A340229 Numbers m such that numbers m, m + 1, m + 2, m + 3 and m + 4 have k, 2k, 4k, 8k and 16k divisors respectively.

1124581, 2101621, 2135701, 3829381, 5801701, 6097381, 6453541, 6535861, 6609781, 6799621, 6972661, 7055317, 7527061, 8281381, 8485502, 8524981, 8883326, 9412981, 9895141, 11455141, 11901781, 12043621, 12929941, 13749061, 14747701, 15150901, 15504661, 15533941

Offset: 1

Author

Jaroslav Krizek, Jan 01 2021

Keywords

• nonn

Comments

Numbers m such that tau(m) = tau(m + 1)/2 = tau(m + 2)/4 = tau(m + 3)/8 = tau(m + 4)/16, where tau(k) = the number of divisors of k (A000005).

Quintuples of [tau(a(n)), tau(a(n) + 1), tau(a(n) + 2), tau(a(n) + 3), tau(a(n) + 4)] = [tau(a(n)), 2*tau(a(n)), 4*tau(a(n)), 8*tau(a(n)), 16*tau(a(n))]: [2, 4, 8, 16, 32], [2, 4, 8, 16, 32], [2, 4, 8, 16, 32], [2, 4, 8, 16, 32], [2, 4, 8, 16, 32], [2, 4, 8, 16, 32], ...

Corresponding values of numbers k: 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 4, ...

Prime terms are in A100365; number 8485502 is the smallest composite term.

Subsequence of A063446, A100363 and A100364.

Examples

			tau(1124581) = 2, tau(1124582) = 4, tau(1124583) = 8, tau(1124584) = 16, tau (1124585) = 32.
		

Crossrefs

Cf. A000005, A063446, A100363, A100364, A100365, A340230.

Programs

Magma

[m: m in [1..10^7] | #Divisors(m) eq #Divisors(m + 1) / 2 and #Divisors(m) eq #Divisors(m + 2) / 4 and #Divisors(m) eq #Divisors(m + 3) / 8 and #Divisors(m) eq #Divisors(m + 4) / 16]

PARI

isok(m) = vector(4, k, numdiv(m+k))/numdiv(m) == [2,4,8,16]; \\ Michel Marcus, Jan 02 2021