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A319045 Length of longest run of consecutive odd numbers having exactly n divisors. 3
1, 3, 1, 8, 1, 8, 1, 17, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n)=1 for n odd, since every number with an odd number of divisors is a square, and no two squares are consecutive odd numbers.

The start of the first run of exactly k consecutive odd numbers having exactly n divisors is A319046(n,k).

From David Wasserman, May 04 2019: (Start)

7 <= a(10) <= 8.

14 <= a(12) <= 59. Dickson's conjecture implies a(12) >= 39. Schinzel's Hypothesis H implies a(12) >= 41. (End)

LINKS

Table of n, a(n) for n=1..9.

EXAMPLE

From David Wasserman, May 04 2019: (Start)

A run of 17 consecutive odd numbers with 8 divisors begins at 237805775327, so a(8) >= 17; a run of 18 or more consecutive odd numbers would include at least two that are multiples of 9, and every multiple of 9 having 8 divisors is also a multiple of 27, but the two multiples of 9 cannot both be multiples of 27, so a(8) = 17.

A run of 5 consecutive odd numbers with 14 divisors begins at 10943266106145622193005970311, so a(14) >= 5. A run of 6 consecutive odd numbers with 14 divisors would include at least two that are multiples of 3, and these two would differ by 6. These must be 3^13, 3^6*p for p prime > 3, or 3*p^6 for p prime > 3. But 3*p^6 = 3 (mod 27), while 3^13 and 3^6*p = 0 (mod 27), so no two of these can differ by 6. Therefore no such run exists, and a(14) = 5. (End)

CROSSREFS

Cf. A119479 (analog for consecutive integers), A319046.

Sequence in context: A179393 A217598 A280207 * A182510 A112420 A010288

Adjacent sequences:  A319042 A319043 A319044 * A319046 A319048 A319049

KEYWORD

nonn,more,hard

AUTHOR

Jon E. Schoenfield, Dec 22 2018

EXTENSIONS

a(6)-a(9) from David Wasserman, Feb 07 2019

STATUS

approved

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Last modified June 15 17:43 EDT 2019. Contains 324142 sequences. (Running on oeis4.)