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 21 results. Next

A152911 Records in A152770.

Original entry on oeis.org

0, 1, 3, 4, 5, 11, 16, 17, 29, 35, 47, 67, 97, 112, 129, 133, 145, 161, 225, 245, 297, 349, 351, 369, 485, 514, 551, 637, 787, 901, 1009, 1033, 1117, 1237, 1333, 1669, 2009, 2061, 2187, 2489, 2615, 3073, 3439, 3449, 4233, 4537, 4809, 5241, 5269, 5377, 6793
Offset: 1

Views

Author

Omar E. Pol, Dec 15 2008

Keywords

Links

Crossrefs

Cf. A152770.

Programs

  • Mathematica
    f[n_] := DivisorSigma[1, n] - DivisorSigma[0, n] - n + 1; t = Array[f, 10000]; lst = {}; a = -1; Do[ If[ t[[n]] > a, a = t[[n]]; AppendTo[lst, a]], {n, 10000}]; lst (* Robert G. Wilson v, Dec 21 2008 *)

Extensions

Extended by Robert G. Wilson v, Dec 21 2008

A152967 Partial sums of A152770.

Original entry on oeis.org

0, 0, 0, 1, 1, 4, 4, 8, 10, 15, 15, 26, 26, 33, 39, 50, 50, 66, 66, 83, 91, 102, 102, 131, 135, 148, 158, 181, 181, 216, 216, 242, 254, 271, 281, 328, 328, 347, 361, 404, 404, 451, 451, 486, 514, 537, 537, 604, 610, 648, 666, 707, 707, 766, 780, 837, 857, 886, 886, 983, 983
Offset: 1

Views

Author

Omar E. Pol, Dec 22 2008

Keywords

Links

Crossrefs

Cf. A152770.

Programs

  • Mathematica
    Accumulate[Array[DivisorSigma[1,#]-DivisorSigma[0,#]-#+1&,70]] (* Harvey P. Dale, Dec 27 2011 *)

Formula

a(n) = Sum_{i=1..n} (i-1)*floor((n-i)/i). - Wesley Ivan Hurt, Sep 13 2017

A152771 a(n) = sigma(n) - 2*d(n) + 1.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 5, 8, 8, 11, 9, 17, 11, 17, 17, 22, 15, 28, 17, 31, 25, 29, 21, 45, 26, 35, 33, 45, 27, 57, 29, 52, 41, 47, 41, 74, 35, 53, 49, 75, 39, 81, 41, 73, 67, 65, 45, 105, 52, 82, 65, 87, 51, 105, 65, 105, 73, 83, 57, 145
Offset: 1

Views

Author

Omar E. Pol, Dec 14 2008

Keywords

Links

Crossrefs

Cf. A000005, A000203, A152770, A152772.

Programs

  • Mathematica
    Table[DivisorSigma[1, n] - 2 DivisorSigma[0, n] + 1, {n, 60}] (* Ivan Neretin, Sep 30 2017 *)
  • PARI
    a(n) = sigma(n) - 2*numdiv(n) + 1; \\ Michel Marcus, Sep 30 2017

Formula

a(n) = A000203(n) - 2*A000005(n) + 1 = A000203(n) - A114003(n) = A088580(n) - A062011(n). - Omar E. Pol, Sep 30 2017
G.f.: Sum_{k>=1} x^(3*k) / (1 - x^k)^2. - Ilya Gutkovskiy, Apr 24 2021

A152864 Deficiency of n, plus the number of proper divisors of n: a(n) = 2n - sigma(n) + d(n) - 1.

Original entry on oeis.org

1, 2, 3, 3, 5, 3, 7, 4, 7, 5, 11, 1, 13, 7, 9, 5, 17, 2, 19, 3, 13, 11, 23, -5, 21, 13, 17, 5, 29, -5, 31, 6, 21, 17, 25, -11, 37, 19, 25, -3, 41, -5, 43, 9, 17, 23, 47, -19, 43, 12, 33, 11, 53, -5, 41, -1, 37, 29, 59, -37, 61, 31
Offset: 1

Views

Author

Omar E. Pol, Dec 14 2008

Keywords

Links

Crossrefs

Cf. A000005, A000203, A005843, A032741, A033879, A152770, A152771, A152772.

