A100365 -id:A100365 - cp's OEIS frontend

A100366 a(n) is the least prime number q such that q,q+1,q+2,q+3,...,q+n-1 have 2,4,8,...,2^n divisors respectively.

2, 5, 193, 613, 1124581, 52071301, 213536830501

Offset: 1

Labos Elemer, Nov 19 2004

a(3), a(4), a(5) are the initial terms of A100363, A100364, A100365 resp.

Any run of 8 or more consecutive integers must include at least one number k of the form 8j+4; in the prime factorization of k, the prime factor 2 appears with multiplicity exactly 2, so the number of divisors of k is divisible by 3 (which is not a power of 2). Thus, there is no term a(8): the sequence is complete, ending with a(7). - Jon E. Schoenfield, Nov 12 2017

			a(4)=613: q=613 (a prime, hence two divisors), q+1 = 614 = 2*307 (4 divisors), q+2 = 615 = 3*5*41 (8 divisors), and q+3 = 616 = 2^3 * 7 * 11 (16 divisors).

Cf. A000005, A063446, A100363, A100364.

a(6)-a(7) from Donovan Johnson, Mar 23 2011

Keywords fini and full added and Example section edited by Jon E. Schoenfield, Nov 12 2017

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.

Original entry on oeis.org

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

Jaroslav Krizek, Jan 01 2021

nonn

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.

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

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

[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]

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

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

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Jan 01 2021

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.

			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.

Cf. A100366 (similar sequence for primes).

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

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

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

cp's OEIS Frontend

A100366 a(n) is the least prime number q such that q,q+1,q+2,q+3,...,q+n-1 have 2,4,8,...,2^n divisors respectively.

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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.

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

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cp's OEIS Frontend

A100366 a(n) is the least prime number q such that q,q+1,q+2,q+3,...,q+n-1 have 2,4,8,...,2^n divisors respectively.

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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.

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Magma

PARI

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.

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