A115231 -id:A115231 - cp's OEIS frontend

A095842 Prime powers having no partition into two prime powers.

1, 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

Offset: 1

Reinhard Zumkeller, Jun 10 2004

nonn

A095840(A095874(a(n))) = 0.
A071330(a(n)) = 0.
Here, "prime powers" is used in the relaxed sense, including 1. The numbers 96721, 121801, 192721, 205379, 226981,... seem to be the smallest composite terms of this sequence, which establishes the difference with the subsequence A115231. - M. F. Hasler, Nov 20 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Subsequence of A071331.

Cf. A000961, A095841.

a095842 n = a095842_list !! (n-1)
a095842_list = filter ((== 0) . a071330) a000961_list
-- Reinhard Zumkeller, Jan 11 2013

isprimepower(n)=ispower(n,,&n);isprime(n)||n==1;
isA095842(n)=if(!isprimepower(n),return(0));forprime(p=2,n\2,if(isprimepower(n-p),return(0)));forprime(p=2,sqrtint(n\2),for(e=1,log(n\2)\log(p),if(isprimepower(n-p^e),return(0))));!isprimepower(n-1)
\\ Charles R Greathouse IV, Jul 06 2011

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

A115233 Primes p which have a unique representation as p = 2^i + q^j where i >= 0, j >= 1, q = odd prime.

Original entry on oeis.org

5, 127, 163, 179, 191, 193, 223, 239, 251, 269, 311, 337, 389, 419, 431, 457, 491, 547, 557, 569, 599, 613, 653, 659, 673, 683, 719, 739, 787, 821, 839, 853, 883, 911, 929, 953, 967, 977, 1117, 1123, 1201, 1229, 1249, 1283, 1289, 1297, 1303, 1327, 1381, 1409, 1423, 1439, 1451, 1471, 1481, 1499

Offset: 1

Reinhard Zumkeller, Jan 17 2006

nonn

			5 = 2+3 belongs to the sequence, but 23 = 2^2+19^1 = 2^4+7^1 does not.

Subsequence of A115232. Cf. A115230, A115231.

Cf. A000079, A061345.

maxp = 1500; Clear[cnt]; cnt[_] = 0;
pp = Prime[Range[PrimePi[maxp]]];
Do[p = 2^i + q^j; If[p <= maxp && PrimeQ[p], cnt[p] = cnt[p] + 1], {i, 0, Log[2, maxp] // Ceiling}, {j, 1, Log[3, maxp] // Ceiling}, {q, Rest[pp]}
];
Select[pp, cnt[#] == 1&] (* Jean-François Alcover, Aug 04 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
Data corrected by Jean-François Alcover, Aug 04 2018

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

A282533 Primes that are the sum of two proper prime powers (A246547) in more than one way.

Original entry on oeis.org

41, 89, 113, 137, 593, 857, 2213

Offset: 1

Altug Alkan, Feb 17 2017

Primes of the form 2^k + p^e in more than one way where p is an odd prime (e > 1, k > 1).
Prime terms in A225103.
29 = 2^4 + 5^2 = 2 + 3^3 is a border case not included in this sequence - Olivier Gérard, Feb 25 2019
a(8) > 10^8 if it exists. - Robert Israel, Feb 17 2017
a(8) > 10^18 if it exists. - Charles R Greathouse IV, Feb 19 2017

			41 = 2^4 + 5^2 = 2^5 + 3^2.
89 = 2^3 + 3^4 = 2^6 + 5^2.
113 = 2^5 + 3^4 = 2^6 + 7^2.
137 = 2^7 + 3^2 = 2^4 + 11^2.
593 = 2^9 + 3^4 = 2^6 + 23^2.
857 = 2^7 + 3^6 = 2^4 + 29^2.
2213 = 2^4 + 13^3 = 2^2 + 47^2.

Cf. A225099, A225102, A225103, A246547.

Cf. A115231 (prime numbers which cannot be written as 2^a + p^b, b>=0)

N = 10^8; % to get all terms <= N
C = sparse(1,N);
for p = primes(sqrt(N))
  C(p .^ [2:floor(log(N)/log(p))]) = 1;
end
R = zeros(1,N);
for k = 2: floor(log2(N))
  R((2^k+1):N) = R((2^k+1):N) + C(1:(N-2^k));
end
P = primes(N);
P(R(P) > 1.5) % Robert Israel, Feb 17 2017

N:= 10^6: # to get all terms <= N
B:= Vector(N):
C:= Vector(N):
for k from 2 to ilog2(N) do B[2^k]:= 1 od:
p:= 2:
do
  p:= nextprime(p);
  if p^2 > N then break fi;
  for k from 2 to floor(log[p](N)) do C[p^k]:= 1 od:
od:
R:= SignalProcessing:-Convolution(B,C):
select(t -> isprime(t) and R[t-1] > 1.5, [seq(i,i=3..N,2)]); # Robert Israel, Feb 17 2017

Select[Prime@ Range[10^3], Function[n, Count[Transpose@{n - #, #}, w_ /; Times @@ Boole@ Map[And[PrimePowerQ@ #, ! PrimeQ@ #] &, w] > 0] >= 2 &@ Range[4, Floor[n/2]]]] (* or *)
With[{n = 10^8}, Keys@ Select[#, Length@ # > 1 &] &@ GroupBy[#, First] &@ SortBy[Transpose@ {Map[Total, #], #}, First] &@ Select[Union@ Map[Sort, Tuples[#, 2]], PrimeQ@ Total@ # &] &@ Flatten@ Map[#^Range[2, Log[#, Prime@ n]] &, Array[Prime@ # &, Floor@ Sqrt@ n]]] (* Michael De Vlieger, Feb 19 2017, latter program Version 10 *)

is(n) = if(!ispseudoprime(n), return(0), my(x=n-1, y=1, i=0); while(y < x, if(isprimepower(x) > 1 && isprimepower(y) > 1, if(i==0, i++, return(1))); y++; x--)); 0 \\ Felix Fröhlich, Feb 18 2017

has(p)=my(t,q); p>40 && sum(k=2,logint(p-9,2), t=2^k; sum(e=2,logint(p-t,3), ispower(p-t,e,&q) && isprime(q)))>1
list(lim)=my(v=List(),t,q); lim\=1; if(lim<9,lim=9); for(k=2,logint(lim-9,2), t=2^k; for(e=2,logint(lim-t,3), forprime(p=3,sqrtnint(lim-t,e), q=t+p^e; if(isprime(q) && has(q), listput(v,q))))); Set(v) \\ Charles R Greathouse IV, Feb 18 2017

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A095842 Prime powers having no partition into two prime powers.

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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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A115233 Primes p which have a unique representation as 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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A282533 Primes that are the sum of two proper prime powers (A246547) in more than one way.

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