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A119479 Length of longest run of consecutive integers having exactly n divisors. 8
1, 2, 1, 3, 1, 5, 1, 7, 1, 3, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

10<=a(12)<=15. If there were 16 such consecutive integers, two would be consecutive multiples of 8. One would have the form 32p and the other the form 8q^2 with odd primes p and q; this implies that 8q^2 is congruent to 24 or 40 (mod 64), which is impossible. It is likely that a(12)=15; this would follow from Dickson's conjecture.

a(14)=3. If there were 4, two would be consecutive even numbers. One would have the form 64p and the other the form 2q^6 with odd primes p and q. Since 2q^6 == 2 (mod 16), this implies that 2q^6 = 64p+2, so p = (q^3-1)(q^3+1)/32 is prime, which is impossible.

a(16)=7. If there were 8, one would be congruent to 4 (mod 8), which is impossible.

Schinzel's conjecture H would imply that:

a(2p) = 3 for all prime p > 3;

a(2pq) = 3 for all primes p, q such that gcd(p-1,q-1) > 4;

a(6p) = 5 for all odd prime p;

a(n) = 7 for all n > 4 such that n is divisible by 4 and nondivisible by 3. - Vladimir Letsko, Jul 18 2016

99949636937406199604777509122843 starts a run of 13 consecutive integers each having 12 divisors. Therefore 13 <= a(12) <= 15. - Vladimir Letsko, Jul 07 2017

LINKS

Table of n, a(n) for n=1..11.

Chris Caldwell, The Prime Glossary: Dickson's conjecture

Vladimir A. Letsko, Table of a(n) for all even n such that exact value of a(n) is proved

Vladimir A. Letsko, Some new results on consecutive equidivisible integers, arXiv:1510.07081 [math.NT], 2015.

Vladimir A. Letsko, Vasilii Dziubenko On consecutive equidivisible integers (in Russian)

FORMULA

a(2n+1) = 1, since numbers with an odd number of divisors must be squares. If n is not divisible by 3, a(2n) <= 7.

CROSSREFS

Cf. A000005, A072507.

Cf. A006558.

Sequence in context: A178810 A319627 A217668 * A130008 A101809 A127203

Adjacent sequences:  A119476 A119477 A119478 * A119480 A119481 A119482

KEYWORD

hard,more,nonn

AUTHOR

Franklin T. Adams-Watters, Jul 26 2006

EXTENSIONS

Edited by Dean Hickerson, Aug 01 2006

STATUS

approved

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Last modified October 17 19:36 EDT 2018. Contains 316293 sequences. (Running on oeis4.)