login
A367167
Let Product_{i = 1..omega(n)} p_i^e_i be the prime factorization of n. Then a(n) = Sum_{i = 1..omega(n)} Product_{j = 1..n} p_i^(e_((i+j-1) mod omega(n) + 1)).
2
0, 2, 3, 4, 5, 12, 7, 8, 9, 20, 11, 30, 13, 28, 30, 16, 17, 30, 19, 70, 42, 44, 23, 78, 25, 52, 27, 126, 29, 90, 31, 32, 66, 68, 70, 72, 37, 76, 78, 290, 41, 126, 43, 286, 120, 92, 47, 210, 49, 70, 102, 390, 53, 78, 110, 742, 114, 116, 59, 300, 61, 124, 210, 64, 130
OFFSET
1,2
COMMENTS
In other words, a(n) is constructed by incrementally shifting the elements of the ordered exponent multiset {e_1,e_2,...,e_k} to the right one place at a time, a total of k = A001221(n) times, and adding the results of these exponents applied to the distinct size ordered primes p_1,p_2,...,p_k which divide n. The first shift gives {e_k,e_1,e_2,...,e_(k-1)}, then {e_(k-1),e_k,...,e_(k-2)} ... and the last yields {e_1,e_2,...,e_k} (which is n itself). At each shift the last exponent of the previous shift becomes the first of the next. There are A001221(n) summands in the computation of a(n), each having the same rad (A007947) and omega (A001221) values as n.
a(n) >= n with equality for n > 1 in A000961.
LINKS
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
FORMULA
a(n) = n for n > 1 in A000961.
a(n) = 2*n for n in A006881, and more generally if A007947(n) = n, a(n) = A001221(n)*n.
Let b(0) = n and let b(n+1) = A105119(b(n)) for n >= 0 and let omega(n) be the number of distinct prime factors of n. Then a(n) = Sum_{i = 1..omega(n)} b(i). - David A. Corneth, Nov 07 2023
EXAMPLE
a(1) = 0, the empty sum.
a(6) = a(2^1*3^1) = 6 + 2^1*3^1 = 6 + 6 = 12.
a(12) = a(2^2*3^1) = 12 + 2^1*3^2 = 12 + 18 = 30.
a(2250) = a(2^1*3^2*5^3) = 2^3*3^1*5^2 + 2^2*3^3*5^1 + 2^1*3^2*5^3 = 3390.
MATHEMATICA
{0}~Join~Array[Total@ Table[Times @@ Power @@@ Transpose@ {#1, RotateRight[#2, k]}, {k, PrimeNu[#]}] & @@ Transpose@ FactorInteger[#] &, 64, 2] (* Michael De Vlieger, Nov 09 2023 *)
PROG
(PARI)
a(n) = {
my(f = factor(n), res = 0);
for(i = 1, #f~,
res+=prod(j = 1, #f~, f[j, 1]^f[(i+j-1)%#f~ + 1, 2])
);
res
} \\ David A. Corneth, Nov 07 2023
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms and revised name from David A. Corneth, Nov 07 2023
STATUS
approved