Records and first position of records in A226378(n), with 0 <= n <= 10^6. by Michael Thomas De Vlieger, St. Louis, MO 201708242030, revised 201709021100. k = record-setting value in A226378. m = first position in A226378 of k. m followed by "a" appear in A002182. m followed by "b" appear in A007416 MN(m) = rev(A054841(m)) = "multiplicity notation" of m. "211" is read as 2^2 * 3^1 * 5^1 = 60. PC(k) = A287352(k) = "pi-code" notation of k. Pi-code is a list of the first differences of indices of prime divisors p of n, e.g., A287352(60) = 1,0,1,1 since 60 = 2 * 2 * 3 * 5. The code is concatenated if all digits in the code are less than 10; if not, the values are delimited by (.)."1011" is read as 2 * 2 * 3 * 5 = 60. Both these notations serve to succinctly illustrate the prime decomposition of m and k respectively. A054841 is useful for numbers that are products of small primes, while A287352 is useful for numbers that are products of more widely separated primes. n m k MN(m) PC(m) 1 0 1 - 0 2 4 a b 2 2 1 3 8 3 3 2 4 12 a b 4 21 10 5 24 a b 6 31 11 6 36 a b 7 22 4 7 48 a b 9 41 20 8 60 a b 10 211 12 9 72 11 32 5 10 108 12 23 101 11 120 a b 16 311 1000 12 180 a b 20 221 102 13 240 a b 22 411 14 14 336 24 4101 1001 15 360 a b 27 321 200 16 480 28 511 103 17 504 30 3201 111 18 576 b 31 62 11 19 720 a b 41 421 13 20 960 b 44 611 104 21 1080 48 331 10001 22 1260 a b 49 2211 40 23 1440 61 521 18 24 1680 a b 63 4111 202 25 2160 66 431 113 26 2520 a b 76 3211 107 27 3360 83 5111 23 28 3780 87 2311 28 29 4320 93 531 29 30 5040 a b 113 4211 30 31 7560 a b 128 3311 1000000 32 10080 a b 153 5211 205 33 12600 159 3221 2.14 34 15120 a b 193 4311 44 35 20160 a b 210 6211 1111 36 25200 a b 223 4221 48 37 27720 a b 229 32111 50 38 30240 262 5311 1.31 39 40320 281 7211 60 40 45360 a b 291 4411 2.23 41 50400 a b 325 5221 303 42 55440 a b 327 42111 2.27 43 60480 b 352 6311 100004 44 75600 369 4321 2.0.11 45 83160 a b 376 33111 1.0.0.14 46 90720 396 5411 10103 47 100800 b 428 6221 1.0.27 48 110880 a b 477 52111 2.0.14 49 131040 486 521101 110000 50 151200 528 5321 100013 51 166320 a b 543 43111 2.40 52 196560 561 431101 232 53 221760 a b 623 62111 4.20 54 262080 637 621101 402 55 277200 a b 670 42211 1.2.16 56 302400 686 6321 1300 57 332640 a b 778 53111 1.76 58 443520 809 72111 140 59 498960 a b 832 44111 1000005 60 524160 836 721101 1043 61 554400 a b 942 52211 1.1.35 62 655200 951 522101 2.64 63 665280 a b 1032 63111 1.0.0.1.12 64 831600 1092 43211 10122 65 982800 1096 432101 1.0.0.32 66 997920 1177 54111 5.23 Remarks and observations: 1. This analysis is based on terms 0 <= n <= 10^6. 2. Though m contains many terms of A002182, {1, 2, 6, 840, 720720, ...} are not found in m, and m contains {0, 8, 72, 108, 336, 480, 504, 576, 960, 1080, 1440, 2160, 3360, 3780, 4320, 12600, 30240, 40320, 60480, 75600, 90720, 100800, 131040, 151200, 196560, 262080, 302400, 443520, 524160, 655200, 831600, 982800, 997920, ...} that are not found in A002182. Conjectures: (All excepting m(0) = 0.) 1. Numbers in m are products of the first several consecutive primes or in the case of m(14), m(17), m(49), m(52), etc., the first primes that are consecutive but include one gap. 2. Outside of m(1), m(2), m(3), m(6), m(9), and m(10), the largest prime factor of m has multiplicity 1. 3. Outside of m(9), the multiplicities of prime factors p of m generally decrease or stay the same as p increases.