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

1,1

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)

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.

Formula

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

Example

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).

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 *)

Crossrefs

Cf. A000217, A128896, A255200, A348185.

Sequence in context: A056733 A256748 A256741 * A246629 A371082 A325376

Adjacent sequences: A349693 A349694 A349695 * A349697 A349698 A349699

Keyword

nonn,changed

Author

Shyam Sunder Gupta, Nov 25 2021

Extensions

Definition clarified by Harvey P. Dale, Oct 20 2023

Status

approved