Problems & Puzzles: Puzzles

 Puzzle 734. "Common Sigma, Uncommon Clique" Numbers My friend Fred Schneider propose the following puzzle: I have come across an interesting sequence while trying to solve a different problem. I was wondering if any readers would like to try to extend it. A239635 (I just submitted it so I think it's still being reviewed) "Common Sigma, Uncommon Clique" Numbers  The n-th term is the minimum number s where exists a set of n numbers which each have sigma = s (Common Sigma) and which are all relatively prime to each other (Uncommon Clique). Here are the first 32 terms: 1, 12, 24, 72, 240, 360, 1440, 1440, 1440, 8640, 10080, 15120, 34560, 45360, 55440, 60480, 60480, 166320, 181440, 211680, 332640, 332640, 332640, 665280, 665280, 665280, 831600, 907200, 1663200, 2494800, 2661120, 2661120 And,  for the first 5 terms, here are the solutions sets (and member factors which show the sets are mutually relatively prime): 1) 1 : 1 = 1 2) 12 : 6 = 2 * 3 11 = 11 3) 24 : 14 = 2 * 7 15 = 3 * 5 23 = 23 4) 72 : 46 = 2 * 23 51 = 3 * 17 55 = 5 * 11 71 = 71 5) 240 : 135 = 3^3 * 5 158 = 2 * 79 203 = 7 * 29 209 = 11 * 19 239 = 239 and similar details on the last (32nd) term I have found: 32) 2661120 : 1126240 = 2^5 * 5 * 7039 1667601 = 3^3 * 13 * 4751 2306549 = 7 * 109 * 3023 2420407 = 11 * 139 * 1583 2498269 = 17 * 223 * 659 2514023 = 19 * 307 * 431 2534629 = 29 * 71 * 1231 2550217 = 23 * 110879 2550773 = 31 * 107 * 769 2566111 = 43 * 83 * 719 2572709 = 41 * 131 * 479 2599021 = 79 * 167 * 197 2605633 = 47 * 55439 2611787 = 53 * 49279 2616709 = 59 * 44351 2631463 = 89 * 29567 2640203 = 127 * 20789 2646157 = 179 * 14783 2647069 = 191 * 13859 2649793 = 239 * 11087 2650309 = 251 * 10559 2650777 = 263 * 10079 2654633 = 439 * 6047 2655337 = 503 * 5279 2657113 = 839 * 3167 2657177 = 863 * 3079 2657659 = 1151 * 2309 2657693 = 1187 * 2239 2657749 = 1259 * 2111 2657833 = 1439 * 1847 2657849 = 1511 * 1759 2661119 = 2661119 Q. Please send some more terms fo this sequence.

Contributions came from Fred Schneider

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Fred wrote:

33) 3991680
34) 3991680
35) 4324320
36) 5765760
37) 6652800
38) 6652800
39) 6652800
40) 8648640
41) 9979200
42) 9979200
43) 9979200
44) 9979200
45) 9979200
46) 17297280
47) 21621600
48) 21621600
49) 28274400
51) 31600800
52) 31600800
55) 43243200
56) 64864800
57) 64864800
59) 64864800
64) 103783680
65) 103783680

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