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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.

%I A319046 #29 May 05 2019 11:52:24
%S A319046 1,23,11,3,9,15,33,91,299,213,1383,3091,8129,81,45,243,3175,2523,
%T A319046 3682662467,164406964254894462023
%N 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.
%C A319046 The number of terms in row n is A319045(n).
%C A319046 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.
%C A319046 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.
%C A319046 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.
%C A319046 T(6,7) <= 7483570769727848971899774228580919;
%C A319046 T(6,7) > 3*10^22. - _David Wasserman_, May 04 2019
%C A319046 10^17 < T(6,8) <= 620228749187663825311276520397486295457519. - _David Wasserman_, Feb 05 2019
%C A319046 Row 7 consists of the single term T(7,1) = 3^6 = 729.
%C A319046 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.
%C A319046 T(8,17) = 237805775327. - _David Wasserman_, Feb 07 2019
%C A319046 T(10,7) <= 3*(7364195527360905184867386522361)^4 - 4 (approx. 8.8*10^123). - _David Wasserman_, May 04 2019
%C A319046 T(12,14) <= 1569073892509234696810905887582957. - _David Wasserman_, May 04 2019
%C A319046 1.7*10^14 < T(14,4) <= 4365641192113347078119. - _David Wasserman_, May 04 2019
%C A319046 T(14, 5) <= 10943266106145622193005970311. - _David Wasserman_, May 04 2019
%e A319046 T(1,1) = 1 because 1 is the first (and only) number having 1 divisor.
%e A319046 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.
%e A319046 T(2,2) = 11 because it is the first odd prime that begins a run of exactly 2 consecutive odd numbers that are prime.
%e A319046 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.)
%e A319046 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.
%e A319046 Table begins:
%e A319046    n  T(n,1), T(n,2), ...
%e A319046   ==  =======================================================
%e A319046    1  1;
%e A319046    2  23, 11, 3;
%e A319046    3  9;
%e A319046    4  15, 33, 91, 299, 213, 1383, 3091, 8129;
%e A319046    5  81;
%e A319046    6  45, 243, 3175, 2523, 3682662467, 164406964254894462023, ...;
%e A319046    7  729;
%e A319046    8  105, 663, 6095, 10503, 35119, 58345, 195831, 247347, 1123281, 943607, 19235031, 148720547, 107473247, 1260718031, 21470685, ...;
%e A319046    9  225;
%e A319046   10  405, 127251, 490219371, ...;
%e A319046   11  59049;
%e A319046   12  315, 2275, 22473, 1389683, 10753975, ...;
%e A319046   13  531441;
%e A319046   14  3645, 26890623, 136349453140621, ...;
%e A319046   15  2025;
%e A319046   16  945, 14875, 155701, 1343013, 4320561, 14906085, 88958433, 376675395, 957171679, ...;
%e A319046   17  43046721;
%e A319046   18  1575, 74725, 732665527, ...;
%e A319046   19  387420489;
%e A319046   20  2835, 244375, 608149373, ...;
%e A319046   21  18225;
%e A319046   22  295245, ...;
%e A319046   23  31381059609;
%e A319046   24  3465, 226525, 3720871, 39198573, ...;
%Y A319046 Cf. A000005, A005408, A292580, A319045.
%K A319046 nonn,more,tabf
%O A319046 1,2
%A A319046 _Jon E. Schoenfield_, Dec 22 2018
%E A319046 T(6,6) and table additions from _David Wasserman_, May 04 2019