A127637 -id:A127637 - cp's OEIS frontend

A076551 Smallest triangular number with n prime factors (counted without multiplicity).

1, 3, 6, 66, 210, 3570, 103740, 930930, 56812470, 1722580860, 32395433070, 265257422430, 91348974206490, 8923863379851420, 330196858581850260, 14193386885746698630, 2378522762792139793830, 351206814022419685159830, 28791787439593010836313310

Offset: 0

Shyam Sunder Gupta, Oct 19 2002

nonn

a(19) <= 8285055066500101241048306610. a(20) <= 120052594044654305809137933570. - Donovan Johnson, Feb 28 2012

			a(0) = 1 because 1 is a triangular number with zero prime factors; a(3) = 66 because 66 = 2*3*11 is a triangular number with 3 distinct prime factors.

Cf. A000217, A127637.

{n=12; v=vector(n); for(k=1,1000000,t=k*(k+1)/2; d=1+omega(t); if(d<=n&&v[d]<1,v[d]=t)); for(j=1,n,print1(v[j],","))}

a(12) from Rick L. Shepherd, May 31 2003
a(13) from Klaus Brockhaus, Aug 02 2006
a(14)-a(16) from Donovan Johnson, Jan 28 2009
a(17) from Donovan Johnson, Feb 07 2009
a(18) from Donovan Johnson, Feb 28 2012

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

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.

cp's OEIS Frontend

A076551 Smallest triangular number with n prime factors (counted without multiplicity).

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A128896 Triangular numbers that are products of three distinct primes.

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

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