The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A000005 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A000005

d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.
(history; published version)

#563 by Joerg Arndt at Fri Jun 21 03:52:14 EDT 2024

FORMULA

Conjecture: a(n) = Sum_{k=1..n} A010051(2*k/gcd(n, k)). - Velin Yanev, Jun 20 2024

KEYWORD

easy,core,nonn,nice,mult,hear,changed

STATUS

proposed

approved

#562 by Velin Yanev at Thu Jun 20 14:10:25 EDT 2024

STATUS

editing

proposed

Discussion

Thu Jun 20

20:56

Kevin Ryde: That's true (and too easy to enter as conjecture).  The isprime() is fairly unnecessary.  You're asking that gcd(n,k)==k, which of course is exactly when k divides n.

Fri Jun 21

03:52

Joerg Arndt: reverting edit

#561 by Velin Yanev at Thu Jun 20 14:09:56 EDT 2024

FORMULA

Conjecture: a(n) = Sum_{k=1..n} A010051(2*k/gcd(n, k)). - Velin Yanev, Jun 20 2024

STATUS

approved

editing

#560 by Michael De Vlieger at Sun May 26 08:30:00 EDT 2024

STATUS

proposed

approved

#559 by Michel Marcus at Sun May 26 04:21:42 EDT 2024

STATUS

editing

proposed

#558 by Michel Marcus at Sun May 26 04:21:37 EDT 2024

REFERENCES

T. Terence Tao, Poincaré's Legacies, Part I, Amer. Math. Soc., 2009, see pp. 31ff for upper bounds on d(n).

LINKS

Terence Tao, <a href="https://terrytao.wordpress.com/wp-content/uploads/2009/01/whatsnew.pdf">Poincaré's Legacies: pages from year two of a mathematical blog</a>, see page 59.

STATUS

approved

editing

#557 by R. J. Mathar at Sun Jan 28 08:46:25 EST 2024

STATUS

editing

approved

#556 by R. J. Mathar at Sun Jan 28 05:35:06 EST 2024

CROSSREFS

Cf. A098198 (Dgf at s=2), A183030 (Dgf at s=3), A183031 (Dgf at s=3).

STATUS

approved

editing

#555 by Peter Bala at Thu Jan 25 05:21:48 EST 2024

FORMULA

a(n) = Sum_{d divides n} mu(core(d))^2 = Sum_{d divides n} core(mu(core(d)), where core(n) = A007913(n). - Peter Bala, Jan 24 2024

KEYWORD

easy,core,nonn,nice,mult,hear,changed

STATUS

editing

approved

#554 by Peter Bala at Wed Jan 24 17:23:56 EST 2024

FORMULA

a(n) = Sum_{d divides n} mu(core(d))^2, = Sum_{d divides n} core(mu(core(d)), where core(n) = A007913(n). - Peter Bala, Jan 24 2024