A115233 -id:A115233 - cp's OEIS frontend

A115231 Primes p which cannot be written in the form 2^i + q^j where i >= 0, j >= 1, q = odd prime.

2, 3, 149, 331, 373, 509, 701, 757, 809, 877, 907, 997, 1019, 1087, 1259, 1549, 1597, 1619, 1657, 1759, 1777, 1783, 1867, 1973, 2293, 2377, 2503, 2579, 2683, 2789, 2843, 2879, 2909, 2999, 3119, 3163, 3181, 3187, 3299, 3343, 3433, 3539, 3643, 3697, 3739, 3779

Offset: 1

Reinhard Zumkeller, Jan 17 2006

nonn

Union with A115232 gives all primes (A000040).

All terms > 3 are in A095842. - M. F. Hasler, Nov 20 2014

			A000040(35) = 149 = 2^7+3*7 = 2^6+5*17 = 2^5+3*3*13 =
2^4+7*19 = 2^3+3*47 = 2^2+5*29 = 2^1+3*7*7 = 2^0+2*2*37, therefore 149 is a term (A115230(35)=0).

David A. Corneth, Table of n, a(n) for n = 1..11532 (terms <= 10^6)

Cf. A095842, A115230, A115232, A115233, A000079, A061345.

maxp = 3779; Complement[pp = Prime[Range[PrimePi[maxp]]], Union[Sort[Reap[Do[p = 2^i + q^j; If[p <= maxp && PrimeQ[p], Sow[p]], {i, 0, Log[2, maxp]//Ceiling}, {j, 1, Log[3, maxp]//Ceiling}, {q, Rest[pp]} ]][[2, 1]]]]] (* Jean-François Alcover, Aug 03 2018 *)

upto(n) = {my(pr = primes(primepi(n)), found = List(), s); for(i = 0, logint(n, 2), s = 2^i; forprime(q = 3, n - 2^i, for(j = 1, logint(n - 2^i, q),
listput(found, s + q^j)))); listsort(found, 1); setminus(Set(pr), Set(found))} \\ David A. Corneth, Aug 03 2018

Recomputed (based on recomputation of A115230) by R. J. Mathar and Reinhard Zumkeller, Apr 29 2010.
Edited by N. J. A. Sloane, Apr 30 2010
2, 3 inserted by David A. Corneth, Aug 03 2018

A115230 Let p = prime(n); a(n) = number of ways to write p = 2^i + q^j where i >= 0, j >= 1, q = odd prime.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 3, 2, 4, 3, 2, 2, 2, 2, 2, 4, 1, 3, 3, 4, 0, 2, 3, 1, 3, 3, 1, 4, 1, 1, 2, 4, 2, 1, 3, 3, 2, 1, 3, 1, 3, 2, 1, 3, 2, 2, 3, 4, 2, 1, 2, 2, 0, 1, 3, 2, 4, 2, 2, 0, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 3, 0, 2, 3, 2, 1, 1, 3, 1, 4

Offset: 1

Reinhard Zumkeller, Jan 17 2006

nonn

			n=25: A000040(25) = 97 = 2^6 + 3*11 = 2^5 + 5*13 = 2^4 + 3^4 = 2^3 + 89^1 = 2^2 + 3*31 = 2^1 + 5*19 = 2^0 + 3*2^5, therefore a(25) = #{[16+81], [8+89]} = 2.

Cf. A115231-A115233, A000079, A061345.

From Reinhard Zumkeller, Apr 30 2010: (Start)
A000035 := proc(n) n mod 2 ; end proc:
A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A036987 := proc(n) A000108(n) mod 2 ; end proc:
A010055 := proc(n) if n = 1 then 1; else numtheory[factorset](n) ; if nops(%) = 1 then 1; else 0; end if; end if: end proc:
A115230 := proc(n) p := ithprime(n) ; add(A036987(k-1)*A000035(p-k)*A010055(p-k), k=1..p-1) ; end proc: seq(A115230(n),n=1..40) ; # R. J. Mathar, Apr 30 2010 (End)

f[p_] := Length@ Table[q = p - 2^exp; If[ PrimeNu@ q == 1, {q}, Sequence @@ {}], {exp, 0, Floor@ Log2@ p}]; Table[ f[ Prime[ n]], {n, 105}] (* Robert G. Wilson v, Oct 05 2014 *)

a(n) = Sum_{k=1..prime(n)-1} A036987(k-1)*A000035(p-k)*A010055(p-k). - Reinhard Zumkeller, Apr 29 2010

Recomputed by Charles R Greathouse IV, Ray Chandler, R. J. Mathar, and Reinhard Zumkeller, Apr 29 2010; thanks to Charles R Greathouse IV, who pointed out that there were many errors in entries of A115230-A115233.
Edited by N. J. A. Sloane, Apr 30 2010
Formula corrected (thanks to R. J. Mathar, who found an error in it) by Reinhard Zumkeller, Apr 30 2010

A115232 Primes p which can be written in the form 2^i + q^j where i >= 0, j >= 1, q = odd prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281

Offset: 1

Reinhard Zumkeller, Jan 17 2006

nonn

a(n)=A000040(n+2) for n <= 32, but A000040(35)=149 is a term of A115231;
A115233 is a subsequence; the union with A115231 gives all primes (A000040);
A006512 and A053703 are subsequences.

Cf. A000079, A061345.

Cf. A006512, A053703, A115231, A115233.

maxp = 281;
Union[Sort[Reap[Do[p = 2^i + q^j; If[p <= maxp && PrimeQ[p], Sow[p]], {i, 0, Log[2, maxp]//Ceiling}, {j, 1, Log[3, maxp]//Ceiling}, {q, Prime[Range[2, PrimePi[maxp]]]}]][[2, 1]]]] (* Jean-François Alcover, Aug 03 2018 *)

Recomputed (based on recomputation of A115230) by R. J. Mathar and Reinhard Zumkeller, Apr 29 2010
Edited by N. J. A. Sloane, Apr 30 2010
Terms a(1)=2 and a(2)=3 removed from Data by Jean-François Alcover, Aug 03 2018

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A115231 Primes p which cannot be written in the form 2^i + q^j where i >= 0, j >= 1, q = odd prime.

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A115230 Let p = prime(n); a(n) = number of ways to write p = 2^i + q^j where i >= 0, j >= 1, q = odd prime.

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A115232 Primes p which can be written in the form 2^i + q^j where i >= 0, j >= 1, q = odd prime.

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