A340230 a(n) is the smallest number m such that numbers m, m + 1, m + 2, ..., m + n - 1 have k, 2*k, 4*k, 8*k, ..., (2^(n-1))*k divisors respectively.

1, 1, 193, 613, 1124581, 52071301, 213536830501

Offset: 1

Author

Jaroslav Krizek, Jan 01 2021

Keywords

• nonn
• more

Comments

a(n) is the smallest number m such that tau(m) = tau(m + 1) / 2 = tau(m + 2) / 4 = tau(m + 3) / 8 = ... = tau(m + n - 1) / 2^(n - 1), where tau(k) = the number of divisors of k (A000005).

Conjecture: a(7) = 213536830501.

Examples

			a(4) = 613 because 613 is the smallest term of 4 consecutive numbers with this property: tau(613) = 2, tau(614) = 4, tau(615) = 8, tau(616) = 16.
		

Crossrefs

Cf. A100366 (similar sequence for primes).

Cf. A000005, A063446, A100363, A100364, A100365, A340229.

Programs

PARI

isok(m, n) = my(nb=numdiv(m)); for (k=1, n-1, if (numdiv(m+k)/nb != 2^k, return(0))); return (1);
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Jan 05 2021

Extensions

a(7), as conjectured by Jaroslav Krizek, from Martin Ehrenstein, Feb 06 2021