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.