

A063440


Number of divisors of nth triangular number.


14



1, 2, 4, 4, 4, 4, 6, 9, 6, 4, 8, 8, 4, 8, 16, 8, 6, 6, 8, 16, 8, 4, 12, 18, 6, 8, 16, 8, 8, 8, 10, 20, 8, 8, 24, 12, 4, 8, 24, 12, 8, 8, 8, 24, 12, 4, 16, 24, 9, 12, 16, 8, 8, 16, 24, 24, 8, 4, 16, 16, 4, 12, 36, 24, 16, 8, 8, 16, 16, 8, 18, 18, 4, 12, 24, 16, 16, 8, 16, 40, 10, 4, 16
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OFFSET

1,2


COMMENTS

a(n) = 4 iff either n is in A005383 or n/2 is in A005384.
a(n) is odd iff n is in A001108.
a(n) = 6 if either n = 18 or n = q^2 where q is in A048161 or n = 2 q^2  1 where q is in A106483.  Robert Israel, Oct 26 2015


LINKS

Ray Chandler, Table of n, a(n) for n=1..10000


FORMULA

a(n) = A000005(A000217(n)).
From Robert Israel, Oct 26 2015: (Start)
a(2k) = A000005(k)*A000005(2k+1).
a(2k+1) = A000005(2k+1)*A000005(k+1).
gcd(a(2k), a(2k+1)) = A000005(2k+1) * A060778(k). (End)


EXAMPLE

a(6) = 4 since 1+2+3+4+5+6 = 21 has four divisors {1,3,7,21}.


MAPLE

seq(numtheory:tau(n*(n+1)/2), n=1..100); # Robert Israel, Oct 26 2015


MATHEMATICA

DivisorSigma[0, #]&/@Accumulate[Range[90]] (* Harvey P. Dale, Apr 15 2019 *)


PROG

(PARI) for (n=1, 10000, write("b063440.txt", n, " ", numdiv(n*(n + 1)/2)) ) \\ Harry J. Smith, Aug 21 2009
(PARI) a(n)=factorback(apply(numdiv, if(n%2, [n, (n+1)/2], [n/2, n+1]))) \\ Charles R Greathouse IV, Dec 27 2014
(PARI) vector(100, n, numdiv(n*(n+1)/2)) \\ Altug Alkan, Oct 26 2015


CROSSREFS

Cf. A000005, A000217.
Cf. A001108, A005383, A005384, A048161, A060778, A081978, A106483.
Sequence in context: A302254 A160409 A035645 * A008497 A220497 A194443
Adjacent sequences: A063437 A063438 A063439 * A063441 A063442 A063443


KEYWORD

nonn,easy


AUTHOR

Henry Bottomley, Jul 24 2001


STATUS

approved



