

A239703


Number of bases b > 1 for which the baseb digital sum of n is b.


10



0, 0, 0, 1, 0, 2, 1, 2, 0, 2, 2, 2, 1, 4, 0, 2, 1, 3, 1, 4, 1, 4, 2, 1, 1, 4, 1, 1, 2, 4, 0, 5, 0, 5, 3, 1, 2, 7, 0, 2, 3, 5, 0, 4, 0, 4, 3, 1, 1, 5, 1, 3, 2, 3, 0, 5, 2, 6, 1, 1, 0, 8, 0, 2, 2, 5, 3, 5, 1, 2, 2, 4, 1, 8, 0, 1, 4, 3, 2, 4, 1, 6, 3, 2, 0, 10, 2
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OFFSET

0,6


COMMENTS

For the definition of the digital sum, see A007953.
For reference, we write digitSum_b(x) for the baseb digital sum of x according to A007953 (with general base b).
The bases counted exclude the special base 1. The base1 expansion of a natural number is defined as 1=1_1, 2=11_1, 3=111_1 and so on. As a result, the base1 digital sum of n is n. The inclusion of base b = 1 would lead to a(1) = 1 instead of a(1) = 0. All other terms remain unchanged.
For odd n > 1 and b := (n + 1)/2 we have digitSum_b(n) = b, and thus a(n) >= 1.
The digitSum_b(n) is < b for bases b which satisfy b > floor((n+1)/2), and thus a(n) <= floor((n+1)/2).
If b is a base such that the baseb digital sum of n is b, then b < n and b  1 is a divisor of n  1, thus the number of such bases is limited by the number of divisors of n  1 (see formula section).
If p < n  1 is a divisor of n  1 which satisfy p >= sqrt(n  1), then digitSum_b(n) = b for b := p + 1. This leads to a lower bound for a(n) (see formula section).
If b  1 is a divisor of n  1, then b is not necessarily a base such that baseb digital sum of n is b. Example: 1, 2, 3, 4, 6, 8, 12, 16, and 24 are the divisors < 48 of 48, but digitSum_2(49) = 3, digitSum_3(49) = 5, digitSum_5(49) = 9, digitSum_7(49) = 1.
a(b*n) > 0 for all b > 1 which satisfy digitSum_b(n) = b.
Example 1: digitSum_2(3) = 2, hence a(2*3) > 0.
Example 2: digitSum_3(5) = 3, hence a(3*5) > 0.
The first n with a(n) = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... are n = 3, 5, 17, 13, 31, 57, 37, 61, 81, 85, ... .


LINKS

Hieronymus Fischer, Table of n, a(n) for n = 0..10000


FORMULA

a(n) = 0, if and only if n is a term of A187813.
a(A187813(n)) = 0.
a(A239708(n)) = 1, for n > 0.
a(A018900(n)) > 0, for n > 0.
a(A079696(n)) > 0, for n > 0.
a(A008864(n)) <= 1, for n > 0.
a(n) <= 1, if n  1 is a prime.
a(n) <= sigma_0(n  1)  1, for n > 1.
a(n) >= floor((sigma_0(n1)1)/2), for n > 1.


EXAMPLE

a(1) = 1, since digitSum_1(1) = 1 and digitSum_b(1) <> b for all b > 1.
a(2) = 0, since digitSum_1(2) = 2 (because of 2 = 11_1), and digitSum_2(2) = 1 (because of 2 = 10_2), and digitSum_b(2) = 2 for all b > 2.
a(3) = 1, since digitSum_1(3) = 3 (because of 3 = 111_1), and digitSum_2(3) = 2 (because of 3 = 11_2), and digitSum_3(3) = 1 (because of 3 = 10_3), and digitSum_b(3) = 3 for all b > 3.
a(5) = 2, since digitSum_1(5) = 5 (because of 5 = 11111_1), and digitSum_2(5) = 2 (because of 5 = 101_2), and digitSum_3(5) = 3 (because of 5 = 12_3), and digitSum_4(5) = 2 (because of 5 = 11_4), and digitSum_5(5) = 1 (because of 5 = 10_5), and digitSum_b(5) = 5 for all b > 5.


PROG

(Smalltalk)
"> Version 1: simple calculation for small numbers.
Answer the number of bases b such that the digital sum of n in base b is b.
Valid for bases b >= 1, thus returning a(1) = 1.
Using digitalSum from A007953.
Usage: n numOfBasesWithAltDigitalSumEQBase
Answer: a(n)"
numOfBasesWithDigitalSumEQBase
 numBases b bmax 
numBases := 0.
bmax := self + 1 // 2.
b := 0.
[b < bmax] whileTrue: [
b := b + 1
(self digitalSum: b) = b
ifTrue: [numBases := numBases + 1]].
^numBases

"> Version 2: accelerated calculation for large numbers.
Answer the number of bases b such that the digital sum of n in base b is b.
Valid for bases b > 1, thus returning a(1) = 0.
Using digitalSum from A007953.
Usage: n numOfBasesWithAltDigitalSumEQBase
Answer: a(n)"
numOfBasesWithDigitalSumEQBase
 numBases div b bsize 
self < 3 ifTrue: [^0].
div := (self  1) divisors.
numBases := 0.
bsize := div size  1.
1 to: bsize do: [ :i  b := (div at: i) + 1.
(self digitalSum: b) = b
ifTrue: [numBases := numBases + 1] ].
^numBases


CROSSREFS

Cf. A007953, A079696, A008864, A187813, A239707, A239708.
Cf. A000040; A000005 (definition of sigma_0(n)).
Sequence in context: A272328 A334956 A335881 * A029338 A240883 A048272
Adjacent sequences: A239700 A239701 A239702 * A239704 A239705 A239706


KEYWORD

nonn


AUTHOR

Hieronymus Fischer, Mar 31 2014


STATUS

approved



