A325662 -id:A325662 - cp's OEIS frontend

A325663 Matula-Goebel numbers of not necessarily regular rooted stars.

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 20, 22, 24, 25, 27, 30, 31, 32, 33, 36, 40, 44, 45, 48, 50, 54, 55, 60, 62, 64, 66, 72, 75, 80, 81, 88, 90, 93, 96, 99, 100, 108, 110, 120, 121, 124, 125, 127, 128, 132, 135, 144, 150, 155, 160, 162, 165, 176

Offset: 1

Gus Wiseman, May 13 2019

nonn

Products of members of A007097.

A rooted star is a rooted tree whose branches are all rooted paths.

			The sequence of rooted stars together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  15: ((o)((o)))
  16: (oooo)
  18: (o(o)(o))
  20: (oo((o)))
  22: (o(((o))))
  24: (ooo(o))
  25: (((o))((o)))
  27: ((o)(o)(o))
  30: (o(o)((o)))

Amiram Eldar, Table of n, a(n) for n = 1..10538 (terms up to A007097(12))

Cf. A007097, A056239, A061775, A109082, A109129, A112798, A196050, A324924, A325614, A325661, A325662.

rpQ[n_]:=n==1||PrimeQ[n]&&rpQ[PrimePi[n]];
Select[Range[100],And@@rpQ/@First/@FactorInteger[#]&]
(* generates terms <= A007097(max) *) seq[max_] := Module[{ps = NestList[Prime@# &, 1, max], psmax, s = {1}, emax, s1, s2}, pmax = Max[ps]; Do[p = ps[[k]]; emax = Floor[Log[p, pmax]]; s1 = p^Range[0, emax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= pmax &]; s = Union[s, s2], {k, 2, Length[ps]}]; s]; seq[7] (* Amiram Eldar, Jul 26 2024 *)

Sum_{n>=1} 1/a(n) = Product_{k>=1} A007097(k)/(A007097(k)-1) = 4.30328607286382284593... . - Amiram Eldar, Jul 26 2024

A325608 Numbers whose factorization into factors prime(i)/i does not have weakly decreasing nonzero multiplicities.

Original entry on oeis.org

147, 245, 294, 357, 490, 511, 539, 588, 595, 637, 681, 714, 735, 845, 847, 853, 867, 903, 980, 1022, 1029, 1043, 1078, 1083, 1135, 1176, 1183, 1190, 1239, 1241, 1267, 1274, 1309, 1362, 1421, 1428, 1445, 1470, 1505, 1519, 1547, 1553, 1563, 1617, 1631, 1690

Offset: 1

Gus Wiseman, May 12 2019

nonn

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example, 147 = q(1)^5 q(2) q(4)^2 has multiplicities (5,1,2), which are not weakly decreasing, so 147 belongs to the sequence.

Cf. A001222, A056239, A112798, A118914.
Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324935.
q-factorization: A324922, A324923, A324924, A324931, A325613, A325614, A325615, A325660, A325662.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Select[Range[1000],!GreaterEqual@@Length/@Split[difac[#]]&]

cp's OEIS Frontend

A325663 Matula-Goebel numbers of not necessarily regular rooted stars.

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