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.

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A143443 a(n) = n * A002321(n).

Original entry on oeis.org

1, 0, -3, -4, -10, -6, -14, -16, -18, -10, -22, -24, -39, -28, -15, -16, -34, -36, -57, -60, -42, -22, -46, -48, -50, -26, -27, -28, -58, -90, -124, -128, -99, -68, -35, -36, -74, -38, 0, 0, -41, -84, -129, -132
Offset: 1

Views

Author

Gary W. Adamson, Aug 15 2008

Keywords

Comments

Equals row sums of triangle A143442.

Examples

			First four terms = (1, 0, -3, -4) = (1*1, 2*0, 3*(-1), 4*(-1)), where the Mertens function A002321 = (1, 0, -1, -1, -2, -1, -2, -2, -2,...)
a(5) = -10 = sum of row 5 terms of triangle A143442: (5 - 5 - 5 + 0 - 5).

Crossrefs

Cf. A002321, A143442.

Programs

  • Mathematica
    Table[n Plus @@ MoebiusMu[Range[n]], {n, 1, 80}] (* Carl Najafi, Aug 17 2011 *)
  • PARI
    a(n) = n*sum(k=1, n, moebius(k)); \\ Michel Marcus, Aug 22 2015
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A143443(n):
    if n == 0:
        return 0
    c, j = n, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A143443(k1)//k1
        j, k1 = j2, n//j2
    return n*(j-c) # Chai Wah Wu, Mar 30 2021

Extensions

More terms from Carl Najafi, Aug 17 2011

A144031 INVERT transform of A002321, Mertens's function.

Original entry on oeis.org

1, 1, 0, -2, -6, -10, -13, -10, 4, 36, 84, 137, 159, 94, -133, -573, -1197, -1788, -1864, -647, 2741, 8784, 16631, 22920, 20769, 87, -49372, -130497, -226511, -286165, -214344, 117678, 822398, 1889427, 3022590, 3465187, 1927286, -3188290, -13016609, -26739085
Offset: 1

Views

Author

Gary W. Adamson, Sep 07 2008

Keywords

Comments

Equals row sums of triangle A144032.

Examples

			Given Mertens's function A002321: (1, 0, -1, -1, -2, ...), apply the INVERT transform.
The first 3 terms of A144031 = (1, 1, 1, ...) which we apply to (-1, 0, 1) as a dot product = 0. (-1, 0, 1) = the first 3 terms of A002321 in reverse. [Comment not clear - A144031 can't be right.]

Crossrefs

Cf. A002321, A144032.

Extensions

More terms from Alois P. Heinz, May 23 2015

A162943 a(n) = 2^(1-A002321(n)).

Original entry on oeis.org

1, 2, 4, 4, 8, 4, 8, 8, 8, 4, 8, 8, 16, 8, 4, 4, 8, 8, 16, 16, 8, 4, 8, 8, 8, 4, 4, 4, 8, 16, 32, 32, 16, 8, 4, 4, 8, 4, 2, 2, 4, 8, 16, 16, 16, 8, 16, 16, 16, 16, 8, 8, 16, 16, 8, 8, 4, 2, 4, 4, 8, 4, 4, 4, 2, 4, 8, 8, 4, 8, 16, 16, 32, 16, 16, 16, 8
Offset: 1

Views

Author

Mats Granvik, Jul 18 2009

Keywords

Crossrefs

Cf. A002321, first column of A162944.

Programs

  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A162943(n):
    if n == 0:
        return 2
    c, j = n, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(4-len(bin(A162943(k1))))
        j, k1 = j2, n//j2
    return 2**(1+c-j) # Chai Wah Wu, Mar 30 2021

A171097 Solutions to the equation M(n) = -2 (M = Mertens's function A002321).

Original entry on oeis.org

5, 7, 8, 9, 11, 12, 14, 17, 18, 21, 23, 24, 25, 29, 34, 37, 42, 46, 51, 52, 55, 56, 61, 67, 68, 70, 77, 86, 89, 90, 103, 104, 106, 122, 127, 128, 130, 133, 137, 142, 154, 157, 170, 171, 172, 178, 209, 211, 212, 241, 242, 243, 244, 245, 247, 248, 251, 252, 257, 259, 260
Offset: 1

Views

Author

Neven Juric (neven.juric(AT)apis-it.hr), Sep 10 2010

Keywords

Links

Crossrefs

Cf. A002321.

Extensions

More terms from Seiichi Manyama, Jan 03 2019

A171098 Solutions to the equation M(n) = -3 (M = Mertens's function A002321).

