A298944 -id:A298944 - cp's OEIS frontend

A298946 a(n) = binomial(2*c-1, c-1) (mod c^4), where c is the n-th composite number.

35, 462, 2339, 4627, 2378, 4238, 5148, 1260, 57635, 85026, 64410, 100509, 163716, 171918, 93876, 309780, 148969, 444220, 370712, 532771, 652200, 938386, 816466, 907874, 569300, 1107298, 2470810, 2953692, 887812, 1341810, 2956584, 1941390, 589961, 6248628

Offset: 1

Felix Fröhlich, Jan 30 2018

nonn

Composites c where a(n) = 1 could be called "Wolstenholme pseudoprimes". Do any such composites exist?

A necessary condition for c to be a "Wolstenholme pseudoprime" would be that it is a term of A228562 or A267824.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A088164, A228562, A244214, A267824, A281302, A298944, A298945.

R:= NULL:
count:= 0: F:= 10;
for n from 4 while count < 100 do
  F:= F * (4*n-2)/n;
  if not isprime(n) then
     count:= count+1;
     R:= R, F mod (n^4);
  fi
od:
R; # Robert Israel, Feb 02 2018

Table[Mod[Binomial[2 c - 1, c - 1], c^4], {c, Select[Range@ 50, CompositeQ]}] (* Michael De Vlieger, Feb 01 2018 *)

forcomposite(c=1, 200, print1(lift(Mod(binomial(2*c-1, c-1), c^4)), ", "))

from sympy import binomial, composite
def A298946(n):
    c = composite(n)
    return binomial(2*c-1,c-1) % c**4 # Chai Wah Wu, Feb 02 2018

A298945 a(n) = F_{c-(5/c)} mod c^2, where c is the n-th composite number, F_i = A000045(i) and (5/c) is the Kronecker symbol.

Original entry on oeis.org

2, 5, 34, 21, 55, 89, 37, 160, 98, 293, 365, 150, 101, 433, 25, 665, 696, 709, 440, 994, 883, 1090, 765, 1241, 230, 1511, 1355, 257, 805, 20, 1382, 289, 2275, 1525, 1414, 821, 1373, 1820, 685, 1504, 2177, 720, 3102, 1302, 1250, 190, 2425, 2178, 2832, 3935

Offset: 1

Felix Fröhlich, Jan 30 2018

nonn

Composites c where a(n) = 0 could be called "Wall-Sun-Sun pseudoprimes" or "Fibonacci-Wieferich pseudoprimes". Do any such composites exist?

Any such c would have to be a term of A241505.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A241505, A298944, A298946.

N:= 100: # to get a(1)..a(N)
count:= 0: R:= NULL:
for n from 4 while count < N do
if not isprime(n) then
  count:= count+1;
  R:= R, combinat:-fibonacci(n - numtheory:-jacobi(5,n)) mod n^2;
fi
od:
R; # Robert Israel, Feb 02 2018

composite[n_Integer] := FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1] ; Array[With[{c = composite@ #}, Mod[Fibonacci[c - KroneckerSymbol[5, c]], c^2]] &, 50] (* Michael De Vlieger, Jan 31 2018, composite function by Robert G. Wilson v at A066277 *)

forcomposite(c=1, 200, print1(lift(Mod(fibonacci(c-kronecker(5, c)), c^2)), ", "))

cp's OEIS Frontend

A298946 a(n) = binomial(2*c-1, c-1) (mod c^4), where c is the n-th composite number.

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