The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A239703 Number of bases b > 1 for which the base-b 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS For the definition of the digital sum, see A007953. For reference, we write digitSum_b(x) for the base-b digital sum of x according to A007953 (with general base b). The bases counted exclude the special base 1. The base-1 expansion of a natural number is defined as 1=1_1, 2=11_1, 3=111_1 and so on. As a result, the base-1 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 base-b 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 base-b 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(n-1)-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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 9 16:50 EDT 2020. Contains 335545 sequences. (Running on oeis4.)