A225103 Numbers that can be represented as a sum of two distinct nontrivial prime powers in two or more ways.

36, 41, 57, 89, 113, 129, 130, 137, 153, 177, 185, 297, 305, 368, 370, 377, 410, 425, 537, 545, 561, 593, 633, 650, 657, 850, 857, 868, 873, 890, 969, 986, 1010, 1130, 1385, 1490, 1690, 1730, 1802, 1865, 1881, 1898, 1970, 2210, 2213, 2217, 2236, 2330, 2337

Offset: 1

Author

Alex Ratushnyak, Apr 28 2013

Keywords

• nonn

Comments

Indices of terms bigger than 1 in A225099.

Nontrivial prime powers are numbers of the form p^k where p is a prime number and k >= 2. That is, A025475 except the first term A025475(1) = 1.

Only 267 of the terms below 2^34 are odd.

Examples

			36 = 32 + 4 = 27 + 9, so 36 is in the sequence.
41 = 32 + 9 = 25 + 16, so 41 is in the sequence.
		

Links

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

Crossrefs

Cf. A025475, A225099, A225102.

Programs

Maple

N:= 10^4: # to get all terms <= N
PP:= [seq(seq(p^k, k=2..floor(log[p](N))),p = select(isprime, [2,seq(i,i=3..floor(sqrt(N)),2)]))]:
npp:= nops(PP):
res:= {}: R:= 'R':
for i from 2 to npp do
  for j from 1 to i-1 do
    q:= PP[i]+PP[j];
    if assigned(R[q]) then res:= res union {q}
    else R[q]:= 1
    fi
od od:
sort(convert(res,list)); # Robert Israel, Feb 17 2017

Mathematica

nn = 2409; p = Sort[Flatten[Table[Prime[n]^i, {n, PrimePi[Sqrt[nn]]}, {i, 2, Log[Prime[n], nn]}]]]; Transpose[Sort[Select[Tally[Flatten[Table[p[[i]] + p[[j]], {i, Length[p] - 1}, {j, i + 1, Length[p]}]]], #[[1]] <= nn && #[[2]] > 1 &]]][[1]] (* T. D. Noe, Apr 29 2013 *)