A343117 - OEIS

A343117

a(n) is the absolute difference between the Pisano periods of prime(n)^2 and prime(n).

3, 16, 80, 96, 100, 336, 576, 324, 1056, 392, 900, 2736, 1600, 3696, 1472, 5616, 3364, 3600, 8976, 4900, 10656, 6084, 13776, 3872, 18816, 5000, 21216, 7632, 11664, 8512, 32256, 16900, 37536, 6348, 21904, 7500, 49296, 53136, 55776, 59856, 31684, 16200, 36100

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OFFSET

1,1

COMMENTS

a(n) = 0 if and only if prime(n) is a Wall-Sun-Sun (Fibonacci-Wieferich) prime.

LINKS

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

Wikipedia, Wall-Sun-Sun prime

FORMULA

a(n) = abs(A343116(n)-A060305(n)) = abs(A001175(A001248(n))-A001175(A000040(n))).

PROG

(PARI) \\ After Charles R Greathouse IV in A001175 (Start)

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

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)

a001175(n)=if(n==1, return(1)); my(k=entry(n)); forstep(i=k, n^2, k, if(fibmod(i-1, n)==1, return(i)))

\\ (End)

a(n) = my(p=prime(n)); abs(a001175(p^2)-a001175(p))

CROSSREFS

Cf. A000040, A001175, A001248, A060305, A343116.

Sequence in context: A329806 A371350 A309915 * A055842 A037773 A037661

Adjacent sequences: A343114 A343115 A343116 * A343118 A343119 A343120

KEYWORD

nonn

AUTHOR

Felix Fröhlich, Apr 05 2021

STATUS

approved