A336595 Numbers whose number of divisors is divisible by 5.

16, 48, 80, 81, 112, 144, 162, 176, 208, 240, 272, 304, 324, 336, 368, 400, 405, 432, 464, 496, 512, 528, 560, 567, 592, 624, 625, 648, 656, 688, 720, 752, 784, 810, 816, 848, 880, 891, 912, 944, 976, 1008, 1040, 1053, 1072, 1104, 1134, 1136, 1168, 1200, 1232

Offset: 1

Author

Amiram Eldar, Jul 26 2020

Keywords

• nonn

Comments

The asymptotic density of this sequence is 1 - zeta(5)/zeta(4) = 0.0419426259... (Sathe, 1945).

Examples

			16 is a term since A000005(16) = 5 is divisible by 5.
		

References

• G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 63.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.
• S. S. Pillai, On a congruence property of the divisor function, J. Indian Math. Soc. (N. S.), Vol. 6, (1942), pp. 118-119.
• L. G. Sathe, On a congruence property of the divisor function, American Journal of Mathematics, Vol. 67, No. 3 (1945), pp. 397-406.

Crossrefs

Cf. A000005, A008587, A059269, A336596.

Cf. A030514, A030628, A113849, A178739, A179665, A030633, A030638, A137488, A137493, A175745.

Programs

Maple

q:= n-> is(irem(numtheory[tau](n), 5)=0):
select(q, [$1..1300])[];  # Alois P. Heinz, Jul 26 2020

Mathematica

Select[Range[1300], Divisible[DivisorSigma[0, #], 5] &]

Formula

A030514 UNION A030628 \ {1} UNION A030633 UNION A030638 UNION A137488 UNION A137493 UNION A175745 UNION A175749 UNION A175752 UNION A175756 UNION ... - R. J. Mathar, May 05 2023