OFFSET
1,1
COMMENTS
Not every term T(k) has the same prime signature as its successor triangular number T(k+1); the first counterexample is the pair (T(52), T(53)) = (1378, 1431) = (2 * 13 * 53, 3^3 * 53), each of which has 8 divisors. The first counterexample in which the two triangular numbers have the same number of distinct prime factors is (T(45630), T(45631)) = (1041071265, 1041116896) = (3^3 * 5 * 13^2 * 45631, 2^5 * 23 * 31 * 45631), each of which has 48 divisors.
LINKS
Muniru A Asiru, Table of n, a(n) for n = 1..10000
EXAMPLE
T(2) = 6 is a term because 6 = 2 * 3 has 4 divisors (1, 2, 3, 6) and T(3) = 10 = 2 * 5 also has 4 divisors (1, 2, 5, 10).
T(17) = 153 is a term because 153 = 3^2 * 17 has 6 divisors (1, 3, 9, 17, 51, 153) and T(18) = 171 = 3^2 * 19 also has 6 divisors (1, 3, 9, 19, 57, 171).
MATHEMATICA
t[n_] := n(n+1)/2; aQ[n_] := DivisorSigma[0, t[n]] == DivisorSigma[0, t[n+1]]; t[Select[Range[100], aQ]] (* Amiram Eldar, Dec 06 2018 *)
PROG
(PARI) lista(nn) = {for (n=1, nn, if (numdiv(t=n*(n+1)/2) == numdiv((n+1)*(n+2)/2), print1(t, ", ")); ); } \\ Michel Marcus, Dec 06 2018
(GAP) T:=List([1..210], n->n*(n+1)/2);; a:=List(Filtered([1..Length(T)-1], i->Tau(T[i])=Tau(T[i+1])), k->T[k]); # Muniru A Asiru, Dec 06 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Dec 05 2018
STATUS
approved