A348185 -id:A348185 - cp's OEIS frontend

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

Shyam Sunder Gupta, Nov 25 2021

nonn

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)

			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).

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.

Cf. A000217, A128896, A255200, A348185.

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

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 *)

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

Definition clarified by Harvey P. Dale, Oct 20 2023

A349539 Smallest number m in a set of at least three consecutive triangular numbers with three distinct prime factors.

Original entry on oeis.org

378, 406, 528, 820, 861, 1953, 2485, 3081, 5050, 5151, 5778, 7750, 9316, 11026, 11175, 18145, 19306, 19503, 36046, 36315, 39621, 92665, 93096, 130816, 131328, 135981, 205120, 326836, 337431, 661825, 816003, 1439056, 1993006, 1995003, 2166321, 2835771

Offset: 1

Shyam Sunder Gupta, Nov 25 2021

nonn

			a(1) = 378 because 378 is the smallest number in the first set of three consecutive triangular numbers with three distinct prime factors, i.e., (378 = 2*3^3*7, 406 = 2*7*29, 435 = 3*5*29).

Cf. A000217, A128896, A348185.

t[n_] := n*(n + 1)/2; q[n_] := PrimeNu[n] == 3; Select[Partition[t /@ Range[3*10^3], 3, 1], AllTrue[#, q] &][[;; , 1]] (* Amiram Eldar, Nov 26 2021 *)

Name clarified by Michel Marcus, Dec 02 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions