A076240 -id:A076240 - cp's OEIS frontend

A263101 a(n) = F(F(n)) mod F(n), where F = Fibonacci = A000045.

0, 0, 1, 2, 0, 5, 12, 5, 33, 5, 1, 0, 232, 233, 55, 5, 1596, 2563, 1, 5, 987, 10946, 28656, 0, 0, 75025, 189653, 89, 1, 6765, 1, 5, 6765, 1, 9227460, 0, 24157816, 1, 63245985, 5, 1, 267914275, 433494436, 4181, 1134896405, 1, 2971215072, 0, 7778741816, 75025

Offset: 1

Alois P. Heinz, Oct 09 2015

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A000045, A002708, A007570, A076240, A127787 (where a(n)=0), A263112, A274996.

F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,
          `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):
a:= n-> p(<<0|1>, <1|1>>, F(n)$2)[1, 2]:
seq(a(n), n=1..50);

F[n_] := MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]];
p[M_, n_, k_] := Mod[#, k]& /@ If[n == 0, {{1, 0}, {0, 1}}, If[EvenQ[n], MatrixPower[p[M, n/2, k], 2], p[M, n - 1, k].M]];
a[n_] := p[{{0, 1}, {1, 1}}, F[n], F[n]][[1, 2]];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 29 2024, after Alois P. Heinz *)

alist(nn)= my(f=fibonacci); [ f(f(n))%f(n) |n<-[1..nn] ]; \\ Ruud H.G. van Tol, Dec 13 2024

a(n) = A007570(n) mod A000045(n).

A076243 Remainder when 3rd-order prime ppp(n) = A038580(n) is divided by n.

Original entry on oeis.org

0, 1, 1, 3, 2, 5, 4, 3, 8, 9, 5, 7, 10, 5, 7, 3, 2, 11, 17, 1, 20, 21, 11, 19, 12, 17, 14, 17, 18, 19, 18, 23, 28, 27, 11, 19, 15, 7, 2, 21, 40, 25, 31, 1, 19, 15, 9, 31, 46, 47, 10, 15, 43, 23, 14, 9, 17, 19, 18, 41, 24, 27, 50, 3, 14, 29, 13, 3, 4, 39, 21, 1, 47, 19, 31, 13, 6, 17

Offset: 1

Labos Elemer, Oct 08 2002

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240, A076241, A076242.

MapIndexed[Mod[#1, First@ #2] &, Nest[Prime, Range@ 79, 3]] (* Michael De Vlieger, Jul 22 2017 *)

a(n) = ppp(n) mod n = A038580(n) mod n.

A076241 Remainder when 2nd order prime pp(n)=A006450(n) is divided by n.

Original entry on oeis.org

0, 1, 2, 1, 1, 5, 3, 3, 2, 9, 6, 1, 10, 9, 1, 1, 5, 13, 8, 13, 10, 5, 17, 5, 9, 1, 23, 27, 19, 17, 27, 3, 14, 15, 19, 13, 31, 17, 16, 31, 38, 37, 35, 27, 31, 21, 28, 17, 12, 47, 43, 43, 39, 31, 26, 45, 13, 1, 17, 23, 17, 53, 11, 15, 1, 53, 10, 25, 64, 41, 38, 41, 68, 33, 59, 63, 65

Offset: 1

Labos Elemer, Oct 08 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240-A076243.

[NthPrime(NthPrime(n)) mod(n): n in [1..100]]; // Vincenzo Librandi, Jul 10 2017

Table[Mod[Prime[Prime[n]], n], {n, 100}] (* Vincenzo Librandi, Jul 10 2017 *)

a(n) = prime(prime(n)) % n; \\ Michel Marcus, Jul 09 2017

a(n) = A006450(n) mod n.

A076242 Remainder when 3rd order prime A038580(n) is divided by n-th prime=A000040(n).

Original entry on oeis.org

1, 2, 1, 3, 6, 10, 5, 8, 17, 19, 27, 31, 38, 35, 28, 39, 17, 17, 10, 38, 68, 63, 13, 55, 48, 4, 74, 100, 37, 29, 47, 121, 115, 136, 105, 28, 128, 109, 159, 90, 114, 31, 151, 4, 86, 108, 81, 147, 149, 189, 185, 119, 231, 166, 88, 238, 197, 233, 64, 186, 258, 111, 128, 260

Offset: 1

Labos Elemer, Oct 08 2002

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240-A076243.

Table[Mod[Prime[Prime[Prime[n]]],Prime[n]],{n,70}] (* Harvey P. Dale, Sep 28 2013 *)

a(n) = Mod[A038580(n), A000040(n)]

A318660 Remainder when A064988(n) is divided by n.

Original entry on oeis.org

0, 1, 2, 1, 1, 3, 3, 3, 7, 3, 9, 9, 2, 9, 10, 1, 8, 3, 10, 19, 1, 5, 14, 15, 21, 19, 17, 13, 22, 15, 3, 19, 23, 7, 12, 9, 9, 11, 10, 17, 15, 3, 19, 15, 5, 19, 23, 21, 44, 13, 40, 5, 29, 51, 11, 11, 50, 37, 41, 15, 39, 9, 47, 25, 61, 3, 63, 55, 1, 1, 69, 27, 2, 27, 5, 71, 65, 69, 6, 11, 58, 45, 16, 9, 54, 57, 23, 45, 16, 15, 60, 11, 77, 69

Offset: 1

Altug Alkan and Antti Karttunen, Sep 08 2018

nonn

Inspired by A064988 and a 'minimum' version of it (A318871).

a(n) = 0 only for n = 1. Numbers n such that a(n) = 1 are 2, 4, 5, 16, 21, 69, 70, 181, 265, 370, 1043, 3760, 4531, ...

			a(6) = prime(2)*prime(3) mod 6 = 15 mod 6 = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A064988, A076240, A318668, A318871.

Table[Mod[If[n == 1, 1, Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 1 :> Prime[p]^e]], n], {n, 94}] (* Michael De Vlieger, Sep 10 2018 *)

A318660(n) = { my(f = factor(n)); for (k=1, #f~, f[k, 1] = prime(f[k, 1]); ); (factorback(f)%n); }; \\ After code in A064988.

a(n) = A064988(n) mod n.

a(A000040(n)) = A076240(n).

cp's OEIS Frontend

A263101 a(n) = F(F(n)) mod F(n), where F = Fibonacci = A000045.

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A076243 Remainder when 3rd-order prime ppp(n) = A038580(n) is divided by n.

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A076241 Remainder when 2nd order prime pp(n)=A006450(n) is divided by n.

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A076242 Remainder when 3rd order prime A038580(n) is divided by n-th prime=A000040(n).

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A318660 Remainder when A064988(n) is divided by n.

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