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A290278 a(n) = Sum_{k=0..n} (A007953(5*k) - A007953(k)). 1
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 (list; graph; refs; listen; history; text; internal format)
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: A228780 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

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Last modified November 20 20:56 EST 2019. Contains 329347 sequences. (Running on oeis4.)