A016089 -id:A016089 - cp's OEIS frontend

A140258 Minimal multiples from A016089 of primes from A129729.

2, 6, 1926, 2471058, 38259378, 41218326, 600917778, 114130755846, 600929334, 28312987734, 342397209654, 722113254, 15559317470256498, 84332966140854, 20543988255894, 1314244621926, 600935058, 6739452314987202

Offset: 1

Max Alekseyev, May 16 2008

nonn

A016089 lists numbers n such that n divide n-th Lucas number A000032(n) while A129729 lists all possible prime divisors of elements of A016089 in the increasing order. This sequence lists minimal multiples from A016089 of primes from A129729.

a(n) = min { A016089(m) : A129729(n)|A016089(m) }

A129729 Primes dividing numbers k such that k divides the k-th Lucas number A000032(k).

Original entry on oeis.org

2, 3, 107, 1283, 8747, 21401, 34667, 46187, 104003, 137387, 138563, 374929, 549547, 2204243, 2771281, 2808107, 11128427, 11223683, 13497443, 14880347, 21747529, 22753547, 23712683, 33697283, 44513387, 46970929, 57395627, 65898683

Offset: 1

Alexander Adamchuk, May 12 2007

nonn

A prime p belongs to this sequence iff for some positive integer m, m*p divides A000032(m*p) or, alternatively, m*p belongs to A016089.

The minimum multiples from A016089 of listed primes are given by A140258.

C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Cf. A000032, A016089, A140258.

Revised and extended by Max Alekseyev, May 16 2008

A338638 a(n) = L(L(n)) mod L(n), where L = Lucas numbers = A000032.

Original entry on oeis.org

1, 0, 1, 3, 1, 1, 0, 1, 1, 7, 4, 1, 199, 1, 4, 843, 1, 1, 0, 1, 29, 123, 4, 1, 3, 199, 4, 39603, 29, 1, 5778, 1, 1, 7, 4, 17622890, 12752043, 1, 4, 39603, 7881196, 1, 5778, 1, 29, 7, 4, 1, 3, 1149851, 28143689044, 7, 29, 1, 0, 312119004790, 6643838879, 7, 4, 1

Offset: 0

Alois P. Heinz, Nov 04 2020

Alois P. Heinz, Table of n, a(n) for n = 0..4795

Cf. A000032, A005371, A016089, A213060, A263101, A338736, A338889.

b:= proc(n) local r, M, p; r, M, p:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
      do if irem(p, 2, 'p')=1 then r:=
        `if`(nargs=1, r.M, r.M mod args[2]) fi;
         if p=0 then break fi; M:=
        `if`(nargs=1, M.M, M.M mod args[2])
      od; (r.<<2, 1>>)[1$2]
    end:
a:= n-> (f-> b(f$2) mod f)(b(n)):
seq(a(n), n=0..60);

Table[Mod[LucasL[LucasL[n]],LucasL[n]],{n,0,60}] (* Harvey P. Dale, Jul 04 2022 *)

a(n) = A005371(n) mod A000032(n).

a(n) = 0 for n in { A016089 }.

A325630 Numbers k such that A000110(k) is divisible by k.

Original entry on oeis.org

1, 2, 35, 16833, 16989, 23684

Offset: 1

Vaclav Kotesovec, Sep 07 2019

No other terms below 50000.
From Amiram Eldar, Jun 20 2024: (Start)
Numbers k such that A166226(k) = 0.
All the terms above 2 are composites since A166226(p) == 2 (mod p) for prime p. (End)
No other terms below 90000. - Michael S. Branicky, Jan 09 2025

			35 is in the sequence because A000110(35) = 35 * 8045720086273150473238297902.

Cf. A000110, A051130, A066562, A073877, A113883, A166226, A280727, A325935.

Cf. A014847, A016089, A023172, A051177, A056848.

Select[Range[1000], Divisible[BellB[#], #] &]

A272318 Integer values of Lucas number A000032(n)/n.

Original entry on oeis.org

1, 3, 321, 3572225067, 44308057022098435739157981016569

Offset: 1

Peter M. Chema, Apr 25 2016

nonn

The digital root of this sequence appears to be alternately 3 and 6, aside from the initial term of "1".

