



0, 4, 3, 6, 4, 6, 3, 4, 0, 0, 4, 12, 15, 22, 24, 30, 31, 36, 36, 40, 39, 42, 40, 42, 39, 40, 36, 36, 31, 30, 33, 40, 42, 48, 49, 54, 54, 58, 57, 60, 58, 60, 57, 58, 54, 54, 49, 48, 42, 40, 42, 48, 49, 54, 54, 58, 57, 60, 58, 60, 57, 58, 54, 54, 49, 48, 42, 40
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OFFSET

0,2


COMMENTS

This sequence is closely related to A289411.
The scatterplots of this sequence and of A289411 have similarites, including the same type of symmetry on the first 10^k terms for k > 0.
For b > 1, let d_b be the digital sum in base b:
 we have for example d_10 = A007953,
 also, d_b(b*n) = d_b(n) for any n >= 0,
 and d_b(n + m) = d_b(n) + d_b(m) iff n and m can be added without carry in base b,
 hence if i divides b and k > 0 and 0 <= n <= b^k1, then d_b(i * n) + d_b(i * (b^k1  n)) = k*(b1) (as i * n and i * (b^k1  n) can be added without carry in base b),
 if i and j divides b and k > 0 and 0 <= n <= b^k1, then d_b(i * n)  d_b(j * n) = d_b(j * (b^k1  n))  d_b(i * (b^k1  n)) (this implies the conjecture about the symmetry of A289411),
 also, if i and j divides b and k > 0, Sum_{m=0..b^k1} (d_b(i * m)  d_b(j * m)) = 0.
For b > 1, i > 0 and j > 0 such that neither i nor j are divisible by b, let G(b,i,j) be the function defined by n > Sum_{k=0..n} (d_b(i*k)  d_b(j*k)):
 we have for example G(10,5,1) = a (this sequence),
 G(b,i,i) = 0, G(b,i,j) = G(b,j,i), G(b,i,j) + G(b,j,k) = G(b,i,k),
 if i and j divide b and k > 0 and 0 <= n <= b^k2, then G(b,i,j)(n) = G(b,i,j)(b^k2  n) (in other words, the sequence G(b,i,j) restricted to the first b^k1 terms is symmetrical), and G(b,i,j)(b^k2) = 0 (in other words, G(b,i,j) has infinitely many zeros).


LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..10000


MATHEMATICA

Block[{nn = 68, k = 5, s}, s = Table[Total@ IntegerDigits[k n]  Total@ IntegerDigits@ n, {n, 0, nn}]; Table[Total@ Take[s, n], {n, nn}]] (* Michael De Vlieger, Jul 31 2017 *)


PROG

(PARI) s = 0; for (n=0, 67, s += sum digits(5*n)  sum digits(n); print1 (s ", "))


CROSSREFS

Cf. A289411.
Sequence in context: A333328 A109339 A071989 * A300894 A328258 A190415
Adjacent sequences: A290275 A290276 A290277 * A290279 A290280 A290281


KEYWORD

nonn,base,look


AUTHOR

Rémy Sigrist, Jul 25 2017


STATUS

approved



