A349696 Smallest number in a set of three consecutive triangular numbers each with three prime factors (counted with multiplicity).

153, 406, 861, 39621, 2166321, 3924201, 11146281, 14804961, 19198306, 73951041, 83417986, 97951006, 209643526, 310415986, 522339681, 526225461, 583333246, 611153241, 801460666, 1601581906, 2520251506, 2690954841, 4455349606, 6681853401, 9895642221, 13878029901

Offset: 1

Author

Shyam Sunder Gupta, Nov 25 2021

Keywords

• nonn

Comments

153 is the only known number in the sequence which is not squarefree.

From Robert Israel, Mar 11 2025: (Start)

Terms are of the form A000217(k) with either

k prime == 5 (mod 12), k + 1 = 6 * prime, k + 2 prime, k + 3 = 4 * prime

or

k = 4 * prime == 4 (mod 12), k + 1 prime, k + 2 = 6 * prime, k + 3 prime.

In particular, for k > 17 (where k = 17 corresponds to a(1) = 153), the primes mentioned above are > 3, and so the terms except for 153 are squarefree. (End)

Examples

			a(1) = 153 because 153 is the smallest number in the first set of three consecutive triangular numbers with three prime factors (counted with multiplicity), i.e., (153 = 3*3*17, 171 = 3*3*19, 190 = 2*5*19).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Shyam Sunder Gupta, Beauty of Number 153, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 15, 399-410.

Crossrefs

Cf. A000217, A128896, A255200, A348185.

Programs

Maple

R:= NULL: count:= 0:
for i from 1 while count < 100 do
  k:= 12*i+4;
  if isprime(k+1) and isprime((k+2)/6) and isprime(k+3) then
     if isprime(k/4) then R:= R, k*(k+1)/2; count:= count+1; fi;
     if isprime((k+4)/4) then R:= R, (k+1)*(k+2)/2; count:= count+1; fi;
  fi;
od:
R; # Robert Israel, Mar 11 2025

Mathematica

t[n_] := n*(n + 1)/2; q[n_] := PrimeOmega[n] == 3; Select[Partition[t /@ Range[10^5], 3, 1], AllTrue[#, q] &][[;; , 1]] (* Amiram Eldar, Nov 25 2021 *)
(#(#+1))/2&/@SequencePosition[PrimeOmega[Accumulate[Range[170000]]],{3,3,3}][[;;,1]] (* Harvey P. Dale, Oct 20 2023 *)

Formula

a(n) = A000217(A255200(n)). - Michel Marcus, Dec 25 2021

Extensions

Definition clarified by Harvey P. Dale, Oct 20 2023