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 A239705 Number of bases b for which the base-b alternate digital sum of n is -b. 5
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 OFFSET 1 COMMENTS For the definition of the alternate digital sum, see A055017 or A225693. For reference: we write altDigitSum_b(x) for the base-b alternate digital sum of x according to A055017. The number of counted bases includes 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 alternate digital sum is 0 if n is even, and is 1 if n is odd. The altDigitSum_b(n) is > -b for bases b that satisfy b > b0 := floor((n - floor(n^(1/3))*(floor(n^(1/3))-1))^(1/3)), and thus a(n) <= b0. If n is the sum of a cube m^3 and an oblong number m*(m-1) (see A002378), then, with b := m, b^3 + b(b-1) = n and b = b0. This implies altDigitSum_b(n) = 0 - (b-1) + 0 - 1 = -b and shows that there are infinitely many n with a base b > 1 such that altDigitSum_b(n) = -b. Consequently, a(n) >= 1 infinitely often (for those n > 1 that are the sum of a cube and an oblong number, i.e., n = 10, 33, 76, 145, 246, ...). Moreover, a(n) >= 1 is also true for n == b(b(b+1)-1) (mod (b+1)b^4), b>1.   Example 1: altDigitSum_2(n) = -2 for n == 10 (mod 48).   Example 2: altDigitSum_3(n) = -3 for n == 33 (mod 324).   Example 3: altDigitSum_4(n) = -4 for n == 76 (mod 1280). If b is a base such that the base-b alternate digital sum of n is -b, then b + 1 is a divisor of n - 1. Thus, the number of such bases is also limited by the number of divisors of n - 1 (see formula section). If b + 1 is a divisor of n - 1, then b is not necessarily a base such that base-b alternate digital sum of n is -b. Example: 2, 4, 8 and 16 are divisors of 32 and altDigitSum_3(33) = -3, but altDigitSum_1(33) = 1, altDigitSum_7(33) = 1, altDigitSum_15(33) = 1. a(b*n) > 0 for all b > 1 that satisfy altDigitSum_b(n) = b.   Example 4: altDigitSum_2(5) = 2, hence a(2*5) > 0.   Example 5: altDigitSum_3(11) = 3, hence a(3*11) > 0. The first n with a(n) = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... are n = 10, 136, 385, 2241, 24781, 26797, 175561, 182401, 374221, 475021, ... . LINKS Hieronymus Fischer, Table of n, a(n) for n = 1..10000 FORMULA a(n^3 + A002378(n-1)) = a(n^3 + n^2 - n) >= 1, n > 1. a(n) = 0, if n - 1 is a prime. A239703(n) = 0 ==> a(n) = 0. a(A187813(n) = 0. a(n) <= floor(sigma_0(n-1)/2). EXAMPLE a(1) = 0, since altDigitSum_1(1) = 1 and altDigitSum_b(1) = 1 > -b for all b > 1. a(2) = 0, since altDigitSum_1(2) = 0 (because of 2 = 11_1), and altDigitSum_2(2) = -1 (because of 2 = 10_2), and altDigitSum_b(2) = 2 > -b for all b > 2. a(3) = 0, since altDigitSum_1(3) = 1 (because of 3 = 111_1), and altDigitSum_2(3) = 0 (because of 3 = 11_2), and altDigitSum_3(3) = -1 (because of 3 = 10_3), and altDigitSum_b(3) = 3 > -b for all b > 3. a(10) = 1, since altDigitSum_1(10) = 0, and altDigitSum_2(10) = -2 (because of 10 = 1010_2), and altDigitSum_3(10) = 2 (because of 10 = 101_3), and altDigitSum_4(10) = 0 (because of 10 = 22_4), and altDigitSum_5(10) = -2 (because of 10 = 20_5), ..., and altDigitSum_b(10) = 10 > -b for all b > 10. PROG (Smalltalk) "> Version 1: simple calculation for small numbers. Answer the number of bases b for which the alternate digital sum of n in base b is -b. Valid for bases b > 0. Using altDigitalSumRight from A055017. Usage: n numOfBasesWithAltDigitalSumEQ0 Answer: a(n)" numOfBasesWithAltDigitalSumEQNegBase      | b q numBases |      self < 10 ifTrue: [^0].      numBases := 0.      q := self cubeRootTruncated.      b := 1.      [b < q] whileTrue:[           (self altDigitalSumRight: b) = 0           ifTrue: [numBases := numBases + 1].           b := b + 1].      ^numBases [by Hieronymus Fischer, May 08 2014] ----------- (Smalltalk) "> Version 2: accelerated calculation for large numbers.    Answer the number of bases b for which the alternate    digital sum of n in base b is -b.    Valid for bases b > 0.    Using altDigitalSumRight from A055017.    Usage: n numOfBasesWithAltDigitalSumEQ0    Answer: a(n)" numOfBasesWithAltDigitalSumEQNegBase    | numBases div b bsize  |    self < 10 ifTrue: [^0].    div := (self - 1) divisors.    numBases := 0.    bsize := div size // 2 + 1.    2 to: bsize do: [ :i | b := (div at: i) - 1.           (self altDigitalSumRight: b) = (b negated)            ifTrue: [numBases := numBases + 1]].    ^numBases [by Hieronymus Fischer, May 08 2014] CROSSREFS Cf. A055017, A225693, A187813. Cf. A239703, A239704, A239706, A239707. Cf. A002378, A008864, A000040, A000005. Sequence in context: A037807 A037817 A297039 * A025468 A025465 A323514 Adjacent sequences:  A239702 A239703 A239704 * A239706 A239707 A239708 KEYWORD nonn AUTHOR Hieronymus Fischer, May 08 2014 STATUS approved

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Last modified June 23 02:16 EDT 2021. Contains 345395 sequences. (Running on oeis4.)