012 *)

a(n) = Sum_{i=1..n+1} Sum_{j=1..n+1} tau(i)*floor((n+1-j)/i). - Ridouane Oudra, Oct 03 2020

A226600 Number of ordered triples (i,j,k) with ijk <= n and i,j,k >= 0.

Original entry on oeis.org

1, 8, 23, 44, 74, 107, 152, 197, 255, 315, 384, 453, 543, 624, 717, 816, 927, 1032, 1158, 1275, 1413, 1548, 1689, 1830, 2004, 2160, 2325, 2497, 2683, 2860, 3067, 3256, 3469, 3676, 3889, 4108, 4360, 4585, 4822, 5065, 5335, 5584, 5863, 6124, 6406, 6694, 6979

Offset: 0

Robert Price, Jun 13 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (terms n = 0..100 from Robert Price)

Cf. A100450, A226357, A226359.

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k <= n, {i, j, k}], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]

a(n) = A003215(n) + A061201(n). - Alois P. Heinz, Jun 13 2013

A077593 Table by antidiagonals where T(n,k) = Sum_{i=1..n} T(floor(n/i),k-1) starting with T(n,0)=1 if n>0 and T(0,0)=0.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 5, 4, 1, 0, 1, 5, 7, 8, 5, 1, 0, 1, 6, 9, 13, 10, 6, 1, 0, 1, 7, 11, 19, 16, 14, 7, 1, 0, 1, 8, 13, 26, 23, 25, 16, 8, 1, 0, 1, 9, 15, 34, 31, 39, 28, 20, 9, 1, 0, 1, 10, 17, 43, 40, 56, 43, 38, 23, 10, 1, 0, 1, 11, 19, 53, 50, 76, 61, 63

Offset: 0

Henry Bottomley, Nov 08 2002

			Rows start:
 0,0,0,0,0,0...;
 1,1,1,1,1,1...;
 1,2,3,4,5,6...;
 1,3,5,7,9,11...;
 1,4,8,13,19,26,...;
 ...

Columns include A057427, A001477, A006218, A061201, A061202, A061203, A061204.
Rows include (with offsets) A000004, A000012, A000027, A005408, A034856, A052905.
Cf. A077593.

T(n, k) = T(n-1, k) + A077592(n, k). Writing m as Sum_{i} p_i^e_i, T(n, k) = Sum_{m=1..n} Product_{i} C(k+e_i-1, e_i).

A175596 Partial products of A007425.

Original entry on oeis.org

1, 3, 9, 54, 162, 1458, 4374, 43740, 262440, 2361960, 7085880, 127545840, 382637520, 3443737680, 30993639120, 464904586800, 1394713760400, 25104847687200, 75314543061600, 1355661775108800, 12200955975979200, 109808603783812800, 329425811351438400, 9882774340543152000, 59296646043258912000, 533669814389330208000, 5336698143893302080000, 96060566590079437440000, 288181699770238312320000, 7780905893796434432640000

Offset: 1

Jonathan Vos Post, Dec 03 2010

nonn

Partial products of the number of ordered factorizations of n as a product of 3 terms.

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = d_4(gcd(i,j)) for 1 <= i,j <= n, where d_4(n) = A007426(n). - Enrique Pérez Herrero, Jan 20 2013

			a(8) = 1 * 3 * 3 * 6 * 3 * 9 * 3 * 10 = 43740 = 2^2 * 3^7 * 5.

Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.

Cf. A000005, A007425, A007426, A061201 (partial sums), A127270, A143354.

Cf. A066843.

Table[Product[Sum[DivisorSigma[0, d], {d, Divisors[k]}], {k, 1, n}], {n, 1, 30}] (* Vaclav Kotesovec, Sep 03 2018 *)

f(n) = sumdiv(n, k, numdiv(k)); \\ A007425
a(n) = prod(k=1, n, f(k)); \\ Michel Marcus, Mar 23 2021

a(n) = Product_{i=1..n} A007425(i).

a(n) = Product_{prime p<=n} Product_{k=1..floor(log_p(n))} (1 + 2/k)^floor(n/p^k). - Ridouane Oudra, Mar 23 2021

A318750 a(n) = Sum_{k=1..n} k * tau_3(k), where tau_3 is A007425.

Original entry on oeis.org

1, 7, 16, 40, 55, 109, 130, 210, 264, 354, 387, 603, 642, 768, 903, 1143, 1194, 1518, 1575, 1935, 2124, 2322, 2391, 3111, 3261, 3495, 3765, 4269, 4356, 5166, 5259, 5931, 6228, 6534, 6849, 8145, 8256, 8598, 8949, 10149, 10272, 11406, 11535, 12327, 13137, 13551

Offset: 1

Vaclav Kotesovec, Sep 02 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..100000
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

Cf. A007425, A034718, A061201, A318413, A318414.

