A276542 Numbers k such that the k-th and (k+1)st triangular numbers have the same number of divisors.

3, 4, 5, 11, 17, 28, 29, 33, 41, 42, 52, 55, 59, 66, 68, 71, 76, 85, 88, 91, 93, 101, 107, 114, 123, 137, 141, 143, 149, 150, 159, 170, 172, 179, 183, 185, 186, 188, 191, 196, 197, 201, 203, 208, 213, 215, 217, 219, 227, 232, 235, 236, 239, 243, 244, 247, 265

Offset

1,1

Comments

The k-th triangular number T(k) = k*(k+1)/2.

The lesser member of each twin-prime pair appears in this sequence. Hence, A001359 is a subset of this sequence.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Example

a(3) = 5; T(5) = 5*(5+1)/2 = 15; T(5+1) = 6*(6+1)/2 = 21; 15 and 21 have 4 divisors each.
a(6) = 28; T(28) = 28*(28+1)/2 = 406; T(28+1) = 29*(29+1)/2 = 435; 406 and 435 have 8 divisors each

Maple

T:= seq(numtheory:-tau(n*(n+1)/2), n=1..1000):
select(t -> T[t]=T[t+1], [$1..999]); # Robert Israel, Apr 09 2017

Mathematica

Select[Range[1000], DivisorSigma[0, #*(# + 1)/2] == DivisorSigma[0, (# + 1)*(# + 1 + 1)/2] &]
SequencePosition[DivisorSigma[0, #]&/@Accumulate[Range[300]], {x_, x_}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 02 2018 *)

Prog

(PARI) k=[]; for(n=1, 1000, a=numdiv(n*(n+1)/2); b=numdiv((n+1)*(n+1+1)/2); if(a==b, k=concat(k, n))); k
(GAP) T:=List([1..270], n->n*(n+1)/2);;  a:=Filtered([1..Length(T)-1], i->Tau(T[i])=Tau(T[i+1])); # Muniru A Asiru, Dec 06 2018
(Magma) [n: n in [1..300] | DivisorSigma(0, n*(n + 1) div 2) eq DivisorSigma(0, (n + 1)*(n + 1 + 1) div 2)]; // Vincenzo Librandi, Dec 06 2018
(Python)
from sympy import divisor_count
for n in range(1, 20):
    if divisor_count(n*(n+1)/2)==divisor_count((n+1)*(n+2)/2):
        print(n, end=', ') # Stefano Spezia, Dec 06 2018

Crossrefs

Cf. A000040, A000217, A001359, A062832.

Cf. A319035 (the corresponding triangular numbers).

Sequence in context: A293713 A280255 A361517 * A318077 A074221 A341785

Adjacent sequences: A276539 A276540 A276541 * A276543 A276544 A276545

Keyword

nonn

Author

K. D. Bajpai, Apr 09 2017

Status

approved