A340159 a(n) is the smallest number m such that numbers m, m + 1, m + 2, ..., m + n - 1 have k, 2*k, 3*k, ..., n*k divisors respectively.

1, 1, 61, 421, 211082, 11238341, 16788951482, 41126483642

Offset: 1

Author

Jaroslav Krizek, Dec 29 2020

Keywords

• nonn
• more
• hard

Comments

a(n) is the smallest number m such that tau(m) = tau(m + 1)/2 = tau(m + 2)/3 = tau(m + 3)/4 = ... = tau(m + n - 1)/n, where tau(k) = the number of divisors of k (A000005).

Corresponding values of tau(a(n)): 1, 1, 2, 2, 4, 4, 4, ...

a(8) <= 41126483642. - David A. Corneth, Dec 31 2020

Any subsequent terms are > 10^11. - Lucas A. Brown, Mar 18 2024

Examples

			a(3) = 61 because 61, 62 and 63 have 2, 4, and 6 divisors respectively and there is no smaller number having this property.
		

Links

• Lucas A. Brown, Python program.

Crossrefs

Cf. A294528 for similar sequence with primes.

Cf. A000005, A063446, A339778, A340157, A340158.

Programs

PARI

isok(m, n) = {my(k=numdiv(m)); for (i=1, n-1, if (numdiv(m+i) != (i+1)*k, return (0));); return(1);}
a(n) = my(m=1); while(!isok(m, n), m++); m; \\ Michel Marcus, Dec 30 2020

Python

# see LINKS

Extensions

a(7) from Jinyuan Wang, Dec 31 2020

a(8) from Lucas A. Brown, Mar 18 2024