Records and First Positions of Records of A113504. by Michael Thomas De Vlieger, St. Louis, MO 201709042100, revised 201709052200. n = index of record setting value used in this document. k = A115212(n) = record-setting value in A113504. m = A115213(n) = first position in A113504 of k. delta = first differences of k, with the first term of delta = first term of k 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. This notation serves to succinctly illustrate the prime decomposition of m. A287352 is useful for numbers that are products of more widely separated primes. n m k delta PC(m) 1 0 1 1 1 2 2 2 1 1 3 4 3 1 10 4 8 5 2 100 5 16 8 3 1000 6 32 13 5 10000 7 64 28 15 100000 8 128 49 21 1000000 9 256 76 27 10000000 10 512 160 84 100000000 11 768 174 14 100000001 12 1024 189 15 1000000000 13 1025 224 35 3.0.10 14 1040 228 4 100023 15 1056 232 4 1000013 16 1088 242 10 1000006 17 1152 246 4 100000010 18 1280 293 47 100000002 19 1536 306 13 1000000001 20 2048 350 44 10000000000 21 2051 353 3 4.58 22 2060 354 1 1.0.2.24 23 2096 355 1 1.0.0.0.31 24 2113 362 7 319 25 2120 363 1 1.0.0.2.13 26 2128 366 3 100034 27 2144 376 10 1.0.0.0.0.18 28 2177 392 16 4.60 29 2178 393 1 11030 30 2192 394 1 1.0.0.0.32 31 2208 395 1 1000017 32 2240 408 13 10000021 33 2305 434 26 3.86 34 2312 435 1 10060 35 2320 437 2 100027 36 2336 439 2 1.0.0.0.0.20 37 2432 455 16 10000007 38 2561 486 31 6.39 39 2564 487 1 1.0.115 40 2568 489 2 1.0.0.1.26 41 2576 492 3 100035 42 2592 503 11 100001000 43 2624 523 20 1.0.0.0.0.0.12 44 2688 546 23 100000012 45 2816 586 40 100000004 46 3073 671 85 4.81 47 3088 673 2 1.0.0.0.43 48 3104 678 5 1.0.0.0.0.24 49 3136 692 14 10000030 50 3200 717 25 100000020 51 3328 746 29 100000005 52 3584 821 75 1000000003 53 4099 858 37 565 54 4116 859 1 101200 55 4120 861 2 1.0.0.2.24 56 4129 864 3 568 57 4144 865 1 100038 58 4192 870 5 1.0.0.0.0.31 59 4225 874 4 3030 60 4240 876 2 1.0.0.0.2.13 61 4256 878 2 1000034 62 4288 882 4 1.0.0.0.0.0.18 63 4353 884 2 2.228 64 4368 886 2 1000122 65 4384 891 5 1.0.0.0.0.32 66 4416 902 11 10000017 67 4480 913 11 100000021 68 4609 923 10 5.76 69 4672 926 3 1.0.0.0.0.0.20 70 4864 929 3 100000007 71 5121 986 57 2.0.102 72 5136 989 3 1.0.0.0.1.26 73 5152 993 4 1000035 74 5184 1002 9 1000001000 75 5248 1026 24 1.0.0.0.0.0.0.12 76 5376 1058 32 1000000012 77 5632 1077 19 1000000004 78 6145 1197 120 3.198 79 6152 1198 1 1.0.0.135 80 6160 1200 2 1000211 81 6176 1207 7 1.0.0.0.0.43 82 6208 1216 9 1.0.0.0.0.0.24 83 6272 1238 22 100000030 84 6400 1315 77 1000000020 85 6656 1372 57 1000000005 86 7168 1521 149 10000000003 87 8195 1568 47 3.2.30 88 8197 1569 1 4.189 89 8199 1613 44 2.0.154 90 8203 1616 3 6.109 91 8205 1618 2 2.1.98 92 8211 1622 4 2232 93 8213 1623 1 14.29 94 8217 1625 2 2.0.3.18 95 8220 1626 1 1.0.1.1.30 96 8227 1632 6 8.76 97 8229 1634 2 2.4.41 98 8233 1635 1 1033 99 8241 1636 1 2.11.6 100 8259 1640 4 2.400 101 8261 1641 1 5.128 102 8265 1642 1 2152 103 8273 1643 1 1038 104 8289 1645 2 2.0.0.61 105 8304 1646 1 1.0.0.0.1.38 106 8323 1653 7 463 107 8325 1654 1 20109 108 8329 1655 1 1045 109 8337 1656 1 2.2.74 110 8344 1657 1 1.0.0.3.31 111 8353 1658 1 1046 112 8368 1659 1 1.0.0.0.98 113 8385 1664 5 2138 114 8400 1665 1 10001101 115 8416 1668 3 1.0.0.0.0.55 116 8451 1669 1 2.0.0.63 117 8528 1670 1 100057 118 8544 1672 2 1.0.0.0.0.1.22 119 8577 1688 16 2.0.160 120 8640 1695 7 1000001001 121 8707 1729 34 1085 122 8769 1730 1 2.10.10 123 8800 1731 1 10000202 124 8833 1736 5 5.0.16 125 8848 1739 3 1.0.0.0.3.18 126 8864 1747 8 1.0.0.0.0.58 127 8961 1772 25 2.8.17 128 8976 1775 3 