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.
%I A128905 #13 Jan 12 2025 12:12:04 %S A128905 20,51,59,60,65,68,69,76,77,83,91,92,105,110,114,115,123,129,131,139, %T A128905 154,156,165,182,185,186,187,194,210,212,221,227,228,235,236,237,246, %U A128905 254,258,265,266,267,273,276,286,290,291,307,309,318,321,322,330,345 %N A128905 Numbers k such that the k-th triangular number has exactly four distinct prime factors. %C A128905 Or, indices of triangular numbers with exactly four distinct prime factors. %F A128905 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. %e A128905 In order of increasing p (the least prime factor of T(k)): %e A128905 a(1) = 20 because T(20) = 210 = 2* 3* 5* 7, %e A128905 a(5) = 65 because T(65) = 2145 = 3* 5*11*13, %e A128905 a(21) = 154 because T(154) = 11935 = 5* 7*11*31, %e A128905 a(45) = 286 because T(286) = 41041 = 7*11*13*41, %e A128905 a(143)= 781 because T(781) = 305371 = 11*17*23*71, %e A128905 a(91) = 493 because T(493) = 121771 = 13*17*19*29, etc. %t A128905 lim=346;tn=Rest[Array[ #*(# - 1)/2 &,lim]];Select[Range[lim-1],PrimeNu[tn[[#]]]==PrimeOmega[tn[[#]]]==4&] (* _James C. McMahon_, Jan 12 2025 *) %Y A128905 Cf. A000217, A068443, A069903, A076551, A127637, A128896. %K A128905 nonn %O A128905 1,1 %A A128905 _Zak Seidov_, Apr 22 2007