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A063440 Number of divisors of n-th triangular number. 13

%I

%S 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,

%T 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,

%U 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

%N Number of divisors of n-th triangular number.

%C a(n) = 4 iff either n is in A005383 or n/2 is in A005384.

%C a(n) is odd iff n is in A001108.

%C 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

%H Ray Chandler, <a href="/A063440/b063440.txt">Table of n, a(n) for n=1..10000</a>

%F a(n) = A000005(A000217(n)).

%F From _Robert Israel_, Oct 26 2015: (Start)

%F a(2k) = A000005(k)*A000005(2k+1).

%F a(2k+1) = A000005(2k+1)*A000005(k+1).

%F gcd(a(2k), a(2k+1)) = A000005(2k+1) * A060778(k). (End)

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

%p seq(numtheory:-tau(n*(n+1)/2), n=1..100); # _Robert Israel_, Oct 26 2015

%t DivisorSigma[0,#]&/@Accumulate[Range[90]] (* _Harvey P. Dale_, Apr 15 2019 *)

%o (PARI) for (n=1, 10000, write("b063440.txt", n, " ", numdiv(n*(n + 1)/2)) ) \\ _Harry J. Smith_, Aug 21 2009

%o (PARI) a(n)=factorback(apply(numdiv,if(n%2,[n,(n+1)/2],[n/2,n+1]))) \\ _Charles R Greathouse IV_, Dec 27 2014

%o (PARI) vector(100, n, numdiv(n*(n+1)/2)) \\ _Altug Alkan_, Oct 26 2015

%Y Cf. A000005, A000217.

%Y Cf. A001108, A005383, A005384, A048161, A060778, A081978, A106483.

%K nonn,easy

%O 1,2

%A _Henry Bottomley_, Jul 24 2001

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Last modified June 17 19:10 EDT 2019. Contains 324198 sequences. (Running on oeis4.)