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 A286068 a(n) = least k such that the prime tower factorizations of k and k+1 both contain the n-th prime. 1
 8, 8, 95, 384, 10240, 57343, 1179647, 4718592, 92274688, 8053063679, 32212254720, 2611340115967, 46179488366591, 184717953466368, 3236962232172544, 243194379878006783, 16717361816799281152, 71481133285624512511, 4869940435459321626624, 82641413450218791239680 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 7 17:21 EDT 2024. Contains 375017 sequences. (Running on oeis4.)