Programs

  • Mathematica
    a[n_] := 2*n - Differences[DivisorSigma[{0, 1}, n]][[1]] - 1; Array[a, 100] (* Amiram Eldar, Apr 07 2024 *)
  • PARI
    a(n) = 2*n - sigma(n) + numdiv(n) - 1; \\ Amiram Eldar, Apr 07 2024

Formula

a(n) = A005843(n)-A000203(n)+A000005(n)-1 = A033879(n)+A032741(n).

A152988 Sum of proper divisors minus the number of proper divisors of Catalan number A000108(n).

Original entry on oeis.org

0, 0, 0, 0, 7, 47, 193, 236, 1579, 4195, 18461, 62143, 275781, 1131909, 7434169, 10522660, 72469339, 268486155, 1442237845, 4284330539, 18146555293, 62021099893, 248289236937, 798007352239, 2832660377605, 11922780595861
Offset: 0

Views

Author

Omar E. Pol, Dec 20 2008, Jan 07 2009

Keywords

Examples

			a(5)=47 because A000108(5)=42 has 7 proper divisors: 1,2,3,6,7,14,21 and 1+2+3+6+7+14+21-7 = 47. - _Emeric Deutsch_, Dec 31 2008

Links

Crossrefs

Cf. A000005, A000108, A000203, A001065, A032741, A152770.

Programs

  • Maple
    with(numtheory): seq(sigma(binomial(2*n, n)/(n+1))-binomial(2*n, n)/(n+1)-tau(binomial(2*n, n)/(n+1))+1, n = 1 .. 27); # Emeric Deutsch, Dec 31 2008
  • Mathematica
    diff[n_] := DivisorSigma[1, n] - DivisorSigma[0, n] - n + 1; Table[diff[ CatalanNumber[n] ], {n, 0, 25}] (* Amiram Eldar, Dec 01 2019 *)

Formula

a(n) = A001065(A000108(n)) - A032741(A000108(n)) = A152770(A000108(n)).

Extensions

Extended by Emeric Deutsch, Dec 31 2008

A152990 Sum of proper divisors minus the number of proper divisors of Fibonacci number A000045(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 8, 17, 14, 0, 245, 0, 40, 499, 542, 0, 2801, 148, 5316, 6771, 286, 0, 110809, 18032, 752, 124327, 155934, 0, 1310617, 2972, 1213164, 1821955, 5166, 2697336, 33280689, 506376, 1416024, 32030851, 106878198, 62156, 295708841, 0
Offset: 1

Views

Author

Omar E. Pol, Dec 20 2008

Keywords

Comments

Note that if a(n) != 0 then Fibonacci number A000045(n) is a composite number (A002808), otherwise A000045(n) is a noncomposite number (A008578). See A152770.

Examples

			a(8)=8 because Fibonacci(8)=21, the proper divisors of 21 are 1,3 and 7; consequently, a(8) = 1 + 3 + 7 - 3 = 8. - _Emeric Deutsch_, Jan 02 2009

Links

Crossrefs

Cf. A000005, A000045, A000203, A001065, A002808, A008578, A032741, A152770.

Programs

  • Magma
    [DivisorSigma(1,f)-f-DivisorSigma(0,f)+1 where f is Fibonacci(n):n in [1..43] ]; // Marius A. Burtea, Feb 18 2020
  • Maple
    with(combinat): with(numtheory): seq(sigma(fibonacci(n))-fibonacci(n)-tau(fibonacci(n))+1, n = 1 .. 45); # Emeric Deutsch, Jan 02 2009

Formula

a(n) = A000203(A000045(n)) - A000005(A000045(n)) - n + 1 = A001065(A000045(n)) - A032741(A000045(n)) = A152770(A000045(n)).

Extensions

Extended by Emeric Deutsch, Jan 02 2009
a(79)-a(80) in b-file corrected by Georg Fischer, Feb 18 2020

A152991 a(n) = sigma(n) - pi(n).

Original entry on oeis.org

1, 2, 2, 5, 3, 9, 4, 11, 9, 14, 7, 23, 8, 18, 18, 25, 11, 32, 12, 34, 24, 28, 15, 51, 22, 33, 31, 47, 20, 62, 21, 52, 37, 43, 37, 80, 26, 48, 44, 78, 29, 83, 30, 70, 64, 58, 33, 109, 42, 78, 57, 83, 38, 104, 56, 104, 64, 74, 43, 151, 44, 78, 86, 109, 66, 126, 49, 107, 77, 125, 52
Offset: 1

Views

Author

Omar E. Pol, Dec 19 2008

Keywords

Comments

a(n) = A000203(n) - A000720(n). - Omar E. Pol, Dec 21 2008

Examples

			a(15) = 24 - 6 = 18 because the sum of the divisors of 15 is 1 + 3 + 5 + 15 = 24 and there are 6 primes not exceeding 15 (2, 3, 5, 7, 11 and 13). - _Emeric Deutsch_, Dec 29 2008

Links

Crossrefs

Cf. A000203, A000720, A152770.

