cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

A330320 a(n) = Sum_{i=1..n} tau(i)*tau(i+1), where tau(n) = A000005(n) is the number of divisors of n.

Original entry on oeis.org

2, 6, 12, 18, 26, 34, 42, 54, 66, 74, 86, 98, 106, 122, 142, 152, 164, 176, 188, 212, 228, 236, 252, 276, 288, 304, 328, 340, 356, 372, 384, 408, 424, 440, 476, 494, 502, 518, 550, 566, 582, 598, 610, 646, 670, 678, 698, 728, 746, 770, 794, 806, 822, 854, 886, 918, 934, 942, 966, 990, 998, 1022, 1064, 1092, 1124, 1140
Offset: 1

Views

Author

N. J. A. Sloane, Dec 11 2019

Keywords

Comments

For background references see A330570.

References

  • József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 61.

Links

Crossrefs

Cf. A000005, A059956, A063123.
Partial sums of A092517.

Programs

  • Mathematica
    Accumulate[a[n_]:=DivisorSum[n+1, DivisorSigma[0, n]&]; Array[a, 66]] (* Vincenzo Librandi, Jan 10 2020 *)
Accumulate[Times@@@Partition[DivisorSigma[0,Range[70]],2,1]] (* Harvey P. Dale, Nov 02 2023 *)
  • PARI
    a(n) = sum(i=1, n, numdiv(i*(i+1))); \\ Michel Marcus, Jan 11 2020

Formula

a(n) ~ (1/zeta(2)) * n * log(n)^2. - Amiram Eldar, Mar 05 2020

A331080 a(n) = Sum_{i=1..n} d_3(i)*d_3(i+1), where d_3(n) = A007425(n).

Original entry on oeis.org

3, 12, 30, 48, 75, 102, 132, 192, 246, 273, 327, 381, 408, 489, 624, 669, 723, 777, 831, 993, 1074, 1101, 1191, 1371, 1425, 1515, 1695, 1749, 1830, 1911, 1974, 2163, 2244, 2325, 2649, 2757, 2784, 2865, 3135, 3225, 3306, 3387, 3441, 3765, 3927, 3954, 4089, 4359, 4467, 4629, 4791, 4845, 4935, 5205, 5475
Offset: 1

Views

Author

N. J. A. Sloane, Jan 10 2020

Keywords

Comments

For background references see A330570.

Links

Crossrefs

Cf. A007425, A331081, A331130.

Programs

  • Mathematica
    f[p_, e_] := (e+1)*(e+2)/2;  s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{t = Array[s, 100]}, Accumulate[Most[t] * Rest[t]]] (* Amiram Eldar, Apr 19 2024 *)

Formula

a(n) = 3 * A331081(n). - Amiram Eldar, Apr 19 2024

A330572 a(n) = Sum_{k = 1..n} [u_2(k)*u_2(n+1-k)], where u_2(k) is the number of unordered factorizations k = i*j (A038548).

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 10, 12, 14, 19, 20, 24, 28, 31, 32, 40, 40, 48, 48, 56, 56, 67, 60, 77, 72, 85, 80, 98, 88, 108, 98, 117, 110, 131, 110, 147, 128, 149, 140, 169, 144, 182, 154, 192, 174, 205, 168, 228, 188, 226, 208, 250, 204, 268, 218, 273, 246, 285, 234, 324
Offset: 0

Views

Author

N. J. A. Sloane, Jan 08 2020

Keywords

Comments

An analog of A055507 for unordered factorizations.
For background references see A330570.

Links

Crossrefs

Cf. A000005, A038548, A055507.
See A330573 for another version.