Cf. A001620, A082633.

Accumulate[Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, 100}]]
(* Asymptotics: *) n^2 * (Log[n]^2 + (6*EulerGamma - 1)*Log[n] + 6*EulerGamma^2 - 3*EulerGamma - 6*StieltjesGamma[1] + 1/2) / 4 (* Vaclav Kotesovec, Sep 09 2018 *)

tau_3(k) = vecprod(apply(e -> (e+1)*(e+2)/2, factor(k)[, 2]));
a(n) = sum(k = 1, n,  k * tau_3(k)); \\ Amiram Eldar, Jan 18 2025

a(n) = Sum_{k=1..n} A034718(k).

a(n) ~ n^2 * (log(n)^2 + (6*g-1)*log(n) + 6*g^2 - 3*g - 6*g1 + 1/2) / 4, where g is the Euler-Mascheroni constant A001620 and g1 is the first Stieltjes constant A082633. - Vaclav Kotesovec, Sep 09 2018

A379357 Numerators of the partial sums of the reciprocals of the 3rd Piltz function d_3(n) (A007425).

Original entry on oeis.org

1, 4, 5, 11, 13, 41, 47, 122, 259, 269, 299, 152, 167, 172, 59, 4, 13, 79, 85, 43, 44, 5, 16, 161, 83, 254, 517, 29, 92, 833, 878, 6191, 6296, 6401, 6506, 26129, 27389, 27809, 28229, 5671, 5923, 5951, 6203, 6245, 6287, 6371, 6623, 33199, 33829, 34039, 34459, 34669

Offset: 1

Amiram Eldar, Dec 21 2024

			Fractions begin with 1, 4/3, 5/3, 11/6, 13/6, 41/18, 47/18, 122/45, 259/90, 269/90, 299/90, 152/45, ...

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 12-13, Theorem 1.2.
József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 59.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, On the asymptotic formulae for some functions connected with powers of the zeta-function, Matematički Vesnik, Vol. 1 (14) (29) (1977), pp. 79-90.

Cf. A007425, A061201, A104528, A379358 (denominators).

f[p_, e_] := (e+1)*(e+2)/2; d3[1] = 1; d3[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[1/d3[n], {n, 1, 100}]]]

d3(n) = vecprod(apply(e -> (e+1)*(e+2)/2, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / d3(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} 1/A007425(k)).

a(n)/A379358(n) = Sum_{i=1..N} b_i * n / log(n)^(i-1/3) + O(n / log(n)^(N+1-1/3)), for any fixed N >= 1, where b_i are constants. The same formula holds (with different constants) for any Piltz function d_k(n), for k >= 2, when 1/3 is replaced by 1/k.

A379358 Denominators of the partial sums of the reciprocals of the 3rd Piltz function d_3(n) (A007425).

Original entry on oeis.org

1, 3, 3, 6, 6, 18, 18, 45, 90, 90, 90, 45, 45, 45, 15, 1, 3, 18, 18, 9, 9, 1, 3, 30, 15, 45, 90, 5, 15, 135, 135, 945, 945, 945, 945, 3780, 3780, 3780, 3780, 756, 756, 756, 756, 756, 756, 756, 756, 3780, 3780, 3780, 3780, 3780, 3780, 756, 756, 3780, 3780, 3780

Offset: 1

Amiram Eldar, Dec 21 2024

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 12-13, Theorem 1.2.
József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 59.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, On the asymptotic formulae for some functions connected with powers of the zeta-function, Matematički Vesnik, Vol. 1 (14) (29) (1977), pp. 79-90.

Cf. A007425, A061201, A104529, A379357 (numerators).

f[p_, e_] := (e+1)*(e+2)/2; d3[1] = 1; d3[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[1/d3[n], {n, 1, 100}]]]

d3(n) = vecprod(apply(e -> (e+1)*(e+2)/2, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / d3(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} 1/A007425(k)).

A134700 Triangle read by rows: A000012 * A051731^2.

Original entry on oeis.org

1, 3, 1, 5, 1, 1, 8, 3, 1, 1, 10, 3, 1, 1, 1, 14, 5, 3, 1, 1, 1, 16, 5, 3, 1, 1, 1, 1, 20, 8, 3, 3, 1, 1, 1, 1, 23, 8, 5, 3, 1, 1, 1, 1, 1, 27, 10, 5, 3, 3, 1, 1, 1, 1, 1, 29, 10, 5, 3, 3, 1, 1, 1, 1, 1, 1, 35, 14, 8, 5, 3, 3, 1, 1, 1, 1, 1, 1, 37, 14, 8, 5, 3, 3, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Nov 07 2007

Left column = A006218.
Row sums = A061201: (1, 4, 7, 13, 16, 25, 28, ...).
A134699 = A127170 * A000012.

