A109453 Cumulative sum of initial digits of n.

1, 3, 6, 10, 15, 21, 28, 36, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 201, 207, 213

Offset: 1

Author

Jonathan Vos Post, Aug 27 2005

Keywords

• base
• easy
• nonn

Links

• A. Cobham, Uniform Tag Sequences, Mathematical Systems Theory, 6 (1972), 164-192.

Crossrefs

Partial sums of A000030.

Programs

Mathematica

f[n_] := Sum[ Floor[i / 10^Floor[Log[10, i]]], {i, n}]; Table[ f[n], {n, 62}] (* or *)
a[0] = 0; a[n_] := a[n] = (a[n - 1] + First[ IntegerDigits[ n]]); Table[ a[n], {n, 62}] (* Robert G. Wilson v, Aug 30 2005 *)
Accumulate[First/@(IntegerDigits/@Range[70])] (* Harvey P. Dale, Jan 11 2016 *)

PARI

a(n) = sum(k=1, n, digits(k)[1]); \\ Michel Marcus, Dec 13 2017

Formula

a(n) = Sum_{i=1..n} A000030(i). a(n) = Sum_{i=1..n} [i / 10^([log_10(i)])] where [] denotes floor and log_10(i) is the logarithm is base 10. a(n+1) = a(n) + first-digit-of(n+1).

Extensions

Corrected by Robert G. Wilson v, Aug 30 2005