A200381 -id:A200381 - cp's OEIS frontend

A200995 Numbers not expressible as a product of Lucas numbers.

5, 10, 13, 15, 17, 19, 20, 23, 25, 26, 30, 31, 34, 35, 37, 38, 39, 40, 41, 43, 45, 46, 50, 51, 52, 53, 55, 57, 59, 60, 61, 62, 65, 67, 68, 69, 70, 71, 73, 74, 75, 78, 79, 80, 82, 83, 85, 86, 89, 90, 91, 92, 93, 95, 97, 100, 101, 102, 103, 104, 105, 106, 107, 109

Offset: 1

Arkadiusz Wesolowski, Jan 07 2013

nonn

a(n+1) - a(n) <= 5 for all n.

			65 = 5*13 is not the product of Lucas numbers, so 65 is in the sequence.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lucas Number
Index entries for Lucas sequences

Cf. A000032. Complement of A200381.

listMultiply[lst_] := Take[Union@Flatten@Table[lst[[i]]*lst[[j]], {i, Length[lst]}, {j, i}], 46]; A200381 = Nest[listMultiply, LucasL@Range[0, 9], 3]; Complement[Range@Last[A200381], A200381]

A201010 Integers that can be written as the product and/or quotient of Lucas numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 23, 24, 27, 28, 29, 31, 32, 33, 36, 38, 41, 42, 44, 46, 47, 48, 49, 54, 56, 57, 58, 62, 63, 64, 66, 69, 72, 76, 77, 81, 82, 84, 87, 88, 92, 93, 94, 96, 98, 99, 107, 108, 112, 114, 116, 121, 123, 124, 126, 128

Offset: 1

Arkadiusz Wesolowski, Jan 08 2013

These numbers do not occur in A178777.
The first 20 terms of this sequence are the same as in A004144 (nonhypotenuse numbers).
Integers of the form A200381(n)/A200381(m) for some m and n.

			19 is in the sequence because Lucas(9)/Lucas(0)^2 = 19.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lucas Number
Index entries for Lucas sequences

Cf. A000032, A200381, A200995, A201011. Subsequence of A178772. Complement of A201012.

maxTerm = 128; Clear[f]; f[lim_] := f[lim] = (luc = LucasL[Range[0, lim]]; luc = Delete[luc, 2];  last = luc[[-1]]; t = {1}; Do[t2 = luc[[n]]^Range[ Floor[ Log[last] / Log[ luc[[n]] ]]]; s = Select[ Union[ Flatten[ Outer[ Times, t, t2]]], # <= last &]; t = Union[t, s], {n, lim}]; maxIndex = Length[A200381 = t]; Reap[ Do[r = A200381[[n]] / A200381[[m]]; If[IntegerQ[r] && r <= maxTerm, Sow[r]], {n, 1, maxIndex}, {m, 1, maxIndex}]][[2, 1]] // Union); f[5]; f[lim = 10]; While[ Print["lim = ", lim]; f[lim] != f[lim-5], lim = lim+5]; f[lim] (* Jean-François Alcover, Jun 24 2015, after script by T. D. Noe in A200381 *)

A201012 Integers that cannot be written as the product and/or quotient of Lucas numbers.

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 26, 30, 34, 35, 37, 39, 40, 43, 45, 50, 51, 52, 53, 55, 59, 60, 61, 65, 67, 68, 70, 71, 73, 74, 75, 78, 79, 80, 83, 85, 86, 89, 90, 91, 95, 97, 100, 101, 102, 103, 104, 105, 106, 109, 110, 111, 113, 115, 117, 118, 119, 120, 122, 125, 127

Offset: 1

Arkadiusz Wesolowski, Jan 08 2013

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lucas Number
Index entries for Lucas sequences

Cf. A000032, A200381, A200995, A201011. Complement of A201010.

A201011 Primes that are Lucas primes, or that can be written as the quotient of Lucas numbers.

Original entry on oeis.org

2, 3, 7, 11, 19, 23, 29, 31, 41, 47, 107, 199, 211, 281, 521, 1103, 2161, 2207, 2521, 3571, 5779, 9349, 9901, 14503, 90481, 103681, 3010349, 11128427, 29134601, 54018521, 261399601, 370248451, 599786069, 6643838879, 10745088481, 10749957121, 10783342081

Offset: 1

Arkadiusz Wesolowski, Jan 08 2013

nonn

			23 is in the sequence because it is prime and Lucas(12)/(Lucas(0)*Lucas(4)) = 23.

Eric Weisstein's World of Mathematics, Lucas Number
Index entries for Lucas sequences

Cf. A000032, A200381, A200995, A201010, A201012. Supersequence of A005479. Subsequence of A178762.

261399601 inserted by Arkadiusz Wesolowski, Feb 05 2013

A308263 Number of ordered factorizations of n into Lucas numbers (beginning at 2) > 1.

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 1, 3, 1, 0, 1, 5, 0, 2, 0, 5, 0, 4, 0, 0, 2, 2, 0, 10, 0, 0, 1, 5, 1, 0, 0, 8, 2, 0, 0, 11, 0, 0, 0, 0, 0, 6, 0, 5, 0, 0, 1, 20, 1, 0, 0, 0, 0, 6, 0, 10, 0, 2, 0, 0, 0, 0, 3, 13, 0, 6, 0, 0, 0, 0, 0, 27, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 18, 0, 0, 2, 10

Offset: 1

Ilya Gutkovskiy, May 17 2019

nonn

Cf. A000032, A102460, A200381 (positions of nonzero terms), A200995 (positions of zeros), A308062.

terms = 88; A[] = 0; Do[A[x] = x + A[x^2] + Sum[A[x^LucasL[k]], {k, 2, 25}] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
f[n_] := f[n] = SeriesCoefficient[x^2 + Sum[x^LucasL[k], {k, 2, 25}], {x, 0, n}]; a[n_] := If[n == 1, n, Sum[If[d < n, f[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 88}]

G.f. A(x) satisfies: A(x) = x + A(x^2) + Sum_{k>=2} A(x^Lucas(k)).

cp's OEIS Frontend

A200995 Numbers not expressible as a product of Lucas numbers.

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A201010 Integers that can be written as the product and/or quotient of Lucas numbers.

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A201012 Integers that cannot be written as the product and/or quotient of Lucas numbers.

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A201011 Primes that are Lucas primes, or that can be written as the quotient of Lucas numbers.

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A308263 Number of ordered factorizations of n into Lucas numbers (beginning at 2) > 1.

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