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 A225693 Alternating sum of digits of n. 15
 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, 8, 7, 6, 5, 4, 3, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A number n is divisible by 11 if and only if a(n) is divisible by 11. For generalizations see Sharpe and Webster, or the links below. The primes p for which the absolute value of the alternating sum of digits of p is also a prime begin: 2, 3, 5, 7, 13, 29, 31, 41, 47, 53, 61, 79, 83, 97, 101, 113, 137, 139, 151. - Jonathan Vos Post, May 27 2013 The above prime sequence is A115261. - Jens Kruse Andersen, Jul 13 2014 Digital sum with alternating signs starting with a positive sign for the most significant digit. - Hieronymus Fischer, Mar 23 2014 a(A135499(n)) = 0; a(A061470(n)) = 1. - Reinhard Zumkeller, Aug 08 2014 LINKS Hieronymus Fischer, Table of n, a(n) for n = 0..10000 Jim Loy, Divisibility Tests Stu Savory, Divisibility by prime numbers under 50 D. Sharpe and R. Webster, Reversing digits: divisibility by 27, 81, and 121, Mathematical Spectrum, 45 (2012/2013), 69-71. FORMULA If n has decimal expansion abc..xyz with least significant digit z, a(n) = a - b + c - d + ... From Hieronymus Fischer, Mar 23 2014: (Start) Formulas for general bases b > 1 (b = 10 for this sequence). Always m := floor(log_b(n)). a(n) = sum_{k>=0} (-1)^k*(floor(n*b^(k-m)) mod b). The sum is finite with floor(log_b(n)) as the highest index. a(n) = (-1)^m*n - (b+1)*sum_{00} (-1)^k*floor(n/b^k)). a(n) = -(-1)^(m-k)*a(n mod b^k) + a(floor(n/b^k)), for 0<=k<=m+1. a(n) = (-1)^m*a(n mod b) + a(floor(n/b)). a(n) = -(-1)^m*a(n mod b^2) + a(floor(n/b^2)). a(n) = (-1)^m*A055017(n). a(n) = A055017(A004086(n)). a(A004086(A004086(n))) = a(n). (End) MAPLE A225693 :=proc(n) local t1, i; t1:=convert(n, base, 10); add((-1)^(i+nops(t1))*t1[i], i=1..nops(t1)); end; [seq(A225693(n), n=0..120)]; MATHEMATICA Table[Total[Times@@@Partition[Riffle[IntegerDigits[n], {1, -1}, {2, -1, 2}], 2]], {n, 0, 90}] (* Harvey P. Dale, Nov 27 2015 *) PROG (Smalltalk) "Version for general bases" "Set base = 10 for this sequence" altDigitalSumLeft: base base > 1 ifTrue:  [m:= self integerFloorLog: base]          ifFalse: [^self \\ 2]. p:=1. s:=0. 1 to: m by: 2 do: [ :k |     p := p*base.     s := s - (self // p) .     p := p*base.     s := s + (self // p) ]. ^(self + ((base + 1)*s)) * (m alternate) "Version for base 10 using altDigitalSumRight from A055017" ^(self A004086) altDigitalSumLeft: 10 [by Hieronymus Fischer, Mar 23 2014] (Haskell) a225693 = f 1 0 where    f _ a 0 = a    f s a x = f (negate s) (s * a + d) x' where (x', d) = divMod x 10 -- Reinhard Zumkeller, May 11 2015, Aug 08 2014 CROSSREFS A055017 is closely related (but less natural). Cf. A061479. Cf. A004086. Cf. A007953, A033999, A031298, A135499, A061470. Cf. A257588. Sequence in context: A241494 A076313 A055017 * A040997 A256851 A247149 Adjacent sequences:  A225690 A225691 A225692 * A225694 A225695 A225696 KEYWORD sign,base,look AUTHOR N. J. A. Sloane, May 27 2013 EXTENSIONS Comment corrected by Jens Kruse Andersen, Jul 13 2014 STATUS approved

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Last modified February 25 23:44 EST 2020. Contains 332270 sequences. (Running on oeis4.)