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 1-5 of 5 results.

A024556 Odd squarefree composite numbers.

Original entry on oeis.org

15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 105, 111, 115, 119, 123, 129, 133, 141, 143, 145, 155, 159, 161, 165, 177, 183, 185, 187, 195, 201, 203, 205, 209, 213, 215, 217, 219, 221, 231, 235, 237, 247, 249, 253, 255, 259, 265, 267, 273
Offset: 1

Views

Author

N. J. A. Sloane, May 22 2000

Keywords

Comments

Composite numbers n such that Sum_{k=1..n-1} floor(k^3/n) = (1/4)*(n-2)*(n^2-1) (equality also holds for all primes). - Benoit Cloitre, Dec 08 2002

Links

Crossrefs

Intersection of A056911 and A071904.
Subsequence of A061346.
Cf. A010051, A046388, A078837.

Programs

  • Haskell
    a024556 n = a024556_list !! (n-1)
a024556_list = filter ((== 0) . a010051) $ tail a056911_list
-- Reinhard Zumkeller, Apr 12 2012
  • Mathematica
    Complement[Select[Range[3,281,2],SquareFreeQ],Prime[Range[PrimePi[281]]]] (* Harvey P. Dale, Jan 26 2011 *)
  • PARI
    is(n)=n>1&&n%2&&!isprime(n)&&issquarefree(n) \\ Charles R Greathouse IV, Apr 12 2012
  • PARI
    forstep(n=3,273,2,k=omega(n);if(k>1&&bigomega(n)==k,print1(n,", "))) \\ Hugo Pfoertner, Dec 19 2018

Formula

a(n) = (Pi^2/4)*n + O(n/log n). - Charles R Greathouse IV, Mar 12 2025

Extensions

More terms from James Sellers, May 22 2000

A053850 Odd numbers divisible by a square > 1.

Original entry on oeis.org

9, 25, 27, 45, 49, 63, 75, 81, 99, 117, 121, 125, 135, 147, 153, 169, 171, 175, 189, 207, 225, 243, 245, 261, 275, 279, 289, 297, 315, 325, 333, 343, 351, 361, 363, 369, 375, 387, 405, 423, 425, 441, 459, 475, 477, 495, 507, 513, 525, 529, 531, 539, 549, 567
Offset: 1

Views

Author

Enoch Haga, Mar 28 2000

Keywords

Comments

Odd n such that sum(k=1,n-1,floor(k^3/n)) is different from (1/4)*(n-2)*(n^2-1) (equality holds for n prime as well as for "1" union "A024556" ). - Benoit Cloitre, Dec 08 2002
Odd nonsquarefree numbers, odd terms of A013929. - Zak Seidov, Aug 16 2006
The asymptotic density of this sequence is 1/2 - 4/Pi^2 = 0.094715... - Amiram Eldar, Nov 21 2020

Links

Crossrefs

Cf. A078837, A013929, A024556.

Programs

  • Mathematica
    Select[Range[1, 500, 2], !SquareFreeQ[#] &] (* Amiram Eldar, Nov 21 2020 *)
  • PARI
    lista(nn) = {forstep(n=1, nn, 2, if (! issquarefree(n), print1(n, ", ")));} \\ Michel Marcus, Jun 06 2014

A361559 a(n) = Sum_{k=1..prime(n)-1} floor(k^5/prime(n)).

Original entry on oeis.org

0, 10, 258, 1740, 20070, 48510, 196920, 350370, 937860, 3075030, 4322160, 10641330, 17925180, 22825110, 35827560, 65816010, 113180910, 133937670, 215070570, 288148140, 331474860, 493573080, 633015810, 899599140, 1387338960, 1700082450, 1876303260, 2272556790, 2494333710
Offset: 1

Views

Author

Michel Marcus, Mar 15 2023

Keywords

Links

Crossrefs

Cf. A078837 (for k^3).

Programs

  • Maple
    a:= n-> (p-> (p-2)*(p-1)*(p+1)*(2*p^2-2*p+3)/12)(ithprime(n)):
seq(a(n), n=1..29);  # Alois P. Heinz, Mar 15 2023
  • PARI
    a(n) = my(p=prime(n)); sum(k=1, p-1, k^5\p);

Formula

a(n) = (p-2)*(p-1)*(p+1)*(2*p^2-2*p+3)/12 where p=prime(n).

A374915 a(n) = (n - 1) * (n - 2) * sigma(n).

Original entry on oeis.org

0, 0, 8, 42, 72, 240, 240, 630, 728, 1296, 1080, 3080, 1848, 3744, 4368, 6510, 4320, 10608, 6120, 14364, 12160, 15120, 11088, 30360, 17112, 25200, 26000, 39312, 22680, 58464, 27840, 58590, 47616, 57024, 53856, 108290, 47880, 79920, 78736, 133380, 65520, 157440
Offset: 1

Views

Author

Seiichi Manyama, Jul 23 2024

Keywords

Links

Crossrefs

Cf. A000203, A002127, A002378, A060043, A078837.

Programs

  • Mathematica
    a[n_]:= (n - 1) * (n - 2) * DivisorSigma[1,n]; Array[a,42] (* Stefano Spezia, Jul 23 2024 *)
  • PARI
    a(n) = (n-1)*(n-2)*sigma(n);

Formula

a(n) = A002378(n-2) * A000203(n).
a(n) >= 8 * A002127(n) and the equal sign only holds if n is 1 or prime.

A380970 a(n) = Sum_{k=1..p-1} floor(k^p/p) where p is prime(n).

Original entry on oeis.org

0, 2, 258, 53820, 12942210870, 11901444483390, 25627001801054931000, 55413915436873048932450, 490667517005738962388828685972, 48588952813858892791005036793649985985110, 303307728036900627681487165427498812641117360, 158544898951978777519612048992784361843596346824881328530
Offset: 1

Views

Author

Michel Marcus, Feb 09 2025

Keywords

Links

Crossrefs

Cf. A078837, A361559.

Programs

  • PARI
    a(n) = my(p=prime(n)); sum(k=1, p-1, k^p\p);

Formula

For n>1, a(n) mod p = (p+1)/2 where p is prime(n).
Showing 1-5 of 5 results.

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

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