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
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
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Jun 13 2017
STATUS
approved