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a(n) = Sum_{prime p|n} p^A001222(n).
2

%I #31 Nov 13 2023 04:46:56

%S 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,

%T 173,27,351,29,160,31,32,130,293,74,97,37,365,178,641,41,378,43,1339,

%U 152,533,47,275,49,133,298,2205,53,97,146,2417,370,845,59,722,61

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

%C The definition implies a(n) >= n, with equality only when n is a term in A000961.

%C 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.

%C 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)).

%H Michael De Vlieger, <a href="/A367140/b367140.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A367140/a367140.png">Log log scatterplot of a(n)</a>, n = 2..2^16.

%H Michael De Vlieger, <a href="/A367140/a367140_1.png">Log log scatterplot of a(n)</a>, 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.

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

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

%e a(1) = 0, the empty sum.

%e a(6) = a(2*3) = 2^2 + 3^2 = 13.

%e a(12) = a(2^2*3) = 2^3 + 3^3 = 8 + 27 = 35.

%e a(18) = a(2^1*3^2) = 2^3 + 3^3 = 35.

%e 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).

%t Table[Function[k, DivisorSum[n, #^k &, PrimeQ]][PrimeOmega[n]], {n, 61}] (* _Michael De Vlieger_, Nov 06 2023 *)

%o (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^bigomega(f)); \\ _Michel Marcus_, Nov 06 2023

%Y Cf. A000961, A001222, A367167.

%K nonn

%O 1,2

%A _David James Sycamore_ and _Michael De Vlieger_, Nov 06 2023