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

1,1

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.

Links

Table of n, a(n) for n=1..28.

Example

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

Prog

(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

Crossrefs

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

Sequence in context: A059767 A069583 A109209 * A100365 A363957 A395926

Adjacent sequences: A340226 A340227 A340228 * A340230 A340231 A340232

Keyword

nonn

Author

Jaroslav Krizek, Jan 01 2021

Status

approved