Programs

  • Maple
    with(numtheory): seq(sigma(n)-pi(n), n = 1 .. 80); # Emeric Deutsch, Dec 29 2008
  • Mathematica
    Table[DivisorSigma[1,n]-PrimePi[n],{n,80}] (* Harvey P. Dale, Oct 20 2021 *)
  • PARI
    a(n) = sigma(n) - primepi(n); \\ Michel Marcus, Jun 18 2019

Extensions

Corrected and extended by Emeric Deutsch, Dec 29 2008

A152757 Numbers k such that the deficiency of k plus the number of proper divisors of k is a prime number (see A152864).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 31, 34, 37, 38, 41, 43, 45, 46, 47, 49, 52, 53, 55, 57, 58, 59, 61, 62, 64, 67, 70, 71, 73, 74, 75, 76, 77, 79, 82, 83, 86, 88, 89, 91, 93, 94, 95, 97, 99, 101, 103, 104, 106, 107, 109, 110, 111
Offset: 1

Views

Author

Omar E. Pol, Dec 14 2008

Keywords

Links

Crossrefs

Cf. A000005, A000203, A033879, A032741, A152770, A152864.

Programs

  • Maple
    with(numtheory): a := proc (n) if isprime(2*n-sigma(n)+tau(n)-1) = true then n else end if end proc: seq(a(n), n = 1 .. 120); # Emeric Deutsch, Jan 08 2009

Extensions

Extended by Emeric Deutsch, Jan 08 2009
Name edited by Jon E. Schoenfield, Jan 06 2019

A152772 a(n) = sigma(n) - 3*d(n) + 3.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 5, 6, 7, 9, 9, 13, 11, 15, 15, 19, 15, 24, 17, 27, 23, 27, 21, 39, 25, 33, 31, 41, 27, 51, 29, 48, 39, 45, 39, 67, 35, 51, 47, 69, 39, 75, 41, 69, 63, 63, 45, 97, 51, 78, 63, 83, 51, 99, 63, 99, 71, 81, 57, 135
Offset: 1

Views

Author

Omar E. Pol, Dec 14 2008

Keywords

Crossrefs

Cf. A000005 (d), A000203 (sigma), A152770, A152771.

Programs

  • Mathematica
    a[n_] := DivisorSigma[1, n] - 3 * DivisorSigma[0, n] + 3; Array[a, 100] (* Amiram Eldar, Apr 07 2024 *)
  • PARI
    a(n) = sigma(n) - 3 * numdiv(n) + 3; \\ Amiram Eldar, Apr 07 2024

A152989 Sum of proper divisors minus the number of proper divisors of triangular number A000217(n).

Original entry on oeis.org

0, 0, 3, 5, 6, 8, 23, 47, 28, 14, 71, 83, 18, 80, 225, 127, 76, 84, 163, 351, 146, 32, 385, 551, 104, 202, 567, 307, 278, 296, 487, 941, 296, 262, 1219, 805, 54, 372, 1549, 933, 476, 498, 631, 1795, 826, 68, 1737, 2221, 534, 946, 1683, 883, 722, 1380, 2469, 2861, 740, 86, 2535
Offset: 1

Views

Author

Omar E. Pol, Dec 20 2008

Keywords

Links

Crossrefs

Cf. A000005, A000203, A000217, A001065, A032741, A152770.

Programs

  • Mathematica
    DivisorSigma[1,#]-DivisorSigma[0,#]-#+1&/@Accumulate[Range[60]] (* Harvey P. Dale, May 06 2014 *)
  • PARI
    a(n) = sigma(n*(n+1)/2) - n*(n+1)/2 - (numdiv(n*(n+1)/2) - 1); \\ Michel Marcus, Jan 28 2014

Formula

a(n) = A001065(A000217(n)) - A032741(A000217(n)) = A152770(A000217(n)).
Showing 1-10 of 21 results. Next

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