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

Author

Seiichi Manyama, Nov 03 2018

Keywords

• nonn

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.

Crossrefs

Cf. A000005, A000594, A076847.

Programs

Magma

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

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 *)

PARI

a(n) = floor(numdiv(n) * n^(11/2)); \\ Amiram Eldar, Jan 07 2025