A196526 a(n) is the number of ways the n-th prime number prime(n) can be written as sum of coprime b and c, in which b is a positive even number and c is an odd number that is -1 or greater, and all odd prime factors of b and c are less than or equal to sqrt(prime(n)).

2, 1, 1, 3, 2, 3, 2, 1, 5, 5, 4, 4, 3, 4, 8, 8, 7, 7, 7, 7, 8, 7, 8, 7, 6, 6, 7, 7, 7, 12, 13, 12, 11, 12, 10, 10, 11, 11, 16, 18, 18, 18, 17, 18, 18, 17, 16, 15, 16, 17, 15, 18, 18, 18, 18, 17, 16, 18, 17, 16, 24, 24, 23, 23, 23, 23, 24, 23, 24, 24, 25, 32, 33, 34, 33, 36, 34, 35, 33, 35, 33, 32, 35, 34, 33, 33, 34, 33, 35, 34, 31, 32, 30, 35, 35, 34, 32, 32, 45

Offset

2,1

Comments

All terms are positive integers, no zero term.

The Mathematica program generates first 99 items and the function AllSplits[n_] can be used to generate a(n) for any n > 1.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 2..1000

Example

n=2: prime(2)=3 = 2 + 1 = 2^2 - 1, so a(2)=2;
n=3: prime(3)=5 = 2^2 + 1, so a(3)=1;
...
n=10: prime(10)=29, int(sqrt(29))=5, 29 = 2+3^3 = 2^2+5^2 = 2^2*5+3^2 = 2^3*3+5 = 2*3*5-1, so a(10)=5.

Mathematica

AllPrimes[k_] :=
Module[{p, maxfactor, pset}, p = Prime[k];
  maxfactor = NextPrime[IntegerPart[Sqrt[p]] + 1, -1];
  If[maxfactor == -2, pset = {2}, p0 = 2; pset = {2};
   While[p0 = NextPrime[p0]; p0 <= maxfactor,
    pset = Union[pset, {p0}]]]; pset];
NextIntegerWithFactor[seed_, fset_] :=
Module[{m, a, l, i, fset1}, m = seed - 1;
  While[m++; If[Mod[m, 2] == 1, m++]; a = FactorInteger[m];
   l = Length[a]; fset1 = {};
   Do[fset1 = Union[fset1, {a[[i]][[1]]}], {i, 1, l}];
   Intersection[fset1, fset] != fset1]; m];
FactorSet[seed_] :=
Module[{fset2, a, l, i}, a = FactorInteger[seed]; l = Length[a];
  fset2 = {}; Do[fset2 = Union[fset2, {a[[i]][[1]]}], {i, 1, l}];
  fset2]; SplitPrime[n_, q0_] :=
Module[{p, pset, q, r, rp, fs, rs, qs}, p = Prime[n];
  pset = AllPrimes[n]; q = q0;
  While[q++; q = NextIntegerWithFactor[q, pset]; r = p - q;
   rp = Abs[r]; fs = FactorSet[rp];
   rs = Complement[pset, FactorSet[q]];
   qs = Intersection[fs,
     rs]; (fs != {1}) && (fs != qs) && (q <= (p + 1))]; {p, q, r} ];
AllSplits[n_] :=
Module[{q, ss, spls}, q = 0; spls = {};
  While[ss = SplitPrime[n, q]; q = ss[[2]];
   If[q <= (Prime[n] + 1), spls = Union[spls, {ss}]];
   q < (Prime[n] + 1)]; spls]; Table[
Length[AllSplits[i]], {i, 2, 100}] (* Lei Zhou *)
zhouAbleCount[n_] := Length[Select[Range[-1, Prime[n], 2], GCD[#, Prime[n] - #] == 1 && FactorInteger[#][[-1, 1]] <= Sqrt[Prime[n]] && (IntegerQ[Log[2, Prime[n] - #]] || FactorInteger[Prime[n] - #][[-1, 1]] <= Sqrt[Prime[n]]) &]]; Table[zhouAbleCount[n], {n, 2, 100}] (* Alonso del Arte, Oct 03 2011 *)

Prog

(Haskell)
a196526 n = length [c | let p = a000040 n,
                        c <- [-1, 1..p-1], let b = p - c,
                        gcd b c == 1,
                        a006530 b ^ 2 < p || p == 3, a006530 c ^ 2 < p]
-- Reinhard Zumkeller, Oct 04 2001

Crossrefs

Cf. A006530 (largest prime factor), A000040.

Sequence in context: A363086 A378881 A060117 * A234504 A112592 A370666

Adjacent sequences: A196523 A196524 A196525 * A196527 A196528 A196529

Keyword

nonn,easy

Author

Lei Zhou, Oct 03 2011

Status

approved