OFFSET
1,2
COMMENTS
The definition implies a(n) >= n, with equality only when n is a term in A000961.
This sequence contains sums of distinct prime powers, but not all such sums are terms (6 is not a term since it cannot be expressed as the sum of powers of distinct primes). If m (a non prime power) is a term it must occur as a(n) = m for some n < m, for if not there is no way it can occur later (if so we would have n > m and a(n) = m, but then a(n) < n; contradiction); see Example.
Some primes occur twice; once as fixed points a(p) = p, and once as a(m) = p for some m < p (e.g. 13 = a(6) = a(13) and 29 = a(10) = a(29)).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 2..2^16.
Michael De Vlieger, Log log scatterplot of a(n), n = 2..2^14, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue.
FORMULA
EXAMPLE
a(1) = 0, the empty sum.
a(6) = a(2*3) = 2^2 + 3^2 = 13.
a(12) = a(2^2*3) = 2^3 + 3^3 = 8 + 27 = 35.
a(18) = a(2^1*3^2) = 2^3 + 3^3 = 35.
15 is expressible as the sum of prime powers (2^2 + 11^1) but it is not a term since it has not occurred prior to a(15), likewise 18 (5 + 13)) is not a term since it has not occurred prior to a(18).
MATHEMATICA
Table[Function[k, DivisorSum[n, #^k &, PrimeQ]][PrimeOmega[n]], {n, 61}] (* Michael De Vlieger, Nov 06 2023 *)
PROG
(PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 1]^bigomega(f)); \\ Michel Marcus, Nov 06 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
David James Sycamore and Michael De Vlieger, Nov 06 2023
STATUS
approved