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

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

Author

David James Sycamore and Michael De Vlieger, Nov 06 2023

Keywords

• nonn

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)).

Examples

			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).
		

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.

Crossrefs

Cf. A000961, A001222, A367167.

Programs

Mathematica

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

PARI

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

Formula

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).