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A063440 Number of divisors of n-th 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 15 11:43 EST 2019. Contains 329999 sequences. (Running on oeis4.)