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 1-6 of 6 results.

A075392 Duplicate of A069901.

Original entry on oeis.org

1, 3, 2, 2, 3, 3, 2, 2, 3, 5, 2, 2, 7, 3, 2, 2, 3, 3, 2, 2, 3, 11, 2, 2, 5, 3, 2, 2, 3, 3, 2, 2, 3, 5, 2
Offset: 1

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Author

Keywords

A069904 Number of prime factors of n-th triangular number (with multiplicity).

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 4, 3, 2, 3, 3, 2, 3, 5, 4, 3, 3, 3, 4, 3, 2, 4, 5, 3, 4, 5, 3, 3, 3, 5, 6, 3, 3, 5, 4, 2, 3, 5, 4, 3, 3, 3, 5, 4, 2, 5, 6, 4, 4, 4, 3, 4, 5, 5, 5, 3, 2, 4, 4, 2, 4, 8, 7, 4, 3, 3, 4, 4, 3, 5, 5, 2, 4, 5, 4, 4, 3, 5, 8, 5, 2, 4, 5, 3, 3, 5, 4, 4, 5
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 10 2002

Keywords

Examples

			A000217(8) = 8*(8+1)/2 = 36 = 2*2*3*3, therefore a(8) = 4.

Links

Crossrefs

Cf. A069901, A069902, A069903, A092405, A101745, A108815, A114435, A114436, A114437, A164977, A240527, A240528, A240529, A278253.

Programs

Formula

a(n) = A001222(A000217(n)).
From Antti Karttunen, Oct 07 2017: (Start)
a(n) = (A001222(n)+A001222(n+1))-1.
a(n) = A001222(A278253(n)). (End)
From Alois P. Heinz, Aug 05 2019: (Start)
a(n) = 2 <=> n in { A164977 }.
a(n) = 3 <=> n in { A108815 }.
a(n) = 4 <=> n in { A114435 }.
a(n) = 5 <=> n in { A114436 }.
a(n) = 6 <=> n in { A114437 }.
a(n) = 7 <=> n in { A240527 }.
a(n) = 8 <=> n in { A240528 }.
a(n) = 9 <=> n in { A240529 }.
a(n) = 10 <=> n im { A101745 }. (End)

A069902 Largest prime factor of n(n+1)/2, the n-th triangular number.

Original entry on oeis.org

1, 3, 3, 5, 5, 7, 7, 3, 5, 11, 11, 13, 13, 7, 5, 17, 17, 19, 19, 7, 11, 23, 23, 5, 13, 13, 7, 29, 29, 31, 31, 11, 17, 17, 7, 37, 37, 19, 13, 41, 41, 43, 43, 11, 23, 47, 47, 7, 7, 17, 17, 53, 53, 11, 11, 19, 29, 59, 59, 61, 61, 31, 7, 13, 13, 67
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 10 2002

Keywords

Comments

Essentially the same as A074399, which has many comments, references and links.

Examples

			A000217(9) = 9*(9+1)/2 = 45 = 3*3*5, therefore a(9) = 5.

Crossrefs

Related properties of triangular numbers: A069901, A069903, A069904.
Cf. A000217, A006530, A074399.

Programs

  • Mathematica
    PrimeFactors[n_]:=Flatten[Table[ #[[1]],{1}]&/@FactorInteger[n]]; Table[PrimeFactors[n*(n-1)/2][[ -1]],{n,2,6!}] (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
(* Second program: *)
Array[FactorInteger[PolygonalNumber[#]][[-1, 1]] &, 66] (* Michael De Vlieger, Sep 14 2023 *)
  • PARI
    \\ written for a(n), n >= 2
a(n)=vecmax(factor(n*(n+1)/2)[,1]) \\ M. F. Hasler, May 02 2015

Formula

a(n) = A006530(A000217(n)).

