A095841 Prime powers having exactly one partition into two prime powers.

2, 3, 127, 163, 179, 191, 193, 223, 239, 251, 269, 311, 337, 343, 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, 1607, 1663, 1681

Offset

1,1

Comments

A095840(A095874(a(n))) = 1.

A071330(a(n)) = 1.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Maple

N:= 10^4: # to get all terms <= N
primepows:= {1, seq(seq(p^n, n=1..floor(log[p](N))),
    p=select(isprime, [2, seq(2*k+1, k=1..(N-1)/2)]))}:
npp:= nops(primepows):
B:= Vector(N, datatype=integer[4]):
for n from 1 to npp do for m from n to npp do
   j:= primepows[n]+primepows[m];
   if j <= N then B[j]:= B[j]+1 fi;
od od:
select(t -> B[t] = 1, primepows); # Robert Israel, Nov 21 2014

Mathematica

max = 2000; ppQ[n_] := n == 1 || PrimePowerQ[n]; pp = Select[Range[max], ppQ]; lp = Length[pp]; Table[pp[[i]] + pp[[j]], {i, 1, lp}, {j, i, lp}] // Flatten // Select[#, ppQ[#] && # <= max&]& // Sort // Split // Select[#, Length[#] == 1&]& // Flatten (* Jean-François Alcover, Mar 04 2019 *)

Prog

(Haskell)
a095841 n = a095841_list !! (n-1)
a095841_list = filter ((== 1) . a071330) a000961_list
-- Reinhard Zumkeller, Jan 11 2013
(PARI) is(n)=if(n<127, return(n==2||n==3)); isprimepower(n) && sum(i=2, n\2, isprimepower(i)&&isprimepower(n-i))==1 \\ naive; Charles R Greathouse IV, Nov 21 2014
(PARI) is(n)=if(!isprimepower(n), return(0)); my(s); forprime(p=2, n\2, if(isprimepower(n-p) && s++>1, return(0))); for(e=2, log(n)\log(2), forprime(p=2, sqrtnint(n\2, e), if(isprimepower(n-p^e) && s++>1, return(0)))); s+(!!isprimepower(n-1))==1 || n==2 \\ faster; Charles R Greathouse IV, Nov 21 2014
(PARI) has(n)=my(s); forprime(p=2, n\2, if(isprimepower(n-p) && s++>1, return(0))); for(e=2, log(n)\log(2), forprime(p=2, sqrtnint(n\2, e), if(isprimepower(n-p^e) && s++>1, return(0)))); s+(!!isprimepower(n-1))==1
list(lim)=my(v=List([2])); forprime(p=2, lim, if(has(p), listput(v, p))); for(e=2, log(lim)\log(2), forprime(p=2, lim^(1/e), if(has(p^e), listput(v, p^e)))); Set(v) \\ Charles R Greathouse IV, Nov 21 2014

Crossrefs

Cf. A000961, A095842.

Intersection of A208247 and A000961.

Cf. A071330, A095840, A095874.

Sequence in context: A258968 A125674 A180533 * A004865 A006286 A371230

Adjacent sequences: A095838 A095839 A095840 * A095842 A095843 A095844

Keyword

nonn

Author

Reinhard Zumkeller, Jun 10 2004

Status

approved