			First few rows of the triangle:
   1;
   3, 1;
   5, 1, 1;
   8, 3, 1, 1;
  10, 3, 1, 1, 1;
  14, 5, 3, 1, 1, 1;
  16, 5, 3, 1, 1, 1, 1;
  20, 8, 3, 3, 1, 1, 1, 1;
  23, 8, 5, 3, 1, 1, 1, 1, 1;
  ...

Cf. A006218, A051731, A061201, A127170, A134699.

row(n) = my(m=matrix(n,n,i,j,!(i%j))); vector(n, k, sum(i=1, n, (m^2)[i, k])); \\ Michel Marcus, Apr 30 2025

A000012 * A051731^2 = A000012 * A127170.

a(45) = 1, a(46) = 27 corrected and more terms from Georg Fischer, Jun 05 2023

More terms from Michel Marcus, Apr 30 2025

A140703 A000012 * A051731^3.

Original entry on oeis.org

1, 4, 1, 7, 1, 1, 13, 4, 1, 1, 16, 4, 1, 1, 1, 25, 7, 4, 1, 1, 1, 28, 7, 4, 1, 1, 1, 1, 38, 13, 4, 4, 1, 1, 1, 1, 44, 13, 7, 4, 1, 1, 1, 1, 1, 53, 16, 7, 4, 4, 1, 1, 1, 1, 1, 56, 16, 7, 4, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, May 24 2008

Row sums = A061202: (1, 5, 9, 19, 23, 39,...).

Leftmost column = A061201: (1, 4, 7, 13, 16, 25, 28,...).

			First few rows of the triangle are:
1;
4, 1;
7, 1, 1;
13, 4, 1, 1;
16, 4, 1, 1, 1;
25, 7, 4, 1, 1, 1;
28, 7, 4, 1, 1, 1, 1;
38, 13, 4, 4, 1, 1, 1, 1;
...

Cf. A051731, A061202, A061201.

A000012 * A051731^3 as infinite lower triangular matrices, where A000012 = [1; 1,1; 1,1,1;...] and A051731 = the inverse Mobius transform.

A319088 a(n) = Sum_{k=1..n} k^2*tau_3(k), where tau_3 is A007425.

Original entry on oeis.org

1, 13, 40, 136, 211, 535, 682, 1322, 1808, 2708, 3071, 5663, 6170, 7934, 9959, 13799, 14666, 20498, 21581, 28781, 32750, 37106, 38693, 55973, 59723, 65807, 73097, 87209, 89732, 114032, 116915, 138419, 148220, 158624, 169649, 216305, 220412, 233408, 247097

Offset: 1

Vaclav Kotesovec, Sep 10 2018

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Stieltjes Constants
Wikipedia, Stieltjes constants

Cf. A007425, A061201, A318750.

Cf. A001620, A082633.

nmax = 50; Accumulate[Table[k^2*Sum[DivisorSigma[0, d], {d, Divisors[k]}], {k, 1, nmax}]]

tau_3(k) = vecprod(apply(e -> (e+1)*(e+2)/2, factor(k)[, 2]));
a(n) = sum(k = 1, n,  k^2 * tau_3(k)); \\ Amiram Eldar, Jan 18 2025

a(n) ~ n^3 * (log(n)^2/6 + (gamma - 1/9)*log(n) + gamma^2 - gamma/3 - g1 + 1/27), where gamma is the Euler-Mascheroni constant A001620 and g1 is the first Stieltjes constant A082633.

cp's OEIS Frontend

A212058 Number of (w,x,y,z) with all terms in {1,...,n} and w>=x*y*z.

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A226600 Number of ordered triples (i,j,k) with i*j*k <= n and i,j,k >= 0.

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A077593 Table by antidiagonals where T(n,k) = Sum_{i=1..n} T(floor(n/i),k-1) starting with T(n,0)=1 if n>0 and T(0,0)=0.

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A175596 Partial products of A007425.

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A318750 a(n) = Sum_{k=1..n} k * tau_3(k), where tau_3 is A007425.

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A379357 Numerators of the partial sums of the reciprocals of the 3rd Piltz function d_3(n) (A007425).

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A379358 Denominators of the partial sums of the reciprocals of the 3rd Piltz function d_3(n) (A007425).

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A134700 Triangle read by rows: A000012 * A051731^2.

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A212058 Number of (w,x,y,z) with all terms in {1,...,n} and w>=xyz.

A226600 Number of ordered triples (i,j,k) with ijk <= n and i,j,k >= 0.