Extensions

Edited by Peter Munn, Sep 14 2023

A069903 Number of distinct prime factors of n-th triangular number.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 4, 3, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 4, 3, 2, 3, 4, 3, 3, 3, 3, 4, 3, 2, 3, 3, 2, 3, 4, 3, 2, 3, 4, 4, 3, 2, 4, 4, 2, 3, 3, 3, 4, 3, 3, 4, 4, 3, 3, 3, 2, 3, 4, 4, 4, 3, 3, 3, 2, 2, 4, 5, 3, 3, 4, 3, 3, 4
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 10 2002

Keywords

Examples

			A000217(11) = 11*(11+1)/2 = 66 = 2*3*11, therefore a(11) = 3.

Links

Crossrefs

Cf. A000217, A001221, A059957, A069901, A069902, A069904, A077761.

Programs

Formula

a(n) = A001221(A000217(n)).
Sum_{k=1..n} a(k) = 2 * n * (log(log(n)) + B - 1/4) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 21 2024

A075753 Smallest prime factor of n-th odd triangular number; a(1) = 1.

Original entry on oeis.org

1, 3, 3, 3, 3, 5, 7, 3, 3, 3, 3, 11, 5, 3, 3, 3, 3, 5, 19, 3, 3, 3, 3, 23, 5, 3, 3, 3, 3, 29, 31, 3, 3, 3, 3, 5, 37, 3, 3, 3, 3, 41, 5, 3, 3, 3, 3, 5, 7, 3, 3, 3, 3, 53, 5, 3, 3, 3, 3, 7, 11, 3, 3, 3, 3, 5, 7, 3, 3, 3, 3, 11, 5, 3, 3, 3, 3, 5, 79, 3, 3, 3, 3, 83, 5, 3, 3, 3, 3, 89, 7, 3, 3, 3, 3, 5, 97, 3, 3, 3
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 11 2003

Keywords

Comments

As A014493(n) = binomial(A042963(n),2) and a(n) = A020639(A014493(n)) > 2 for n > 1, A014493(n) is a good binomial coefficient.

Links

Crossrefs

Cf. A014493, A020639, A042963, A069901.

Programs

  • Mathematica
    FactorInteger[#][[1, 1]]&/@Select[Accumulate[Range[200]], OddQ] (* Harvey P. Dale, Jul 30 2016 *)
  • PARI
    a(n) = if(n == 1, 1, factor((2*n-1)*(2*n-1-(-1)^n)/2)[1,1]); \\ Amiram Eldar, May 16 2025

Formula

From Amiram Eldar, May 16 2025: (Start)
a(n) = A020639(A014493(n)).
a(n) = A069901(A042963(n)). (End)

Extensions

a(1) inserted by Amiram Eldar, May 16 2025

A226442 a(n) = smallest index m such that smallest prime factor of m-th triangular number is prime(n).

Original entry on oeis.org

3, 2, 10, 13, 22, 298, 526, 37, 46, 58, 61, 73, 82, 3397, 2866, 106, 3481, 3721, 5293, 5041, 7081, 157, 166, 178, 193, 10201, 14317, 23326, 23761, 226, 17398, 262, 19042, 277, 24286, 38806, 313, 45802, 29893, 346, 358, 32761, 382, 46126, 52993, 397, 421, 68461
Offset: 1

Views

Author

Zak Seidov, Jun 06 2013

Keywords

Comments

Or, smallest proper divisor of m-th triangular number is prime(n).
The curve is bimodal. Why? - T. D. Noe, Jun 07 2013

Examples

			3rd triangular number, A000217(3) = 6 = 2*3, 2nd triangular number, A000217(2) = 3,
10th triangular number, A000217(10) = 55 = 5*11, 13th triangular number, A000217(13) = 91 =7*13.

Crossrefs

Cf. A000040, A000217, A069901.

Programs

  • Mathematica
    nn = 50; t = Table[0, {nn}]; tri = 1; n = 1; found = 0; While[found < nn, n++; tri = tri + n; p = FactorInteger[tri][[1, 1]]; pi = PrimePi[p]; If[pi <= nn && t[[pi]] == 0, t[[pi]] = n; found++]]; t (* T. D. Noe, Jun 07 2013 *)

Formula

A069901(a(n)) = A000040(n).
Showing 1-6 of 6 results.

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