cp's OEIS Frontend

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

%I A349696 #26 Mar 14 2025 11:01:45
%S A349696 153,406,861,39621,2166321,3924201,11146281,14804961,19198306,
%T A349696 73951041,83417986,97951006,209643526,310415986,522339681,526225461,
%U A349696 583333246,611153241,801460666,1601581906,2520251506,2690954841,4455349606,6681853401,9895642221,13878029901
%N A349696 Smallest number in a set of three consecutive triangular numbers each with three prime factors (counted with multiplicity).
%C A349696 153 is the only known number in the sequence which is not squarefree.
%C A349696 From _Robert Israel_, Mar 11 2025: (Start)
%C A349696 Terms are of the form A000217(k) with either
%C A349696   k prime == 5 (mod 12), k + 1 = 6 * prime, k + 2 prime, k + 3 = 4 * prime
%C A349696 or
%C A349696   k = 4 * prime == 4 (mod 12), k + 1 prime, k + 2 = 6 * prime, k + 3 prime.
%C A349696 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)
%H A349696 Robert Israel, <a href="/A349696/b349696.txt">Table of n, a(n) for n = 1..10000</a>
%H A349696 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_15">Beauty of Number 153</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 15, 399-410.
%F A349696 a(n) = A000217(A255200(n)). - _Michel Marcus_, Dec 25 2021
%e A349696 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).
%p A349696 R:= NULL: count:= 0:
%p A349696 for i from 1 while count < 100 do
%p A349696   k:= 12*i+4;
%p A349696   if isprime(k+1) and isprime((k+2)/6) and isprime(k+3) then
%p A349696      if isprime(k/4) then R:= R, k*(k+1)/2; count:= count+1; fi;
%p A349696      if isprime((k+4)/4) then R:= R, (k+1)*(k+2)/2; count:= count+1; fi;
%p A349696   fi;
%p A349696 od:
%p A349696 R; # _Robert Israel_, Mar 11 2025
%t A349696 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 *)
%t A349696 (#(#+1))/2&/@SequencePosition[PrimeOmega[Accumulate[Range[170000]]],{3,3,3}][[;;,1]] (* _Harvey P. Dale_, Oct 20 2023 *)
%Y A349696 Cf. A000217, A128896, A255200, A348185.
%K A349696 nonn
%O A349696 1,1
%A A349696 _Shyam Sunder Gupta_, Nov 25 2021
%E A349696 Definition clarified by _Harvey P. Dale_, Oct 20 2023