OFFSET
2,1
COMMENTS
From A.H.M. Smeets, Dec 14 2019: (Start)
a(n)-a(n-1) >= 2 due to the fact that n = 10_n, so there is an increment of at least 2. If n can be written as a perfect power m^s, an additional +1 comes to it for the representation of n in each base m.
For instance, for n = 729 we have 729 = 3^6 = 9^3 = 27^2, so there is an additional increment of 3. For n = 1296 we have 1296 = 6^4 = 36^2, so there is an additional increment of 2. For n = 4096 we have 4096 = 2^12 = 4^6 = 8^4 = 16^3= 64^2, so there is an additional increment of 5. (End)
LINKS
A.H.M. Smeets, Table of n, a(n) for n = 2..20000
Vaclav Kotesovec, Plot of a(n)/(2*n) for n = 2..1000000
FORMULA
a(n) = Sum_{i=2..n} floor(log_i(i*n)); a(n) ~ 2*n. - Vladimir Shevelev, Jun 03 2011 [corrected by Vaclav Kotesovec, Apr 05 2021]
From Ridouane Oudra, Nov 13 2019: (Start)
a(n) = Sum_{i=1..n-1} floor(n^(1/i));
a(n) = n - 1 + Sum_{i=1..floor(log_2(n))} floor(n^(1/i) - 1);
a(n) = n - 1 + A255165(n). (End)
If n is in A001597 then a(A001597(m)) - a(A001597(m)-1) = 2 + A253642(m), otherwise a(n) - a(n-1) = 2. - A.H.M. Smeets, Dec 14 2019
EXAMPLE
5 = 101_2 = 12_3 = 11_4 = 10_5. Thus a(5) = 3+2+2+2 = 9.
MAPLE
A043000 := proc(n) add( nops(convert(n, base, b)), b=2..n) ; end proc: # R. J. Mathar, Jun 04 2011
MATHEMATICA
Table[Total[IntegerLength[n, Range[2, n]]], {n, 2, 60}] (* Harvey P. Dale, Apr 23 2019 *)
PROG
(Magma) [&+[Floor(Log(i, i*n)):k in [2..n]]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019
(Python)
def count(n, b):
c = 0
while n > 0:
n, c = n//b, c+1
return c
n = 0
while n < 50:
n = n+1
a, b = 0, 1
while b < n:
b = b+1
a = a + count(n, b)
print(n, a) # A.H.M. Smeets, Dec 14 2019
(PARI) a(n)=sum(b=2, n, #digits(n, b)) \\ Jeppe Stig Nielsen, Dec 14 2019
(PARI) a(n)= n-1 +sum(b=2, n, logint(n, b)) \\ Jeppe Stig Nielsen, Dec 14 2019
(PARI) a(n) = {2*n-2+sum(i=2, logint(n, 2), sqrtnint(n, i)-1)} \\ David A. Corneth, Dec 31 2019
(PARI) first(n) = my(res = vector(n)); res[1] = 2; for(i = 2, n, inc = numdiv(gcd(factor(i+1)[, 2]))+1; res[i] = res[i-1]+inc); res \\ David A. Corneth, Dec 31 2019
CROSSREFS
KEYWORD
nonn,easy,base
AUTHOR
STATUS
approved