A327737 a(n) is the sum of the lengths of the base-b expansions of n for all b with 1 <= b <= n.
1, 4, 7, 11, 14, 17, 20, 24, 28, 31, 34, 37, 40, 43, 46, 51, 54, 57, 60, 63, 66, 69, 72, 75, 79, 82, 86, 89, 92, 95, 98, 102, 105, 108, 111, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 155, 158, 161, 164, 167, 170, 173, 176, 179, 182, 185
Offset: 1
Examples
a(5) = 14 because 5 has the following representations in bases 1 to 5: 11111, 101, 12, 11, 10 giving a total length of 5+3+2+2+2 = 14. a(12) = 37 because 12 in bases 1 through 12 is 1...1 (12 1's), 1100, 110, and for bases 4 through 12 we get a 2-digit number, for a total length of 12+4+3+9*2 = 37. - _N. J. A. Sloane_, Sep 23 2019
Crossrefs
Cf. A043000.
Programs
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Go
package main import ( "fmt" "strconv" ) func main() { // Due to limitations in strconv, this will only work for the first 36 terms for i := 1; i <= 36; i++ { count := i for base := 2; base <= i; base++ { count += len(strconv.FormatInt(int64(i), base)) } fmt.Printf("%d, ", count) } } -
PARI
a(n) = my(i=n); for(b=2, n, i+=#digits(n, b)); i \\ Felix Fröhlich, Sep 23 2019
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Python
def count(n,b): c = 0 while n > 0: n, c = n//b, c+1 return c n = 0 while n < 60: n = n+1 a, b = n, 1 while b < n: b = b+1 a = a + count(n,b) print(n,a) # A.H.M. Smeets, Sep 23 2019
Formula
a(n) = A043000(n) + n. - A.H.M. Smeets, Sep 23 2019
Extensions
More terms from Felix Fröhlich, Sep 23 2019