A069903 -id:A069903 - cp's OEIS frontend

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

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

Reinhard Zumkeller, Apr 10 2002

nonn

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

Antti Karttunen, Table of n, a(n) for n = 1..20000

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

Array[Plus@@Last/@FactorInteger[ #*(#+1)/2]&,33] (* Vladimir Joseph Stephan Orlovsky, Feb 28 2010 *)

A069904(n) = bigomega((n*(n+1))/2); \\ Antti Karttunen, Oct 07 2017

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

Reinhard Zumkeller, Apr 10 2002

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

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

Related properties of triangular numbers: A069901, A069903, A069904.

Cf. A000217, A006530, A074399.

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 *)

\\ written for a(n), n >= 2
a(n)=vecmax(factor(n*(n+1)/2)[,1]) \\ M. F. Hasler, May 02 2015

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

Edited by Peter Munn, Sep 14 2023

A128896 Triangular numbers that are products of three distinct primes.

Original entry on oeis.org

66, 78, 105, 190, 231, 406, 435, 465, 561, 595, 741, 861, 903, 946, 1378, 1653, 2211, 2278, 2485, 3081, 3655, 3741, 4371, 4465, 5151, 5253, 5995, 6441, 7021, 7503, 8515, 8911, 9453, 9591, 10011, 10153, 10585, 11026, 12561, 13366, 14878, 15051, 15753

Offset: 1

Zak Seidov, Apr 20 2007

nonn

			a(1)=T(11)=66=2*3*11, a(2)=T(12)=78=2*3*13, a(3)=T(14)=105=3*5*7, a(4)=T(19)=190=2*5*19, a(5)=T(21)=231=3*7*11, a(6)=T(28)=406=2*7*29.
T(15) = 120 = 2^3*3*5. The triangular 120 has three prime factors but is not a product of these factors. Thus, 120 is not in this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000217, A068443, A069903, A076551, A127637.

Select[Table[n(n+1)/2,{n,1,210}],Transpose[FactorInteger[ # ]][[2]]=={1,1,1}&]
Select[Accumulate[Range[200]],PrimeNu[#]==PrimeOmega[#]==3&] (* Harvey P. Dale, Apr 23 2017 *)

a(n) = T(k) = k*(k+1)/2 = p*q*r for some k,p,q,r, where T(k) is triangular number and p, q, r are distinct primes.

Equals A000217 INTERSECT A007304 and A075875 INTERSECT A121478. - R. J. Mathar, Apr 22 2007

Name clarified by Tanya Khovanova, Sep 06 2022

A069901 Smallest prime factor of n-th triangular number.

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, 2, 19, 3, 2, 2, 3, 3, 2, 2, 3, 23, 2, 2, 5, 3, 2, 2, 3, 3, 2, 2, 3, 29, 2, 2, 31, 3, 2, 2, 3, 3, 2, 2, 3, 5, 2, 2, 37, 3, 2, 2, 3, 3, 2, 2, 3, 41, 2, 2, 5, 3, 2, 2, 3, 3, 2, 2, 3, 5, 2, 2, 7, 3, 2

Offset: 1

Reinhard Zumkeller, Apr 10 2002

Or, a(1) = 1, then the smallest nontrivial k (>1) which divides the sum of (next n) numbers from k+1 to k+n or smallest k > 1 that divides nk + n(n+1)/2. - Amarnath Murthy, Sep 22 2002. For example, a(7) = 4, which is the smallest nontrivial number that divides the sum 5+6+...+11, of 7 numbers.

			A000217(10) = 10*(10+1)/2 = 55 = 5*11, therefore a(10) = 5.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000217, A020639, A069902, A069903, A069904.

FactorInteger[#][[1,1]]&/@Accumulate[Range[100]] (* Harvey P. Dale, Apr 05 2014 *)

a(n) = if (n==1, 1, vecmin(factor(n*(n+1)/2)[,1]));

a(n) = A020639(A000217(n)).
a(4*k-1) = a(4*k) = 2.
From Zak Seidov, Jun 06 2013: (Start)
a(n) = 3 for n = {2, 5, 6, 9} + 12*k;
a(n) = 5 for n = {10, 25, 34, 49} + 60*k;
a(n) = 7 for n = {13, 97, 118, 133, 181, 202, 217, 238, 286, 301, 322, 406} + 420*k, etc. (End)

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of Franklin T. Adams-Watters

A128905 Numbers k such that the k-th triangular number has exactly four distinct prime factors.

Original entry on oeis.org

20, 51, 59, 60, 65, 68, 69, 76, 77, 83, 91, 92, 105, 110, 114, 115, 123, 129, 131, 139, 154, 156, 165, 182, 185, 186, 187, 194, 210, 212, 221, 227, 228, 235, 236, 237, 246, 254, 258, 265, 266, 267, 273, 276, 286, 290, 291, 307, 309, 318, 321, 322, 330, 345

Offset: 1

Zak Seidov, Apr 22 2007

nonn

Or, indices of triangular numbers with exactly four distinct prime factors.

			In order of increasing p (the least prime factor of T(k)):
  a(1)  =  20 because T(20)  =    210 =  2* 3* 5* 7,
  a(5)  =  65 because T(65)  =   2145 =  3* 5*11*13,
  a(21) = 154 because T(154) =  11935 =  5* 7*11*31,
  a(45) = 286 because T(286) =  41041 =  7*11*13*41,
  a(143)= 781 because T(781) = 305371 = 11*17*23*71,
  a(91) = 493 because T(493) = 121771 = 13*17*19*29, etc.

Cf. A000217, A068443, A069903, A076551, A127637, A128896.

lim=346;tn=Rest[Array[ #*(# - 1)/2 &,lim]];Select[Range[lim-1],PrimeNu[tn[[#]]]==PrimeOmega[tn[[#]]]==4&] (* James C. McMahon, Jan 12 2025 *)

a(n)=k and T(k)=k*(k+1)/2=p*q*r*s for some k, p, q, r, s where T(k) is a triangular number and p, q, r, s are distinct primes.

A375658 The smallest triangular number that begins a run of at least n consecutive triangular numbers with the same number of distinct prime factors.

Original entry on oeis.org

1, 6, 6, 6, 6, 6, 6, 6, 103327500, 1977230170, 214596675756, 2796022463625, 32291440570378, 32291440570378, 549091407981061, 4680505857819681, 96914999755056255, 96914999755056255

Offset: 1

Shyam Sunder Gupta, Aug 23 2024

			a(8) = 6 because 6 is the smallest triangular number that begins a run of 8 consecutive triangular numbers (6, 10, 15, 21, 28, 36, 45, 55) with the same number of distinct prime factors, i.e. 2.
a(9) = 103327500 because 103327500 is the smallest triangular number that begins a run of 9 consecutive triangular numbers (103327500, 103341876, 103356253, 103370631, 103385010, 103399390,  103413771, 103428153, 103442536) with the same number of distinct prime factors, i.e. 5.

Cf. A000217, A069903, A076551.

Cf. A375657.

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A128896 Triangular numbers that are products of three distinct primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A069901 Smallest prime factor of n-th triangular number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A128905 Numbers k such that the k-th triangular number has exactly four distinct prime factors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A375658 The smallest triangular number that begins a run of at least n consecutive triangular numbers with the same number of distinct prime factors.

Views

Author

Keywords

Examples

Crossrefs