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 1-4 of 4 results.

A075383 Rearrangement of natural numbers so that next n numbers are each divisible by n.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 8, 12, 16, 20, 5, 10, 15, 25, 30, 18, 24, 36, 42, 48, 54, 7, 14, 21, 28, 35, 49, 56, 32, 40, 64, 72, 80, 88, 96, 104, 27, 45, 63, 81, 90, 99, 108, 117, 126, 50, 60, 70, 100, 110, 120, 130, 140, 150, 160, 11, 22, 33, 44, 55, 66, 77, 121, 132, 143, 154, 84
Offset: 1

Views

Author

Amarnath Murthy, Sep 22 2002

Keywords

Comments

Integer permutation with inverse A096780: a(A096780(n))=A096780(a(n))=n; A096781(n) = a(a(n)). - Reinhard Zumkeller, Jul 09 2004
Primes occur in natural order: a(A072205(n)) = A000040(n). - Reinhard Zumkeller, Jun 22 2009
n = greatest common divisor of row n. - Reinhard Zumkeller, Nov 30 2015

Examples

			   1;
   2,  4;
   3,  6,  9;
   8, 12, 16, 20;
   5, 10, 15, 25, 30;
  18, 24, 36, 42, 48, 54;
   7, 14, 21, 28, 35, 49, 56;
  32, ...

Links

Crossrefs

Cf. A075384, A075385, A075386, A075387, A075388, A075389, A075390.
Cf. A000040, A072205, A096780, A096781, A263896 (central terms).

Programs

  • Haskell
    import Data.List ((\\))
a075383 n k = a075383_tabl !! (n-1) !! (k-1)
a075383_row n = a075383_tabl !! (n-1)
a075383_tabl = f 1 [1..] where
   f x zs = ys : f (x + 1) (zs \\ ys) where
            ys = take x $ filter ((== 0) . (`mod` x)) zs
a075383_list = concat a075383_tabl
-- Reinhard Zumkeller, Nov 30 2015
  • Mathematica
    row[1] = {1}; row[n_] := row[n] = (For[rows = Join[row /@ Range[n-1]]; ro = {}; k = n, Length[ro] < n, k = k+n, If[FreeQ[rows, k], AppendTo[ro, k]]]; ro); Array[row, 12] // Flatten (* Jean-François Alcover, Apr 28 2017 *)

Extensions

More terms from Sascha Kurz, Jan 28 2003

A071988 Triple Peano sequence: a list of triples (x,y,z) starting at (1,1,1); then x'=x+1, y'=y+x, z'=z+y, for x only ranging over the primes.

Original entry on oeis.org

2, 2, 2, 3, 4, 4, 5, 11, 15, 7, 22, 42, 11, 56, 176, 13, 79, 299, 17, 137, 697, 19, 172, 988, 23, 254, 1794, 29, 407, 3683, 31, 466, 4526, 37, 667, 7807, 41, 821, 10701, 43, 904, 12384, 47, 1082, 16262, 53, 1379, 23479, 59, 1712, 32568, 61, 1831, 36051, 67, 2212
Offset: 1

Views

Author

Roger L. Bagula, Jun 17 2002

Keywords

Comments

a(3k+1) are the primes (A000040), by definition.
a(3k+2) are A072205. Second terms are (n^2+n+2)/2 by induction (for n prime).
a(3k) are A072206. Third terms are (n^3+5*n+6)/6 by induction (for n prime).

Examples

			x'=x+1=1+1=2, y'=y+x=1+1=2, z'=z+y=1+1=2.

Crossrefs

Cf. A072205 & A072206.

Programs

  • Mathematica
    seq[n_Integer?Positive] := Module[{fn01 = 1, fn10 = 1, fnout = 1}, Do[{fn10, fn01, fnout} = {fn10 + 1, fn01 + fn10, fn01 + fnout}, {n - 1}]; {fn10, fn01, fnout}]; Flatten[ Table[ seq[ Prime[n]], {n, 1, 100}]]
  • PARI
    a(n)=subst([x,x*(x-1)/2+1,(x^3-3*x^2+8*x)/6],x, prime(1+(n-1)\3))[1+(n-1)%3]

Extensions

Edited by Robert G. Wilson v, Jul 03 2002

A279911 a(n) = Sum_{i=1..n} denominator(n^i/i).

Original entry on oeis.org

0, 1, 2, 4, 6, 11, 10, 22, 22, 31, 28, 56, 36, 79, 58, 72, 86, 137, 80, 172, 112, 145, 148, 254, 146, 261, 208, 274, 230, 407, 182, 466, 342, 375, 360, 448, 322, 667, 456, 528, 444, 821, 384, 904, 592, 635, 676, 1082, 574, 1051, 692, 924, 836, 1379, 732, 1154, 912, 1153
Offset: 0

Views

Author

Wesley Ivan Hurt, Dec 22 2016

Keywords

Links

Crossrefs

Cf. A279912.

Programs

  • Magma
    [0] cat [&+[Denominator(n^i/i):i in [1..n]]:n in [1..60]]; // Marius A. Burtea, Jul 29 2019
  • Maple
    A279911:=n->add(denom(n^i/i), i=1..n): seq(A279911(n), n=0..100);
  • PARI
    a(n) = sum(k=1, n, if(gcd(n,k) == 1, k, denominator(n^k/k))); \\ Daniel Suteu, Jul 28 2019
  • PARI
    a(n) = sum(k=1, n, if(gcd(n,k) == 1, k, vecmax(select(d->gcd(d, n) == 1, divisors(k))))); \\ Daniel Suteu, Jul 28 2019
  • PARI
    a(n) = my(f=factor(n)[,1]); sum(k=1, n, if(gcd(n, k) == 1, k, gcd(vector(#f, j, k / f[j]^valuation(k, f[j]))))); \\ Daniel Suteu, Jul 29 2019

Formula

From Daniel Suteu, Jul 28 2019: (Start)
a(prime(n)) = A072205(n).
a(p^k) = (p^(2*k+1) + p + 2) / (2*(p+1)), for prime powers p^k.
a(n) = Sum_{k=1..n} gcd(m, k), where m = A095996(n).
a(n) = Sum_{k=1..n} f(n,k), where f(n,k) is the largest divisor d of k for which gcd(d, n) = 1. (End)
a(n) = Sum_{1<=k<=n, gcd(n,k)=1} phi(k)*floor(n/k). - Ridouane Oudra, May 24 2023

A320440 Row sums of A225043.

Original entry on oeis.org

0, 2, 4, 8, 11, 20, 22, 32, 31, 52, 56, 80, 79, 100, 94, 128, 137, 176, 172, 208, 193, 244, 254, 320, 266, 340, 283, 400, 407, 332, 466, 512, 499, 580, 569, 680, 667, 724, 745, 848, 821, 872, 904, 976, 1021, 1060, 1082, 1280, 1093, 1312, 1330, 1360, 1379, 1472, 1479, 1584, 1543
Offset: 0

Views

Author

Nathan M Epstein, Jan 09 2019

Keywords

Links

Crossrefs

Cf. A000040, A062173, A072205, A225043.

Programs

  • Mathematica
    a[n_]:=Sum[Mod[Binomial[n, k],n+1], {k,0, n}]; Array[a, 100, 0] (* Stefano Spezia, Jan 09 2019 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n,k) % (n+1)); \\ Michel Marcus, Jan 09 2019

Formula

a(A000040(n)) = A072205(n) for n > 0.
Showing 1-4 of 4 results.

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