1000132 129 8992 1782 7 1.0.0.0.0.59 130 9024 1807 25 1.0.0.0.0.0.1.13 131 9088 1857 50 1.0.0.0.0.0.0.19 132 9219 1948 91 2.2.81 133 9249 1949 1 2.439 134 9256 1950 1 1.0.0.5.18 135 9264 1954 4 1.0.0.0.1.42 136 9281 1964 10 1149 137 9288 1965 1 1.0.0.1.0.0.12 138 9296 1968 3 1.0.0.0.3.19 139 9312 1970 2 1.0.0.0.0.1.23 140 9345 1973 3 2.1.1.20 141 9348 1974 1 10165 142 9352 1976 2 1.0.0.3.35 143 9360 1977 1 10001013 144 9376 1979 2 1.0.0.0.0.61 145 9408 1985 6 100000120 146 9504 1991 6 100001003 147 9536 1997 6 1.0.0.0.0.0.34 148 9600 2017 20 1000000110 149 9729 2089 72 2076 150 9730 2090 1 1.2.1.30 151 9732 2091 1 1.0.1.139 152 9760 2092 1 1.0.0.0.0.2.15 153 9792 2101 9 100000105 154 9856 2107 6 100000031 155 9984 2110 3 1000000014 156 10243 2116 6 1255 157 10305 2117 1 2.0.1.47 158 10306 2118 1 1.686 159 10308 2119 1 1.0.1.147 160 10312 2121 2 1.0.0.208 161 10320 2124 3 1.0.0.0.1.1.11 162 10336 2125 1 1000061 163 10369 2126 1 1272 164 10376 2127 1 1.0.0.210 165 10432 2128 1 1.0.0.0.0.0.37 166 10497 2133 5 2.487 167 10528 2136 3 1.0.0.0.0.3.11 168 10560 2141 5 100000112 169 10624 2144 3 1.0.0.0.0.0.0.22 170 10753 2158 14 1311 171 10760 2159 1 1.0.0.2.54 172 10768 2162 3 1.0.0.0.121 173 10784 2171 9 1.0.0.0.0.67 174 10816 2195 24 10000050 175 10880 2229 34 100000024 176 11008 2245 16 1.0.0.0.0.0.0.0.13 177 11265 2374 129 2.1.130 178 11296 2375 1 1.0.0.0.0.70 179 11328 2376 1 1.0.0.0.0.0.1.15 180 11392 2383 7 1.0.0.0.0.0.0.23 181 11520 2414 31 10000000101 182 11776 2439 25 1000000008 183 12291 2479 40 2.5.46 184 12293 2480 1 8.110 185 12297 2481 1 2.563 186 12305 2483 2 3.6.19 187 12312 2484 1 10010006 188 12321 2488 4 2.0.10.0 189 12353 2489 1 5.183 190 12384 2496 7 1.0.0.0.0.1.0.12 191 12417 2508 12 2.568 192 12448 2511 3 1.0.0.0.0.76 193 12480 2512 1 100000113 194 12545 2526 14 3.3.38 195 12576 2528 2 1.0.0.0.0.1.30 196 12672 2536 8 1000000103 197 12801 2539 3 2.5.47 198 12808 2540 1 1.0.0.251 199 12816 2543 3 1.0.0.0.1.0.22 200 12832 2550 7 1.0.0.0.0.78 201 12864 2560 10 1.0.0.0.0.0.1.17 202 12928 2583 23 1.0.0.0.0.0.0.25 203 13056 2603 20 1000000015 204 13313 2714 111 1581 205 13328 2718 4 1000303 206 13344 2724 6 1.0.0.0.0.1.32 207 13376 2739 15 10000043 208 13440 2764 25 1000000111 209 13568 2787 23 1.0.0.0.0.0.0.0.15 210 13824 2883 96 100000000100 211 14337 3098 215 2.0.0.0.0.15 212 14340 3099 1 1.0.1.1.49 213 14344 3101 2 1.0.0.4.33 214 14352 3104 3 1000143 215 14368 3107 3 1.0.0.0.0.86 216 14400 3123 16 1000001010 217 14464 3132 9 1.0.0.0.0.0.0.29 218 14592 3187 55 1000000016 219 14848 3261 74 1000000009 220 15360 3525 264 100000000011 Remarks and observations: 1. This analysis is based on 2^14 terms of A113504. 2. The integer powers of 2 in m are {2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, ...}. Curiously, larger powers do not seem to enter m. Are there any other powers of 2 in m? 3. Primes in m: {2, 2113, 4099, 4129, 8233, 8273, 8329, 8353, 8707, 9281, 10243, 10369, 10753, 13313, ...}. 4. Odds in m: {1025, 2051, 2113, 2177, 2305, 2561, 3073, 4099, 4129, 4225, 4353, 4609, 5121, 6145, 8195, 8197, 8199, 8203, 8205, 8211, 8213, 8217, 8227, 8229, 8233, 8241, 8259, 8261, 8265, 8273, 8289, 8323, 8325, 8329, 8337, 8353, 8385, 8451, 8577, 8707, 8769, 8833, 8961, 9219, 9249, 9281, 9345, 9729, 10243, 10305, 10369, 10497, 10753, 11265, 12291, 12293, 12297, 12305, 12321, 12353, 12417, 12545, 12801, 13313, 14337, ...}. 5. First Differences of k have an interesting compression-rarefaction oscillation that appears to roughly culminate at a high and collapse down to a low level quite irregularly.