A075029 a(n) is the smallest number k such that the number of divisors of the n numbers from k through k+n-1 are decreasing.

1, 4, 45, 80, 28974, 28974, 8489103, 80314575, 5196065775, 77506573550, 166622964149389, 489289301397948

Offset: 1

Author

Amarnath Murthy, Sep 02 2002

Keywords

• nonn
• more

Comments

a(11) > 10^12. - Donovan Johnson, Oct 13 2009

a(11) > 10^13. - Giovanni Resta, Jul 25 2013

a(13) > 2.64*10^15. - Jud McCranie, Mar 27 2019

Examples

			a(4) = 80, as tau(80) = 10 > tau(81) = 5 > tau(82) = 4 > tau(83) = 2, and no smaller sequence of 4 consecutive integers satisfies such a chain of inequalities.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 45, p. 17, Ellipses, Paris 2008.

Crossrefs

Cf. A075028, A075031.

Cf. A000005 (numdiv).

Programs

PARI

n=2; for(k=1, 100, while(1, m=n; s=1; while(numdiv(m)=k, print1(n-1, ", "); break); n=n+1))

PARI

apply( {A075029(n)=for(k=4^(n-1),oo, my(d=numdiv(k)); d(d=numdiv(k+j)) || [k+=j-1, next(2)]); return(k))}, [1..6]) \\ For illustration - becomes slow for n >= 7. - M. F. Hasler, May 26 2025

Extensions

Edited by Ralf Stephan, Mar 21 2003

Definition corrected by Leroy Quet, Feb 23 2008

2 more terms from Lekraj Beedassy, Jul 13 2008

a(9)-a(10) from Donovan Johnson, Oct 13 2009

a(11)-a(12) from Jud McCranie, Mar 27 2019

Definition changed by Robert Israel, May 26 2025