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A053031 Numbers with 1 zero in Fibonacci numbers mod m. 5
1, 2, 4, 11, 19, 22, 29, 31, 38, 44, 58, 59, 62, 71, 76, 79, 101, 116, 118, 121, 124, 131, 139, 142, 151, 158, 179, 181, 191, 199, 202, 209, 211, 229, 236, 239, 242, 251, 262, 271, 278, 284, 302, 311, 316, 319, 331, 341, 349, 358, 359, 361, 362, 379, 382, 398 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: m is on this list iff m is an odd number all of whose factors are on this list or m is 2 or 4 times such an odd number.

A001176(a(n)) = A128924(a(n),1) = 1. - Reinhard Zumkeller, Jan 16 2014

Also numbers n such that A001175(n) = A001177(n). - Daniel Suteu, Aug 08 2018

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

M. Renault, Fibonacci sequence modulo m

MATHEMATICA

With[{s = {1}~Join~Table[Count[Drop[NestWhile[Append[#, Mod[Total@ Take[#, -2], n]] &, {1, 1}, If[Length@ # < 3, True, Take[#, -2] != {1, 1}] &], -2], 0], {n, 2, 400}]}, Position[s, 1][[All, 1]] ] (* Michael De Vlieger, Aug 08 2018 *)

PROG

(Haskell)

a053031 n = a053031_list !! (n-1)

a053031_list = filter ((== 1) . a001176) [1..]

-- Reinhard Zumkeller, Jan 16 2014

(PARI) entryp(p)=my(k=p+[0, -1, 1, 1, -1][p%5+1], f=factor(k)); for(i=1, #f[, 1], for(j=1, f[i, 2], if((Mod([1, 1; 1, 0], p)^(k/f[i, 1]))[1, 2], break); k/=f[i, 1])); k

entry(n)=if(n==1, return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i, 1]>1e14, entryp(f[i, 1]^f[i, 2]), entryp(f[i, 1])*f[i, 1]^(f[i, 2]-1))); if(f[1, 1]==2&&f[1, 2]>1, v[1]=3<<max(f[1, 2]-2, 1)); lcm(v)

fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]

is(n)=fibmod(entry(n)+1, n)==1 \\ Charles R Greathouse IV, Dec 14 2016

CROSSREFS

Cf. A001176, A053032.

Cf. A053029 (with 4 zeros), A053030 (with 2 zeros).

Sequence in context: A056394 A056395 A288621 * A018674 A076518 A139785

Adjacent sequences:  A053028 A053029 A053030 * A053032 A053033 A053034

KEYWORD

nonn

AUTHOR

Henry Bottomley, Feb 23 2000

STATUS

approved

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Last modified March 19 17:38 EDT 2019. Contains 321330 sequences. (Running on oeis4.)