|
| |
|
|
A242404
|
|
a(n) = Sum over i and j with j<=i and i=1.. sopf(n) of binomial(d(i), d(j)), where d(i) are the prime divisors of n and sopf(n) = sum of the distinct primes dividing n (A008472).
|
|
1
|
|
|
|
0, 1, 1, 1, 1, 5, 1, 1, 1, 12, 1, 5, 1, 23, 12, 1, 1, 5, 1, 12, 37, 57, 1, 5, 1, 80, 1, 23, 1, 26, 1, 1, 167, 138, 23, 5, 1, 173, 288, 12, 1, 62, 1, 57, 12, 255, 1, 5, 1, 12, 682, 80, 1, 5, 464, 23, 971, 408, 1, 26, 1, 467, 37, 1, 1289, 226, 1, 138, 1773, 55
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,6
|
|
|
COMMENTS
|
This sequence is very rich in properties. We cite a few.
If n = p^m where p is prime, then a(n) = 1 => a(A025475(n)) = 1.
If n = 2^i*3^j, i and j >= 1, then a(n) = 5 => a(A033845(n)) = 5.
We observe a very interesting property with a fractal structure if a(n) = 23.
a(n) = 23 if n belongs to the set E = {n} = {14, 28, 35, 56, 98, 112, 175, 196, 224, 245, 392, 448, 686, ...} and we observe that E1 = {n/7} = {2, 4, 5, 8, 14, 16, 25, 28, 32, 35, 56, 64, 98, 112, 125, 128, 175, 196, 224, 245, 256, ...} = E2 union E3 union E4 where:
E2 = {2, 4, 8, 16, 32, 64, 128, ...} are powers of 2 (A000079),
E3 = {5, 25, 125, 625, 3125, ...} are powers of 5 (A000351),
E4 = {14, 28, 35, 56, 98, 112, 175, 196, 224, 245, ...} = E, a surprising result: E1 contains E => this shows a fractal structure.
But you can continue with other values of a(n) in order to find similar properties. For example, a(n) = 55 if n belongs to the set F = {n} = {70, 140, 280, 350, 490, 560, 700, 980, 1120, ...} => F1 = {n/70} = { 1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 25, 28, 32, 35, 40, 49, 50, 56, 64, 70, 80, ...} = F2 union F3 union F4 union F5 where:
F2 = {2, 4, 8, 16, 32, 64, 128, ...} are powers of 2 (A000079),
F3 = {5, 25, 125, 625, 3125, ...} are powers of 5 (A000351),
F4 = {1, 7, 49, 343, 2401, ...} are powers of 7 (A000420)
and F5 = {10, 14, 20, 28, 35, 40, 50, 56, 70, 80, 98, 100, 112, 140, ...} contains F but also E.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
a(10) = 12 because 12 = 2^2*3 => sopf(12) = 2+3=5 and a(10) = sum{j<=i=1..5} binomial(d(i),d(j)) = binomial(2,2)+ binomial(5,2)+ binomial(5,5) = 12.
|
|
|
MAPLE
|
with(numtheory):for n from 1 to 200 do:x:=factorset(n):n1:=nops(x): r:=sum('sum('binomial(x[j], x[i])', 'j'=i..n1)', 'i'=1..n1):printf(`%d, `, r):od:
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
