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 A317705 Matula-Goebel numbers of series-reduced powerful rooted trees. 15
 1, 4, 8, 16, 32, 49, 64, 128, 196, 256, 343, 361, 392, 512, 784, 1024, 1372, 1444, 1568, 2048, 2401, 2744, 2809, 2888, 3136, 4096, 5488, 5776, 6272, 6859, 8192, 9604, 10976, 11236, 11552, 12544, 16384, 16807, 17161, 17689, 19208, 21952, 22472, 23104, 25088 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A positive integer n is a Matula-Goebel number of a series-reduced powerful rooted tree iff either n = 1 or n is a powerful number (meaning its prime multiplicities are all greater than 1) whose prime indices are all Matula-Goebel numbers of series-reduced powerful rooted trees, where a prime index of n is a number m such that prime(m) divides n. LINKS EXAMPLE The sequence of Matula-Goebel numbers of series-reduced powerful rooted trees together with the corresponding trees begins:     1: o     4: (oo)     8: (ooo)    16: (oooo)    32: (ooooo)    49: ((oo)(oo))    64: (oooooo)   128: (ooooooo)   196: (oo(oo)(oo))   256: (oooooooo)   343: ((oo)(oo)(oo))   361: ((ooo)(ooo))   392: (ooo(oo)(oo))   512: (ooooooooo)   784: (oooo(oo)(oo)) MATHEMATICA powgoQ[n_]:=Or[n==1, And[Min@@FactorInteger[n][[All, 2]]>1, And@@powgoQ/@PrimePi/@FactorInteger[n][[All, 1]]]]; Select[Range[1000], powgoQ] (* Gus Wiseman, Aug 31 2018 *) (* Second program: *) Nest[Function[a, Append[a, Block[{k = a[[-1]] + 1}, While[Nand[AllTrue[#[[All, -1]], # > 1 & ], AllTrue[PrimePi[#[[All, 1]] ], MemberQ[a, #] &]] &@ FactorInteger@ k, k++]; k]]], {1}, 44] (* Michael De Vlieger, Aug 05 2018 *) CROSSREFS Cf. A000081, A001694, A061775, A111299, A214577, A276625, A277098, A303431. Cf. A317102, A317707, A317708, A317709, A317710, A317711, A317712, A317717, A317718, A317719. Sequence in context: A293780 A048168 A175341 * A318692 A291441 A331967 Adjacent sequences:  A317702 A317703 A317704 * A317706 A317707 A317708 KEYWORD nonn AUTHOR Gus Wiseman, Aug 04 2018 EXTENSIONS Rewritten by Gus Wiseman, Aug 31 2018 STATUS approved

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Last modified September 17 16:32 EDT 2021. Contains 347487 sequences. (Running on oeis4.)