A319045 -id:A319045 - cp's OEIS frontend

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

Jon E. Schoenfield, Dec 22 2018

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

			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, ...;

Cf. A000005, A005408, A292580, A319045.

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

A325116 Length of longest run of consecutive even integers having exactly n divisors.

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 2, 1, 3, 1, 1, 1

Offset: 1

David Wasserman, Mar 27 2019

The start of the first run of exactly k consecutive even integers having exactly n divisors is A325117(n,k).
If m does not divide n, then a(n) < 2^(m-1). Proof: 2^(m-1) consecutive even integers must include one congruent to 2^(m-1) mod 2^m, and the number of divisors of this one is a multiple of m.
For m > 2, if 2*(m-1) does not divide n, then a(n) < 2^(m-1). Proof: 2^(m-1) consecutive even integers must include two congruent to 2^(m-2) mod 2^(m-1). These are 2^(m-2)*x and 2^(m-2)*(x+2) for some odd x, and x and x+2 each have n/(m-1) divisors. If 2*(m-1) does not divide n, then n/(m-1) is odd. Any number with an odd number of divisors is a square, so x and x+2 are squares, but squares cannot differ by 2.
14 <= a(24) <= 15. Dickson's conjecture implies a(24)=15.

Cf. A119479 (with consecutive integers), A319045 (with consecutive odd integers).

Cf. A325117.

A323743 Table read by rows: row n lists the numbers k for which there exist only finitely many runs of n consecutive integers whose number-of-divisors function sums to k.

Original entry on oeis.org

1, 3, 4, 5, 5, 7, 8, 9, 8, 9, 11, 12, 13, 14, 15, 10, 13, 15, 17, 18, 19, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 16, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 20, 22, 24, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39

Offset: 1

Jon E. Schoenfield, Apr 02 2019

Row n lists the numbers k such that
  0 < |{m : Sum_j={m..m+n-1} tau(j) = k}| < infinity
where tau(j) = A000005(j) is the number of divisors of j.

			There is only one number with exactly 1 divisor (namely, k=1), but there are infinitely many numbers with j divisors for every j >= 2, so row 1 consists only of the single term 1.
The sequence of values tau(k) for k >= 1 is A000005, which begins 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ..., from which the sums of two consecutive terms are 1+2=3, 2+2=4, 2+3=5, 3+2=5, 2+4=6, 4+2=6, 2+4=6, 4+3=7, 3+4=7, ...; no number j < 3 appears as such a sum, every j >= 6 appears infinitely many times as such a sum, and each j in {3,4,5} appears as such a sum only finitely many times, so row 2 is {3, 4, 5}.
Row 3 does not contain 6 as a term because there exists no run of 3 consecutive numbers whose sum of tau values is exactly 6.
The first six rows of the table are as follows:
  row 1: {1};
  row 2: {3, 4, 5};
  row 3: {5, 7, 8, 9};
  row 4: {8, 9, 11, 12, 13, 14, 15};
  row 5: {10, 13, 15, 17, 18, 19};
  row 6: {14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27}.

Cf. A000005, A005237, A006558, A048892, A072507, A100366, A119479, A141621, A284596, A284597, A292580, A319037, A319045, A319046.

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A325116 Length of longest run of consecutive even integers having exactly n divisors.

Views

Author

Keywords

Comments

Crossrefs

A323743 Table read by rows: row n lists the numbers k for which there exist only finitely many runs of n consecutive integers whose number-of-divisors function sums to k.

Views

Author

Keywords

Comments

Examples

Crossrefs