A117988 -id:A117988 - cp's OEIS frontend

A117986 Number of functions f:[n]->[n] such that f[(xy) mod n]=[f(x)f(y)] mod n for all x,y in [n], for n=1,2,3,... Here [n] denotes {0,1,2,...,n-1}.

1, 3, 4, 6, 6, 35, 8, 50, 20, 55, 12, 160, 14, 75, 160, 194, 18, 195, 20, 256, 220, 115, 24, 3936, 102, 135, 164, 352, 30, 5301, 32, 770, 340, 175, 352, 2496, 38, 195, 400, 6396, 42, 7353, 44, 544, 928, 235, 48, 15456, 296, 1015, 520, 640, 54, 1635, 544, 8856

Offset: 1

John W. Layman, Apr 07 2006

nonn

If, instead, the modular functional equation f[(x+y) mod n]=[f(x)+f(y)] mod n is considered, it is found that for each n=1,2,3,... there appears to be exactly n functions with the desired property. See A117987 and A117988 for results on other modular functional equations.

			For n=5 the six functions are (0,0,0,0,0), (0,1,1,1,1), (1,1,1,1,1), (0,1,4,4,1), (0,1,3,2,4), (0,1,2,3,4). For the 5th of these, (0,1,3,2,4), the x=2, y=3 case is verified by the calculations f(2*3 mod 4) = f(1) = 1 and f(2)*f(3) mod 5 = 3*2 mod 5 = 1.

Rémy Sigrist, C++ program for A117986

Cf. A117987, A117988.

Apparently, a(p) = p + 1 for any prime number p. - Rémy Sigrist, Sep 19 2019

More terms from Rémy Sigrist, Sep 19 2019

A117987 Number of functions f:[n]->[n] such that f[(2x) mod n]=[2f(x)] mod n for all x in [n], for n=1,2,3,... Here [n] denotes {0,1,2,...,n-1}.

Original entry on oeis.org

1, 2, 3, 8, 5, 24, 49, 128, 27, 160, 11, 1536, 13, 6272, 10125, 32768, 289, 13824, 19, 163840, 64827, 22528, 529, 6291456, 125, 106496, 729, 102760448, 29, 331776000, 887503681, 2147483648, 107811, 37879808, 300125, 3623878656, 37, 9961472

Offset: 1

John W. Layman, Apr 11 2006

nonn

See A117986 and A117988 for results on other modular functional equations.

Cf. A117986, A117988.

{ A117987(n) = my(m,r); m=n\2^valuation(n,2); r=2^(n-m); fordiv(znorder(Mod(2,m)),d, r *= gcd(m,2^d-1)^(sumdiv(d,q, moebius(d\q)*gcd(m,2^q-1) )\d); ); r } /* Max Alekseyev, Jun 11 2009 */

For n = 2^t * m with odd m, a(n) = 2^(n-m) * \sum_{d|A007733(n)} gcd(m,2^d-1)^{ \sum_{q|d} moebius(d/q) * gcd(m,2^q-1) / d }. - Max Alekseyev, Jun 11 2009

Extended by Max Alekseyev, Jun 11 2009

cp's OEIS Frontend

A117986 Number of functions f:[n]->[n] such that f[(xy) mod n]=[f(x)f(y)] mod n for all x,y in [n], for n=1,2,3,... Here [n] denotes {0,1,2,...,n-1}.

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A117987 Number of functions f:[n]->[n] such that f[(2x) mod n]=[2f(x)] mod n for all x in [n], for n=1,2,3,... Here [n] denotes {0,1,2,...,n-1}.

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cp's OEIS Frontend

A117986 Number of functions f:[n]->[n] such that f[(x*y) mod n]=[f(x)*f(y)] mod n for all x,y in [n], for n=1,2,3,... Here [n] denotes {0,1,2,...,n-1}.

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A117987 Number of functions f:[n]->[n] such that f[(2*x) mod n]=[2*f(x)] mod n for all x in [n], for n=1,2,3,... Here [n] denotes {0,1,2,...,n-1}.

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PARI

Formula

Extensions

A117986 Number of functions f:[n]->[n] such that f[(xy) mod n]=[f(x)f(y)] mod n for all x,y in [n], for n=1,2,3,... Here [n] denotes {0,1,2,...,n-1}.

A117987 Number of functions f:[n]->[n] such that f[(2x) mod n]=[2f(x)] mod n for all x in [n], for n=1,2,3,... Here [n] denotes {0,1,2,...,n-1}.