A276542 - OEIS

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

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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).