Original entry on oeis.org

13, 19, 20, 30, 33, 43, 44, 45, 47, 48, 49, 50, 53, 54, 71, 72, 74, 75, 76, 78, 82, 85, 105, 107, 108, 119, 120, 121, 131, 132, 138, 141, 173, 177, 179, 180, 185, 187, 188, 189, 206, 207, 208, 246, 258, 271, 272, 274, 275, 276, 278, 279, 280, 302, 311, 312, 314, 315
Offset: 1

Views

Author

Neven Juric (neven.juric(AT)apis-it.hr), Sep 10 2010

Keywords

Links

Crossrefs

Cf. A002321.

Extensions

More terms from Seiichi Manyama, Jan 03 2019

A171235 Solutions to the equation M(n) = -4 (M = Mertens's function A002321).

Original entry on oeis.org

31, 32, 73, 79, 80, 81, 83, 84, 109, 111, 112, 118, 139, 140, 174, 175, 176, 181, 183, 184, 186, 190, 205, 273, 277, 281, 301, 313, 317, 319, 320, 322, 433, 454, 482, 485, 486, 489, 490, 493, 497, 502, 505, 511, 512, 513, 618, 622, 643, 644, 699, 700, 703, 704, 706
Offset: 1

Views

Author

Neven Juric (neven.juric(AT)apis-it.hr), Sep 03 2010

Keywords

Links

Crossrefs

Cf. A002321.

Extensions

More terms from Seiichi Manyama, Jan 04 2019

A171236 Solutions to the equation M(n) = -5 (M = Mertens's function A002321).

Original entry on oeis.org

110, 113, 115, 116, 117, 182, 191, 192, 194, 203, 204, 282, 299, 300, 318, 434, 437, 453, 455, 456, 458, 459, 460, 462, 473, 478, 481, 483, 484, 487, 488, 491, 492, 494, 495, 496, 498, 501, 503, 504, 506, 507, 508, 510, 619, 620, 621, 645, 698, 701, 702, 705, 710, 711, 712, 743
Offset: 1

Views

Author

Neven Juric (neven.juric(AT)apis-it.hr), Sep 28 2010

Keywords

Links

Crossrefs

Cf. A002321.

Extensions

More terms from Seiichi Manyama, Jan 04 2019

A171374 Solutions to the equation M(n) = -6 (M = Mertens's function A002321).

Original entry on oeis.org

114, 193, 195, 196, 202, 283, 284, 298, 435, 436, 438, 447, 448, 451, 452, 457, 461, 463, 464, 466, 469, 471, 472, 474, 475, 476, 477, 479, 480, 499, 500, 509, 646, 649, 650, 697, 1033, 1037, 1042, 1045, 1051, 1052, 1053, 1055, 1056, 1063, 1064, 1081, 1085, 1145, 1147
Offset: 1

Views

Author

Neven Juric (neven.juric(AT)apis-it.hr), Sep 28 2010

Keywords

Links

Crossrefs

Cf. A002321.

Programs

  • Mathematica
    Position[Accumulate[MoebiusMu[Range[1200]]], -6]//Flatten (* Vincenzo Librandi, Jan 05 2019 *)

Extensions

More terms from Seiichi Manyama, Jan 05 2019

A171383 Solutions to the equation M(n) = -7 (M = Mertens's function A002321).

Original entry on oeis.org

197, 198, 201, 285, 287, 288, 289, 291, 292, 295, 296, 297, 439, 440, 441, 446, 449, 450, 465, 467, 468, 470, 647, 648, 651, 652, 695, 696, 1034, 1035, 1036, 1038, 1041, 1054, 1065, 1067, 1068, 1079, 1080, 1086, 1142, 1143, 1144, 1146, 1153, 1182, 1183, 1184, 1186, 1189
Offset: 1

Views

Author

Neven Juric (neven.juric(AT)apis-it.hr), Sep 28 2010

Keywords

Links

Crossrefs

Cf. A002321.

Programs

  • Mathematica
    Position[Accumulate[MoebiusMu[Range[1200]]], -7]//Flatten (* Vincenzo Librandi, Jan 05 2019 *)

Extensions

More terms from Seiichi Manyama, Jan 05 2019

A171391 Solutions to the equation M(n) = -8 (M = Mertens's function A002321).

Original entry on oeis.org

199, 200, 286, 290, 293, 294, 442, 445, 653, 655, 656, 657, 690, 694, 1039, 1040, 1066, 1069, 1073, 1077, 1078, 1087, 1088, 1089, 1141, 1185, 1187, 1188, 1601, 1602, 1605, 1691, 1692, 1718, 1719, 1720, 1722, 1726, 1729, 1731, 1732, 1735, 1736, 1737, 1739, 1740, 1765, 1767, 1768, 1778
Offset: 1

Views

Author

Neven Juric (neven.juric(AT)apis-it.hr), Sep 28 2010

Keywords

Links

Crossrefs

Cf. A002321.

Extensions

More terms from Seiichi Manyama, Jan 05 2019
Previous Showing 21-30 of 152 results. Next

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