Programs

  • Maple
    u2:= proc(n) option remember; if issqr(n) then (numtheory:-tau(n)+1)/2 else numtheory:-tau(n)/2 fi end proc:
f:= proc(n) local k; add(u2(k)*u2(n+1-k),k=1..n) end proc:
map(f, [$0..100]); # Robert Israel, Dec 05 2022
  • Mathematica
    s[n_] := s[n] = Ceiling[DivisorSigma[0, n] / 2]; a[n_] := Sum[s[k] * s[n+1-k], {k, 1, n}]; Array[a, 100, 0] (* Amiram Eldar, Apr 19 2024 *)

Extensions

Offset corrected by Robert Israel, Dec 05 2022

A330573 a(n) = Sum_{k = 1..ceiling(n/2)} [u_2(k)*u_2(n+1-k)], where u_2(k) is number of unordered factorization k = i*j (A038548).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 5, 8, 7, 10, 10, 14, 14, 16, 16, 22, 20, 26, 24, 30, 28, 34, 30, 43, 36, 43, 40, 51, 44, 56, 49, 63, 55, 66, 55, 78, 64, 75, 70, 89, 72, 93, 77, 98, 87, 103, 84, 122, 94, 115, 104, 127, 102, 136, 109, 141, 123, 143, 117, 170, 128, 153, 138, 174, 138, 183, 143, 183, 161, 189, 152, 224, 163, 200, 180
Offset: 0

Views

Author

N. J. A. Sloane, Jan 08 2020

Keywords

Comments

An analog of A055507 for unordered factorizations.
For background references see A330570.

Links

Crossrefs

Cf. A000005, A038548, A055507.
See A330572 for another version.

Programs

  • Mathematica
    s[n_] := s[n] = Ceiling[DivisorSigma[0, n]/2]; a[n_] := Sum[s[k]*s[n + 1 - k], {k, 1, Ceiling[n/2]}]; Array[a, 100, 0] (* Amiram Eldar, Apr 19 2024*)

Extensions

Offset and name corrected by Amiram Eldar, Apr 19 2024

A331073 a(n) = Sum_{k = 1..n} [d_3(k)*d_3(n+1-k)], where d_3 = A007425.

Original entry on oeis.org

0, 1, 6, 15, 30, 51, 72, 114, 128, 207, 204, 315, 294, 465, 402, 622, 510, 834, 636, 1023, 834, 1251, 918, 1602, 1074, 1767, 1392, 2117, 1428, 2523, 1620, 2781, 2088, 3003, 2000, 3882, 2238, 3729, 2886, 4350, 2688, 5055, 2928, 5313, 3768, 5343, 3420, 7032, 3678, 6264, 4800, 7347, 4230, 8278, 4584, 8394
Offset: 0

Views

Author

N. J. A. Sloane, Jan 10 2020

Keywords

Comments

For background references see A330570.

Links

Crossrefs

Cf. A007425.
See A331074 for another version.

Programs

  • Mathematica
    f[p_, e_] := (e+1)*(e+2)/2; s[1] = 1; s[n_] := s[n] = Times @@ f @@@ FactorInteger[n]; a[n_] := Sum[s[k] * s[n+1-k], {k, 1, n}]; Array[a, 100, 0] (* Amiram Eldar, Apr 19 2024 *)

Extensions

Offset corrected by Amiram Eldar, Apr 19 2024

A331074 a(n) = Sum_{k = 1..ceiling(n/2)} [d_3(k)*d_3(n+1-k)], where d_3 = A007425.

Original entry on oeis.org

0, 1, 3, 12, 15, 30, 36, 75, 64, 108, 102, 198, 147, 237, 201, 361, 255, 435, 318, 552, 417, 630, 459, 963, 537, 888, 696, 1099, 714, 1302, 810, 1503, 1044, 1506, 1000, 2103, 1119, 1869, 1443, 2337, 1344, 2568, 1464, 2697, 1884, 2676, 1710, 3966, 1839, 3150, 2400, 3714, 2115, 4189, 2292, 4359, 2943, 4098
Offset: 0

Views

Author

N. J. A. Sloane, Jan 10 2020

Keywords

Comments

For background references see A330570.

Links

Crossrefs

Cf. A007425.
See A331073 for another version.

