Records and first position of records in A252665 by Michael Thomas De Vlieger, St. Louis, MO 201708281445, revised 201708311730. k = record-setting value in A252665 m = first position in A252665 of k. m followed by "a" or "b" appear in A002182 or A007416 respectively. A054841(m) = multiplicity notation of m. "211" is read as 2^2 * 3 * 5 = 60. n m k A054841(m) 1 1 a b 1 0 2 4 a b 2 2 3 8 3 3 4 12 a b 4 21 5 16 b 5 4 6 24 a b 7 31 7 36 a b 9 22 8 48 a b 12 41 9 72 16 32 10 96 18 51 11 120 a b 21 311 12 144 b 28 42 13 216 30 33 14 240 a b 37 411 15 288 43 52 16 360 a b 51 321 17 432 53 43 18 480 59 511 19 576 b 66 62 20 720 a b 92 421 21 1080 103 331 22 1440 150 521 23 2160 188 431 24 2520 a b 189 3211 25 2880 b 235 621 26 3600 b 239 422 27 4320 312 531 28 5040 a b 351 4211 29 7200 396 522 30 7560 a b 400 3311 31 8640 493 631 32 10080 a b 593 5211 33 14400 b 628 622 34 15120 a b 751 4311 35 20160 a b 947 6211 36 25200 a b 954 4221 37 30240 1283 5311 38 40320 1433 7211 39 50400 a b 1632 5221 40 60480 b 2063 6311 41 75600 2074 4321 42 80640 2088 8211 43 90720 2446 5411 44 100800 b 2629 6221 45 120960 3143 7311 46 151200 3582 5321 47 181440 b 3952 6411 48 201600 4008 7221 49 221760 a b 4121 62111 50 241920 4602 8311 51 302400 5803 6321 52 362880 6046 7411 53 443520 6323 72111 54 453600 6899 5421 55 554400 a b 7217 52211 56 604800 8898 7321 57 665280 a b 9203 63111 58 831600 9235 43211 59 887040 9309 82111 60 907200 b 11220 6421 61 1108800 b 11765 62211 62 1209600 13097 8321 63 1330560 14183 73111 64 1663200 16199 53211 65 1814400 17261 7421 66 1995840 b 17908 64111 67 2217600 18143 72211 68 2419200 18612 9321 69 2661120 20952 83111 70 3326400 26513 63211 71 3991680 27676 74111 72 4989600 31630 54211 73 6350400 b 32249 6422 74 6652800 41033 73211 75 8648640 a b 42833 631111 76 9979200 b 51904 64211 Remarks and observations: 1. This analysis is based on terms 1 <= m <= 10^7. 2. Though m contains many terms of A002182, {2, 6, 60, 180, 840, 1260, 1680, 27720, 45360, 55440, 83160, 110880, 166320, 277200, 332640, 498960, 720720, 1081080, 1441440, 2162160, 2882880, 3603600, 4324320, 6486480, 7207200, ...} are not found in m, and m contains terms that are not found in A002182. The trend across A218320 and A252665 is that increasingly many terms 1 <= m <= 10^7 are not in A002182. A few more terms m (i.e., 16, 144, 576, ...) appear in A007416. Conjectures: 1. Numbers in m are products of the first several consecutive primes. 2. The largest prime factor of m generally has multiplicity 1. 3. The multiplicities of prime factors p of m generally decrease or stay the same as p increases.