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

Pierre Deligne, La conjecture de Weil. I, Publications Mathématiques de l’Institut des Hautes Scientifiques, Vol. 43 (1974), pp. 273-307; alternative link.

Srinivasa Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc., Vol. 22, No. 9 (1916), pp. 159-184.

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

Mathematica

a[n_] := Floor[DivisorSigma[0, n] * n^(11/2)]; Array[a, 32] (* Amiram Eldar, Jan 07 2025 *)

Prog

(Magma) [Floor(NumberOfDivisors(n)*n^(11/2)): n in [1..32]]; // Marius A. Burtea, Oct 24 2019
(PARI) a(n) = floor(numdiv(n) * n^(11/2)); \\ Amiram Eldar, Jan 07 2025

Crossrefs

Cf. A000005, A000594, A076847.

Sequence in context: A232588 A097372 A263170 * A376706 A101243 A173483

Adjacent sequences: A321300 A321301 A321302 * A321304 A321305 A321306

Keyword

nonn

Author

Seiichi Manyama, Nov 03 2018

Status

approved