A367167 -id:A367167 - cp's OEIS frontend

0, 2, 3, 4, 5, 13, 7, 8, 9, 29, 11, 35, 13, 53, 34, 16, 17, 35, 19, 133, 58, 125, 23, 97, 25, 173, 27, 351, 29, 160, 31, 32, 130, 293, 74, 97, 37, 365, 178, 641, 41, 378, 43, 1339, 152, 533, 47, 275, 49, 133, 298, 2205, 53, 97, 146, 2417, 370, 845, 59, 722, 61

Offset: 1

David James Sycamore and Michael De Vlieger, Nov 06 2023

nonn

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)).

			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).

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.

Cf. A000961, A001222, A367167.

Table[Function[k, DivisorSum[n, #^k &, PrimeQ]][PrimeOmega[n]], {n, 61}] (* Michael De Vlieger, Nov 06 2023 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^bigomega(f)); \\ Michel Marcus, Nov 06 2023

a(n) = n for n a term >1 in A000961.

For m != n, a(m) = a(n) iff A001222(m) = A001222(n) and A000961(m) = A000961(n).

cp's OEIS Frontend

A367140 a(n) = Sum_{prime p|n} p^A001222(n).

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