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A119479
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Length of longest run of consecutive integers having exactly n divisors.
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11
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1, 2, 1, 3, 1, 5, 1, 7, 1, 3, 1, 15, 1, 3, 1, 7, 1, 5, 1, 7, 1, 3, 1
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OFFSET
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1,2
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COMMENTS
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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. On the other hand Dmitry Petukhov found a run of 15 consecutive integers each having 12 divisors. It starts with 66387422053662391209161093722597723545. - Vladimir Letsko, Apr 07 2022
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
One of any 32 consecutive integers is divisible by 16 but not by 32. The number of divisors of such an integer is divisible by 5. Therefore a(24) <= 31 and a(48) <= 31.
768369049267672356024049141254832375543516 starts a run of 17 consecutive integers each having 24 divisors. Hence 17 <= a(24) <= 31.
17668887847524548413038893976018715843277693308027547 starts a run of 20 consecutive integers each having 48 divisors. Therefore 20 <= a(48) <= 31. (End)
Using Dmitry Petukhov's programs, Eugene Zhilitsky found a chain of 13 consecutive numbers with 36 divisors each. It starts with 1041358820322424595598704771003665679363657167077976401029442221233039097. Hence 13 <= a(36) <= 15. (End)
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LINKS
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FORMULA
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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.
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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