A321303 a(n) = floor(d(n) * n^(11/2)) where d(n) is the number of divisors of n.

1, 90, 841, 6144, 13975, 76188, 88934, 370727, 531441, 1264911, 1068291, 5171875, 2677431, 8049412, 11764186, 20971520, 11708440, 48100548, 21586130, 85865010, 74862807, 96690707, 61735233, 312069853, 146484375, 242333472, 298236431, 546412244, 220911835, 1064772651, 318800733, 1138875187

Offset

1,2

Comments

|tau(n)| <= d(n) * n^(11/2) where tau(n) is Ramanujan function. So |tau(n)| <= a(n).

Ramanujan conjectured in 1916 that |tau(p)| <= 2 * p^(11/2) for all primes p and Pierre Deligne proved this conjecture in 1974. [Wikipedia] - Bernard Schott, Oct 24 2019

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Tau Function

Wikipedia, Ramanujan tau function

Maple

f:= n -> floor(numtheory:-tau(n)*n^(11/2)):
map(f, [$1..100]); # Robert Israel, Oct 23 2019

Prog

(Magma) [Floor(NumberOfDivisors(n)*n^(11/2)): n in [1..32]]; // Marius A. Burtea, Oct 24 2019

Crossrefs

Cf. A000005, A000594, A076847.

Sequence in context: A232588 A097372 A263170 * A101243 A173483 A202960

Adjacent sequences: A321300 A321301 A321302 * A321304 A321305 A321306

Keyword

nonn

Author

Seiichi Manyama, Nov 03 2018

Status

approved