A331701 Prime powers (A025475) that can be represented as a sum of two prime powers.

8, 9, 16, 25, 32, 64, 81, 125, 128, 256, 512, 1024, 2048, 4096, 5041, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset

1,1

Comments

A000079 is a subsequence, starting from the 4th term, 2^3.

The subsequence of odd terms begins: 9, 25, 81, 125, 5041.

Links

Table of n, a(n) for n=1..36.

Example

9 = 8 + 1.
25 = 16 + 9.
81 = 49 + 32.
125 = 121 + 4.
5041 = 71^2 = 4913 + 128 = 17^3 + 2^7.

Mathematica

Select[#, Last@ # == 1 &][[All, 1]] &@ Fold[Function[{s, k}, Append[s, If[And[! PrimeQ@ k, DivisorSigma[1, k]*EulerPhi[k] > (k - 1)^2], {k, If[AnyTrue[IntegerPartitions[k, {2}], SubsetQ[s[[All, 1]], #] &], 1, 0]}, Nothing]]], {}, Range[10^4]] (* Michael De Vlieger, Jan 31 2020 *)

Prog

(Python)
from sympy import isprime
TOP = 10**5
primePowers={}
primePowers[1]=1
for x in range(2, TOP):
    if isprime(x):
        p = pp = x
        while pp < TOP**2:
            pp *= p
            primePowers[pp] = 1
a=[]
pps = sorted(primePowers.keys())[:]
for pp in pps:
    for p in pps:
        if p*2 > pp: break
        if (pp-p) in primePowers:
            print(pp)
            a.append(pp)
            break
print(sorted(a))

Crossrefs

Cf. A025475, A000079.

Sequence in context: A192636 A265731 A227646 * A145820 A227647 A175053

Adjacent sequences: A331698 A331699 A331700 * A331702 A331703 A331704

Keyword

nonn

Author

Alex Ratushnyak, Jan 25 2020

Status

approved