A290278 a(n) = Sum_{k=0..n} (A007953(5*k) - A007953(k)).

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

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^k-1, then d_b(i * n) + d_b(i * (b^k-1 - n)) = k*(b-1) (as i * n and i * (b^k-1 - n) can be added without carry in base b),

- if i and j divides b and k > 0 and 0 <= n <= b^k-1, then d_b(i * n) - d_b(j * n) = d_b(j * (b^k-1 - n)) - d_b(i * (b^k-1 - n)) (this implies the conjecture about the symmetry of A289411),

- also, if i and j divides b and k > 0, Sum_{m=0..b^k-1} (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^k-2, then G(b,i,j)(n) = G(b,i,j)(b^k-2 - n) (in other words, the sequence G(b,i,j) restricted to the first b^k-1 terms is symmetrical), and G(b,i,j)(b^k-2) = 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