A subsequence of A181885. For instance, a(2)=A181885(6), a(3)=A181885(18), a(4)=A181885(54); a(5)=A181885(162); and, a(6)=A181885(486). Also 6, 18, 54, 162 and 486 are consecutive terms of the Pinot sequence A008776. Is this a coincidence?

Andrew Howroyd, Table of n, a(n) for n = 1..9

Cf. A000032, A008776, A016089, A167745, A181885, A167745 and A016089.

LucasL[#]/# & /@ Range@ 1200 /. _Rational -> Nothing (* Version 10.2, or *)
Select[Array[LucasL[#]/# &, {1200}], IntegerQ] (* Michael De Vlieger, Apr 25 2016 *)

a(n) = A000032(A016089(n))/n. - Michel Marcus, Apr 25 2016

A372898 Numbers k that divide the k-th Padovan number.

Original entry on oeis.org

1, 2, 4, 16, 25, 27, 59, 69, 101, 167, 173, 211, 223, 271, 307, 317, 347, 387, 422, 449, 463, 593, 599, 607, 634, 691, 694, 719, 809, 821, 829, 844, 853, 877, 883, 898, 926, 991, 997, 1097, 1117, 1151, 1163, 1181, 1197, 1198, 1231, 1319, 1369, 1388, 1451, 1453, 1481

Offset: 1

Amiram Eldar, May 16 2024

Numbers k such that k | A000931(k).

			2 is a term since A000931(2) = 0 is divisible by 2.
27 is a term since A000931(27) = 351 = 13 * 27 is divisible by 27.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000931.

Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

With[{m = 1500}, Position[LinearRecurrence[{0, 1, 1}, {0, 0, 1}, m]/Range[m], _?IntegerQ] // Flatten]

lista(kmax) = {my(p1 = 0, p2 = 0, p3 = 1, p4); print1("1, 2, "); for(k = 4, kmax, p4 = p1 + p2; if(!(p4 % k), print1(k, ", ")); p1 = p2; p2 = p3; p3 = p4);}

A372899 Numbers k that divide the k-th companion Pell number.

Original entry on oeis.org

1, 2, 6, 18, 54, 66, 162, 198, 486, 594, 726, 1314, 1458, 1782, 2178, 2838, 3222, 3942, 4374, 5346, 5778, 5874, 6534, 7986, 8514, 8646, 9666, 11826, 13122, 14454, 16038, 17334, 17622, 19602, 23958, 25542, 25938, 28998, 31218, 35442, 35478, 39366, 43362, 48114

Offset: 1

Amiram Eldar, May 16 2024

nonn

Numbers k such that k | A002203(k).

			2 is a term since A002203(2) = 6 = 2 * 3 is divisible by 2.
6 is a term since A002203(6) = 198 = 6 * 33 is divisible by 6.

Amiram Eldar, Table of n, a(n) for n = 1..327

Cf. A002203.

Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Select[Range[50000], Divisible[LucasL[#, 2], #] &]

lista(kmax) = {my(p1 = 2, p2 = 6, p3); print1("1, 2, "); for(k = 3, kmax, p3 = p1 + 2*p2; if(!(p3 % k), print1(k, ", ")); p1 = p2; p2 = p3);}

A372900 Numbers k that divide the k-th term of Narayana's cows sequence.

Original entry on oeis.org

1, 6, 12, 52, 390, 650, 663, 2077, 11479, 31671, 41158, 43508, 104894, 123682, 127370, 170819, 175075, 191516, 266247, 274378, 327159, 341638, 366903, 383847, 733985, 1236087, 1755063, 1763775, 2277964, 2364654, 3165126, 6726156, 7007823, 7221084, 10903815

Offset: 1

Amiram Eldar, May 16 2024

nonn

Numbers k such that k | A000930(k).

			6 is a term since A000930(6) = 6 is divisible by 6.
12 is a term since A000930(12) = 60 = 5 * 12 is divisible by 12.

Amiram Eldar, Table of n, a(n) for n = 1..43

Cf. A000930.

Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

With[{m = 50000}, Position[LinearRecurrence[{1, 0, 1}, {1, 1, 2}, m]/Range[m], _?IntegerQ] // Flatten]

lista(kmax) = {my(nc1 = 1, nc2 = 1, nc3 = 2, nc4); print1("1, "); for(k = 4, kmax, nc4 = nc1 + nc3; if(!(nc4 % k), print1(k, ", ")); nc1 = nc2; nc2 = nc3; nc3 = nc4);}

A372901 Numbers k that divide the k-th central Delannoy number.

Original entry on oeis.org

1, 3, 9, 21, 27, 81, 171, 189, 217, 243, 297, 351, 729, 903, 1547, 2187, 3591, 3661, 4131, 5499, 5967, 6019, 6561, 7533, 8001, 11997, 13203, 14217, 15309, 17181, 19683, 20601, 22113, 22599, 23529, 24297, 25659, 26163, 26319, 26487, 28441, 30051, 33021, 37179, 37791

Offset: 1

Amiram Eldar, May 16 2024

nonn

Numbers k such that k | A001850(k).

			3 is a term since A001850(3) = 63 = 3 * 21 is divisible by 3.
9 is a term since A001850(9) = 1462563 = 9 * 162507 is divisible by 9.

Amiram Eldar, Table of n, a(n) for n = 1..307

Cf. A001850.

Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

Select[Range[1000], Divisible[LegendreP[#, 3], #] &]

lista(kmax) = {my(cd0 = 1, cd1 = 3, cd2); print1("1, "); for(k = 2, kmax, cd2 = (3*(2*k-1)*cd1 - (k-1)*cd0)/k; if(!(cd2 % k), print1(k, ", ")); cd0 = cd1; cd1 = cd2);}

A372902 Numbers k that divide the k-th large Schröder number.

Original entry on oeis.org

1, 2, 6, 33, 42, 154, 198, 258, 270, 342, 850, 1170, 1666, 1806, 2295, 2574, 3262, 3366, 3834, 4070, 4654, 4970, 5439, 6006, 6118, 6162, 6699, 7095, 7254, 7497, 7595, 10241, 11475, 12642, 14014, 15345, 17470, 17670, 18018, 19845, 22446, 23994, 24570, 24651, 25245, 25974, 26334

Offset: 1

Amiram Eldar, May 16 2024

nonn

Numbers k such that k | A006318(k).

			2 is a term since A001850(2) = 6 = 2 * 3 is divisible by 2.
6 is a term since A001850(6) = 1806 = 6 * 301 is divisible by 6.

Amiram Eldar, Table of n, a(n) for n = 1..434

Cf. A006318.

Similar sequences: A014847 (Catalan), A016089 (Lucas), A023172 (Fibonacci), A051177 (partition), A232570 (tribonacci), A246692 (Pell), A266969 (Motzkin).

seq[kmax_] := Module[{sc0 = 1, sc1 = 2, sc2, s = {1}}, Do[sc2 = ((6*k-3)*sc1 - (k-2)*sc0)/(k+1); If[Divisible[sc2, k], AppendTo[s, k]]; sc0 = sc1; sc1 = sc2, {k, 2, kmax}]; s]; seq[27000]

lista(kmax) = {my(sc0 = 1, sc1 = 2, sc2); print1(1, ", "); for(k = 2, kmax, sc2 = ((6*k-3)*sc1 - (k-2)*sc0)/(k+1); if(!(sc2 % k), print1(k, ", ")); sc0 = sc1; sc1 = sc2);}

cp's OEIS Frontend

A140258 Minimal multiples from A016089 of primes from A129729.

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A129729 Primes dividing numbers k such that k divides the k-th Lucas number A000032(k).

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A338638 a(n) = L(L(n)) mod L(n), where L = Lucas numbers = A000032.

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A325630 Numbers k such that A000110(k) is divisible by k.

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Mathematica

A272318 Integer values of Lucas number A000032(n)/n.

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A372898 Numbers k that divide the k-th Padovan number.

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Mathematica

PARI

A372899 Numbers k that divide the k-th companion Pell number.

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Mathematica

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A372900 Numbers k that divide the k-th term of Narayana's cows sequence.

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Mathematica

PARI

A372901 Numbers k that divide the k-th central Delannoy number.

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