A319046 Irregular table read by rows: T(n,k) is the start of the first run of exactly k consecutive odd numbers having exactly n divisors, or 0 if no such run exists.

1, 23, 11, 3, 9, 15, 33, 91, 299, 213, 1383, 3091, 8129, 81, 45, 243, 3175, 2523, 3682662467, 164406964254894462023

Offset: 1

Author

Jon E. Schoenfield, Dec 22 2018

Keywords

• nonn
• more
• tabf

Comments

The number of terms in row n is A319045(n).

For each odd n, row n contains only one term, i.e., T(n,1), since every number with an odd number of divisors is a square, and no two squares are consecutive odd numbers.

If n is prime, every number having n divisors is of the form p^(n-1) where p is an odd prime, so T(n,1) = 3^(n-1) if n is an odd prime.

Row 6 cannot cannot contain more than eight terms, because every number with six divisors is of the form p^5 or p^2 * q where p and q are distinct primes, and in any run of nine or more consecutive odd numbers, at least three would include be divisible by 3, of which at least two would not be divisible by 9 but would differ by at most 12; for any such pair of numbers (3*p1^2, 3*p2^2), p1^2 and p2^2 would differ by at most 4, and no such pair of primes (p1, p2) exists.

T(6,7) <= 7483570769727848971899774228580919;

T(6,7) > 3*10^22. - David Wasserman, May 04 2019

10^17 < T(6,8) <= 620228749187663825311276520397486295457519. - David Wasserman, Feb 05 2019

Row 7 consists of the single term T(7,1) = 3^6 = 729.

Row 8 cannot have more than 17 terms (see A319045); its first 15 terms are 105, 663, 6095, 10503, 35119, 58345, 195831, 247347, 1123281, 943607, 19235031, 148720547, 107473247, 1260718031, and 21470685.

T(8,17) = 237805775327. - David Wasserman, Feb 07 2019

T(10,7) <= 3*(7364195527360905184867386522361)^4 - 4 (approx. 8.8*10^123). - David Wasserman, May 04 2019

T(12,14) <= 1569073892509234696810905887582957. - David Wasserman, May 04 2019

1.7*10^14 < T(14,4) <= 4365641192113347078119. - David Wasserman, May 04 2019

T(14, 5) <= 10943266106145622193005970311. - David Wasserman, May 04 2019

Examples

			T(1,1) = 1 because 1 is the first (and only) number having 1 divisor.
T(2,1) = 23 because it is the first odd number having 2 divisors (i.e., the first prime) that is not part of a run of two or more consecutive odd numbers that are prime.
T(2,2) = 11 because it is the first odd prime that begins a run of exactly 2 consecutive odd numbers that are prime.
T(2,3) = 3 because it is the first (and only) number that begins a run of 3 consecutive odd numbers all of which are prime. (There exists no run of more than 3 consecutive odd numbers that are all prime, so T(2,3) is the last term in row 2.)
T(4,8) = 8129 because {8129 = 11*739, 8131 = 47*173, 8133 = 3*2711, 8135 = 5*1627, 8137 = 79*103, 8139 = 3*2713, 8141 = 7*1163, 8143 = 17*479} is the first run of 8 consecutive odd numbers with 4 divisors.
Table begins:
   n  T(n,1), T(n,2), ...
  ==  =======================================================
   1  1;
   2  23, 11, 3;
   3  9;
   4  15, 33, 91, 299, 213, 1383, 3091, 8129;
   5  81;
   6  45, 243, 3175, 2523, 3682662467, 164406964254894462023, ...;
   7  729;
   8  105, 663, 6095, 10503, 35119, 58345, 195831, 247347, 1123281, 943607, 19235031, 148720547, 107473247, 1260718031, 21470685, ...;
   9  225;
  10  405, 127251, 490219371, ...;
  11  59049;
  12  315, 2275, 22473, 1389683, 10753975, ...;
  13  531441;
  14  3645, 26890623, 136349453140621, ...;
  15  2025;
  16  945, 14875, 155701, 1343013, 4320561, 14906085, 88958433, 376675395, 957171679, ...;
  17  43046721;
  18  1575, 74725, 732665527, ...;
  19  387420489;
  20  2835, 244375, 608149373, ...;
  21  18225;
  22  295245, ...;
  23  31381059609;
  24  3465, 226525, 3720871, 39198573, ...;
		

Crossrefs

Cf. A000005, A005408, A292580, A319045.

Extensions

T(6,6) and table additions from David Wasserman, May 04 2019