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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 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.)