A286068 a(n) = least k such that the prime tower factorizations of k and k+1 both contain the n-th prime.

8, 8, 95, 384, 10240, 57343, 1179647, 4718592, 92274688, 8053063679, 32212254720, 2611340115967, 46179488366591, 184717953466368, 3236962232172544, 243194379878006783, 16717361816799281152, 71481133285624512511, 4869940435459321626624, 82641413450218791239680

Offset

1,1

Comments

The prime tower factorization of a number is defined in A182318.

Two consecutive numbers cannot have a common prime factor; however, their prime tower factorizations can share a prime number.

For example, the prime tower factorizations of 8 and 9, that is, 2^3 and 3^2, share the prime numbers 2 and 3.

We can also find triples of consecutive numbers whose prime tower factorizations share a prime number:

- if n is an odd squarefree number > 1, then the prime tower factorizations of n^2-1, n^2 and n^2+1 share the prime number 2,

- the prime tower factorizations of 5344, 5345 and 5346 share the prime number 5.

Also, the prime tower factorizations of:

- 342, 343, 344 and 345 share the prime number 3,

- 99125, 99126, 99127, 99128 and 99129 share the prime number 3,

- 72470 ... 72480 share the prime number 2,

- 1674274 ... 1674288 share the prime number 2.

Are there tuples of more than 15 consecutive numbers with such a property?

Links

Table of n, a(n) for n=1..20.

Rémy Sigrist, Illustration of the first terms

Formula

a(1) = 8.

If prime(n) = 4*k+1, then a(n) = 2^(4*k+1)*(2*k+1)-1.

If prime(n) = 4*k+3, then a(n) = 2^(4*k+3)*(2*k+1).

To prove the formula for n > 1:

- we use Fermat's little theorem: 2^p = 2 mod p,

- we check that there are no lower values near a multiple of 2^p,

- we check that the given value is less than 3^p - 1.

Example

See illustration of first terms in Links section.

Prog

(PARI) a(n) = my (p=prime(n)); if (p==2, return (8), my (k = p\4); if (p % 4 == 1, return (2^p*(2*k+1)-1), return (2^p*(2*k+1))))

Crossrefs

Cf. A182318.

Sequence in context: A186984 A298962 A082798 * A228071 A192386 A119932

Adjacent sequences: A286065 A286066 A286067 * A286069 A286070 A286071

Keyword

nonn

Author

Rémy Sigrist, Jun 13 2017

Status

approved