A330570 -id:A330570 - cp's OEIS frontend

A330571 Square of number of unordered factorizations of n as n = i*j.

1, 1, 1, 4, 1, 4, 1, 4, 4, 4, 1, 9, 1, 4, 4, 9, 1, 9, 1, 9, 4, 4, 1, 16, 4, 4, 4, 9, 1, 16, 1, 9, 4, 4, 4, 25, 1, 4, 4, 16, 1, 16, 1, 9, 9, 4, 1, 25, 4, 9, 4, 9, 1, 16, 4, 16, 4, 4, 1, 36, 1, 4, 9, 16, 4, 16, 1, 9, 4, 16, 1, 36, 1, 4, 9, 9, 4, 16, 1, 25, 9, 4, 1, 36, 4, 4, 4, 16, 1, 36, 4, 9, 4, 4, 4, 36

Offset: 1

N. J. A. Sloane, Jan 08 2020

nonn

Unordered analog of A035116.

For background references see A330570.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Equals A038548(n)^2.

Cf. A035116.

a[n_] := Ceiling[DivisorSigma[0, n] / 2]^2; Array[a, 100] (* Amiram Eldar, Apr 19 2024 *)

a(n) = (numdiv(n) \/ 2)^2; \\ Amiram Eldar, Apr 19 2024

A331072 a(n) = Sum_{k = 1..n} u_3(k), where u_3 = A034836.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 12, 14, 16, 17, 21, 22, 24, 26, 30, 31, 35, 36, 40, 42, 44, 45, 51, 53, 55, 58, 62, 63, 68, 69, 74, 76, 78, 80, 88, 89, 91, 93, 99, 100, 105, 106, 110, 114, 116, 117, 126, 128, 132, 134, 138, 139, 145, 147, 153, 155, 157, 158, 168, 169, 171, 175, 182, 184, 189, 190, 194, 196, 201, 202, 214, 215, 217

Offset: 1

N. J. A. Sloane, Jan 10 2020

nonn

For background references see A330570.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A034836.

A096276 has the same initial terms, but is a different sequence.

s[1] = 1; s[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[IntegerQ[Surd[n, 3]], 1/3, 0] + (Times @@ ((e + 1)*(e + 2)/2))/6 + (Times @@ (Floor[e/2] + 1))/2]; Accumulate[Array[s, 100]] (* Amiram Eldar, Apr 19 2024 *)

A331077 a(n) = Sum_{k = 1..n} [d(k)*d_3(k)], where d = A000005, d_3 = A007425.

Original entry on oeis.org

1, 7, 13, 31, 37, 73, 79, 119, 137, 173, 179, 287, 293, 329, 365, 440, 446, 554, 560, 668, 704, 740, 746, 986, 1004, 1040, 1080, 1188, 1194, 1410, 1416, 1542, 1578, 1614, 1650, 1974, 1980, 2016, 2052, 2292, 2298, 2514, 2520, 2628, 2736, 2772, 2778, 3228, 3246, 3354, 3390, 3498, 3504, 3744, 3780, 4020, 4056

Offset: 1

N. J. A. Sloane, Jan 10 2020

nonn

For background references see A330570.

Amiram Eldar, Table of n, a(n) for n = 1..10000
E. C. Titchmarsh, Some problems in the analytic theory of numbers, The Quarterly Journal of Mathematics, Vol. 1 (1942), pp. 129-152.

Cf. A000005, A007425.

f[p_, e_] := (e+1)^2*(e+2)/2; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[s, 100]] (* Amiram Eldar, Apr 19 2024 *)

lista(nmax) = {my(s = 0); for(n = 1, nmax, s += vecprod(apply(e -> (e+1)^2*(e+2)/2, factor(n)[,2])); print1(s, ", "));} \\ Amiram Eldar, Apr 19 2024

a(n) ~ c * n * log(n)^5 /5!, where c = Product_{p prime} ((1-1/p)^2*(1+2/p)) = 0.286747428434478734107... (Titchmarsh, 1942). - Amiram Eldar, Apr 19 2024

A331078 a(n) = Sum_{k = 1..n} [u_2(k)*u_3(k)], where u_2 = A038548, u_3 = A034836.

Original entry on oeis.org

1, 2, 3, 7, 8, 12, 13, 19, 23, 27, 28, 40, 41, 45, 49, 61, 62, 74, 75, 87, 91, 95, 96, 120, 124, 128, 134, 146, 147, 167, 168, 183, 187, 191, 195, 235, 236, 240, 244, 268, 269, 289, 290, 302, 314, 318, 319, 364, 368, 380, 384, 396, 397, 421, 425, 449, 453, 457, 458, 518, 519, 523, 535, 563, 567, 587, 588, 600, 604, 624

Offset: 1

N. J. A. Sloane, Jan 10 2020

nonn

For background references see A330570.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A038548, A034836

s[1] = 1; s[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Ceiling[(Times @@ (e + 1)) / 2]*(If[IntegerQ[Surd[n, 3]], 1/3, 0] + (Times @@ ((e + 1)*(e + 2)/2))/6 + (Times @@ (Floor[e/2] + 1))/2)]; Accumulate[Array[s, 100]] (* Amiram Eldar, Apr 19 2024 *)

A331079 a(n) = Sum_{i=1..n} u_2(i)*u_2(i+1), where u_2(n) = A038548(n).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 20, 22, 25, 28, 30, 34, 40, 43, 46, 49, 52, 58, 62, 64, 68, 76, 80, 84, 90, 93, 97, 101, 104, 110, 114, 118, 128, 133, 135, 139, 147, 151, 155, 159, 162, 171, 177, 179, 184, 194, 200, 206, 212, 215, 219, 227, 235, 243, 247, 249, 255, 261

Offset: 1

N. J. A. Sloane, Jan 10 2020

nonn

For background references see A330570.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A038548, A330320.

With[{s = Table[Ceiling[DivisorSigma[0, n] / 2], {n, 1, 100}]}, Accumulate[Most[s] * Rest[s]]] (* Amiram Eldar, Apr 19 2024 *)

cp's OEIS Frontend

A330571 Square of number of unordered factorizations of n as n = i*j.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A331072 a(n) = Sum_{k = 1..n} u_3(k), where u_3 = A034836.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A331077 a(n) = Sum_{k = 1..n} [d(k)*d_3(k)], where d = A000005, d_3 = A007425.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A331078 a(n) = Sum_{k = 1..n} [u_2(k)*u_3(k)], where u_2 = A038548, u_3 = A034836.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A331079 a(n) = Sum_{i=1..n} u_2(i)*u_2(i+1), where u_2(n) = A038548(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica