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