Programs

  • Mathematica
    f[p_, e_] := (e+1)*(e+2)/2; s[1] = 1; s[n_] := s[n] = Times @@ f @@@ FactorInteger[n]; a[n_] := Sum[s[k] * s[n+1-k], {k, 1, Ceiling[n/2]}]; Array[a, 100, 0] (* Amiram Eldar, Apr 19 2024 *)

Extensions

Offset and name corrected by Amiram Eldar, Apr 19 2024

A331075 a(n) = Sum_{k = 1..n} [u_3(k)*u_3(n+1-k)], where u_3 = A034836.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 10, 12, 16, 21, 22, 28, 32, 37, 36, 49, 48, 58, 56, 74, 68, 85, 72, 104, 88, 113, 98, 138, 110, 146, 122, 170, 146, 179, 142, 220, 166, 213, 186, 254, 190, 266, 204, 300, 236, 293, 224, 368, 252, 334, 290, 402, 278, 413, 294, 450, 354, 429, 320, 547
Offset: 0

Views

Author

N. J. A. Sloane, Jan 10 2020

Keywords

Comments

For background references see A330570.

Links

Crossrefs

Cf. A034836.
See A331076 for another version.

Programs

  • Mathematica
    s[1] = 1; s[n_] := 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]; a[n_] := Sum[s[k] * s[n+1-k], {k, 1, n}]; Array[a, 100, 0] (* Amiram Eldar, Apr 19 2024 *)

Extensions

Offset corrected by Amiram Eldar, Apr 19 2024

A331076 a(n) = Sum_{k = 1..ceiling(n/2)} [u_3(k)*u_3(n+1-k)], where u_3 = A034836.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 5, 8, 8, 11, 11, 16, 16, 19, 18, 29, 24, 31, 28, 39, 34, 43, 36, 60, 44, 57, 49, 71, 55, 75, 61, 93, 73, 90, 71, 118, 83, 107, 93, 135, 95, 135, 102, 152, 118, 147, 112, 202, 126, 169, 145, 203, 139, 211, 147, 233, 177, 215, 160, 286
Offset: 0

Views

Author

N. J. A. Sloane, Jan 10 2020

Keywords

Comments

For background references see A330570.

Links

Crossrefs

Cf. A034836.
See A331075 for another version.

Programs

  • Mathematica
    s[1] = 1; s[n_] := 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]; a[n_] := Sum[s[k] * s[n+1-k], {k, 1, Ceiling[n/2]}]; Array[a, 100, 0] (* Amiram Eldar, Apr 19 2024 *)

Extensions

Offset and name corrected by Amiram Eldar, Apr 19 2024

A331081 a(n) = Sum_{i=1..n} d_3(i)*d_3(i+1)/3, where d_3(n) = A007425(n).

Original entry on oeis.org

1, 4, 10, 16, 25, 34, 44, 64, 82, 91, 109, 127, 136, 163, 208, 223, 241, 259, 277, 331, 358, 367, 397, 457, 475, 505, 565, 583, 610, 637, 658, 721, 748, 775, 883, 919, 928, 955, 1045, 1075, 1102, 1129, 1147, 1255, 1309, 1318, 1363, 1453, 1489, 1543, 1597, 1615, 1645, 1735, 1825, 1915, 1942, 1951, 2005, 2059
Offset: 1

Views

Author

N. J. A. Sloane, Jan 10 2020

Keywords

Comments

For background references see A330570.

Links

Crossrefs

Cf. A007425, A331080.

Programs

  • Mathematica
    f[p_, e_] := (e+1)*(e+2)/2;  s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{t = Array[s, 100]}, Accumulate[Most[t] * Rest[t]] / 3] (* Amiram Eldar, Apr 19 2024 *)

Formula

a(n) = A331080(n) / 3. - Amiram Eldar, Apr 19 2024

A331130 a(n) = Sum_{i=1..n} u_3(i)*u_3(i+1), where u_3(n) = A034836(n).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 13, 19, 23, 25, 29, 33, 35, 39, 47, 51, 55, 59, 63, 71, 75, 77, 83, 95, 99, 105, 117, 121, 126, 131, 136, 146, 150, 154, 170, 178, 180, 184, 196, 202, 207, 212, 216, 232, 240, 242, 251, 269, 277, 285, 293, 297, 303, 315, 327, 339, 343, 345, 355, 365
Offset: 1

Views

Author

N. J. A. Sloane, Jan 10 2020

Keywords

Comments

For background references see A330570.

Links

Crossrefs

Cf. A034836, A331080.

Programs

  • Mathematica
    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]; With[{t = Array[s, 100]}, Accumulate[Most[t] * Rest[t]]] (* Amiram Eldar, Apr 19 2024 *)
Showing 1-10 of 15 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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