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A325117 Irregular table read by rows: T(n,k) is the start of the first run of exactly k consecutive even integers having exactly n divisors. 1
2, 4, 14, 0, 6, 16, 12, 18, 64, 24, 40, 182, 36, 48, 1024, 60, 198, 348, 9050, 25180, 25658650, 584558736346, 4096, 192, 144, 120, 918, 5430, 65536, 180, 17298, 262144, 240, 6640, 4413038, 576, 3072, 4194304, 360, 3400, 19548, 134044, 182644, 7126044, 359208340, 16074693138, 419893531348, 214932235538 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

The number of terms in row n is A325116(n).

2.46*10^12 <= T(24,11) <= 299005907036986132.

T(24,14) <= 1010085195622895590495442.

T(30,3) <= 1359906389476760004389052496.

5.17*10^12 < T(36, 6) <= 13707985134823441146.

T(36, 7) <= 1678936725442128595619270138.

LINKS

Table of n, a(n) for n=2..49.

EXAMPLE

T(4,3) = 6 because 6, 8, and 10 each have 4 divisors.

T(4,2) = 0. The runs 6, 8 and 8, 10 are excluded because they are part of a longer run, and there are no other consecutive even integers with 4 divisors.

T(18, 3) does not exist. This follows from the theorem: If m = 2 mod 4, and m has 18 divisors, then m-2 does not have 18 divisors.

Proof: Let d be the number of divisors function (A000005). Recall that it is multiplicative with d(p^i)=i+1. If m = 2 mod 4 and has 18 divisors, then m/2 is odd and has 9 divisors, so m=2*r^2 for some odd r. Then m-2=2(r-1)(r+1). r-1 and r+1 are even and one of them is divisible by 4, so 2^4 divides m-2. r-1 and r+1 have no prime factors in common except 2, so if they are both divisible by odd primes, call them s and t, then m-2 is divisible by 2^4*s*t and has at least 20 divisors, contrary to hypothesis. Therefore either r-1 or r+1 is a power of 2; call it 2^j. Then the exponent of 2 in m-2 is j+2, so j+3 divides 18, so j is 3 or 6. This leaves 4 possibilities for m-2: 2*6*8, 2*8*10, 2*62*64, or 2*64*66. Of these, only 2*62*64 has 18 divisors, and 2*62*64+2 does not have 18 divisors.

T(36, 11) does not exist. Proof: Suppose 11 consecutive even numbers with 36 divisors exist. Name them n_i where n_i = i (mod 32). n_16 and n_24 cannot have 36 divisors, so the 11 numbers are n_26 through n_14. Then n_8 is 8*x^2 for some odd x. Suppose 3 | x. Then 9 | n_8, so n_2 and n_14 are divisible by 3 but not 9, and by 2 but not 4. So n_2 = 6*y^2 and n_14 = 6*z^2 for some y and z, and z^2 = y^2 + 2, which is impossible. Therefore 3 doesn't divide x. Therefore x^2 = 1 (mod 3), and n_8 = 2 (mod 3). So 3 | n_6. Suppose n_6 = 0 (mod 9). Then n_26 = 6 (mod 9). So n_26 is divisible by 3 but not 9, and by 2 but not 4. So n_26 = 6*y^2. y^2 = 1 (mod 4), so n_26 = 6 (mod 8), but by definition n_26 = 2 (mod 8), a contradiction. Therefore n_6 != 0 (mod 9). Suppose n_6 = 3 (mod 9). Then n_6 is divisible by 3 but not 9, and by 2 but not 4. So n_6 = 6*y^2. y^2 = 1 (mod 3), so n_6 = 6 (mod 9), a contradiction. Therefore n_6 != 3 (mod 9), so n_6 = 6 (mod 9). Then n_26 = 3 (mod 9). So n_26 and n_6 are divisible by 3 but not 9, and by 2 but not 4. So n_26 = 6*y^2 and n_6 = 6*z^2 for some y and z, and z^2 = y^2 + 2, which is impossible.

In the table below, the following notation will be used for terms with unknown values: F: k consecutive even integers with n divisors have been found. D: Dickson's Conjecture implies the existence of k consecutive even integers with n divisors. H: Schinzel's Hypothesis H implies the existence of k consecutive even integers with n divisors. ?: It has not been proven that k consecutive even integers with n divisors do not exist. A semicolon indicates than no further terms exist.

Table begins:

   n  T(n,1), T(n,2), ...

  ==  =======================================================

   2  2;

   3  4;

   4  14, 0, 6;

   5  16;

   6  12, 18;

   7  64;

   8  24, 40, 182;

   9  36;

  10  48;

  11  1024;

  12  60, 198, 348, 9050, 25180, 25658650, 584558736346;

  13  4096;

  14  192;

  15  144;

  16  120, 918, 5430;

  17  65536;

  18  180, 17298;

  19  262144;

  20  240, 6640, 4413038;

  21  576;

  22  3072;

  23  4194304;

  24  360, 3400, 19548, 134044, 182644, 7126044, 359208340, 16074693138, 419893531348, 214932235538, F, D, D, F, D;

  25  1296;

  26  12288;

  27  900;

  28  960, 640062, 32858781246;

  29  268435456;

  30  720, 110796496, F;

  31  1073741824;

  32  840, 18088, 180726;

  33  9216;

  34  196608;

  35  5184;

  36  1260, 41650, 406780, 3237731546, 3651712573692, F, F, ?, ?, ?;

CROSSREFS

Cf. A292580 (analog for consecutive integers), A319046 (analog for consecutive odd integers), A325116.

Sequence in context: A327443 A056678 A218655 * A193232 A189486 A131758

Adjacent sequences:  A325114 A325115 A325116 * A325118 A325119 A325120

KEYWORD

nonn,tabf,more

AUTHOR

David Wasserman, Mar 27 2019

STATUS

approved

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Last modified July 8 04:16 EDT 2020. Contains 335504 sequences. (Running on oeis4.)