# Two sequences with sod: more terms?

Jonathan Post jvospost3 at gmail.com
Mon Feb 11 00:00:15 CET 2008

```Also known as "digital sum" and "digit sum." More important, I think,
is the general case (10 being an arbitrary base due to rule-of-thumb
with two human thumbs worth of fingers):

A digit sum s_b(n) is a sum of the base-b digits of n, which can be
implemented in Mathematica as

DigitSum[n_, b_:10] := Total[IntegerDigits[n, b]]

That is itself, I hasten to add, a special case of s_b(n,k) =  sum of
the k-th powers of (base-b digits of n).

All of these occur sometimes in OEIS, and often get slapped with a
"less" by those who quite reasonably disdain "base" sequences.

Also, in the interests of international understanding, what is called
"Murphy's Law" in the USA is called "Sod's Law" in the United Kingdom,
somehow overlapping "Finagle's law."

PRIMARY REFERENCE:

Cooper, Topher and Weisstein, Eric W. "Digit Sum." From MathWorld--A
Wolfram Web Resource. http://mathworld.wolfram.com/DigitSum.html

SECONDARY REFERENCES:

Allouche, J.-P. "Series and Infinite Products Related to Binary
Expansions of Integers." 1992.
http://algo.inria.fr/seminars/sem92-93/allouche.ps.

Allouche, J.-P. and Shallit, J. "The Ring of k-Regular Sequences."
Theor. Comput. Sci. 98, 163-197, 1992.

Fujiwara, M. and Ogawa, Y. Introduction to Truly Beautiful
Mathematics. Tokyo: Chikuma Shobo, 2005.

Grabner, P. J.; Herendi, T.; and Tichy, R. F. "Fractal Digital Sums
and Codes." Appl. Algebra Engrg. Comm. Comput. 8, 33-39, 1997.

Shallit, J. O. "On Infinite Products Associated with Sums of Digits."
J. Number Th. 21, 128-134, 1985.

Sloane, N. J. A. Sequences A000120/M0105, A007953, A053735, A053737,
A053824, A053827, A053828, A053829, A053830, A100044, A100045, and
A100046 in "The On-Line Encyclopedia of Integer Sequences."

Sondow, J. "Problem 11222." Amer. Math. Monthly 113, 459, 2006.

Trott, M. The Mathematica GuideBook for Programming. New York:
Springer-Verlag, p. 218, 2004. http://www.mathematicaguidebooks.org/.

Best,

prof. Jonathan Vos Post

[now primarily involved in computational biomathematics research
project, paper to be submitted to "Nature" by myself and Dr. Thomas L.
Vander Laan, M.D., F.A.C.S., with software modeling in SBML (Systems
Biology Markup Language] at Beckman Institute, of Caltech, with
applications to be clinically tested at USC Medical School; all of
which can be considered a clinical application of the very beautiful
theory papers by Ian Stewart and Martin Golubitsky on the Groupoid
formalism in Biological Network Dynamics, which were supported by 3
NSF grants, and highlighted by a Bull. Am. Math. Soc. paper July 2006]

On 2/10/08, Richard Guy <rkg at cpsc.ucalgary.ca> wrote:
> Glad that someone besides myself was bewildered
> by `sod'.   Can we find something that's more
> self-explicative and less offensive?  `digit-sum'
> or  `digitsum'  may not be too cumbersome?   R.
>
> On Sun, 10 Feb 2008, hv at crypt.org wrote:
>
> > zak seidov <zakseidov at yahoo.com> wrote:
> > :%N A000001 Numbers n such that sod(n^2}=10. Multiples
> > :of 10 are omitted.
> > [...]
> > :%N A000001 Numbers n such that n and n^2 have the same
> > :sod=10. Multiples of 10 are omitted.
> >
> > I hope you'll add an explanation of "sod" as well - my initial guess
> > was "sum of divisors", but that clearly isn't what is intended here.
> > None of mathworld, planetmath or wikipedia illuminate.
> >
> > It isn't until I search on OEIS itself that I find "sum of digits".
> > For the few extra letters required, I'd suggest spelling it out.
> > I found that OEIS has 45 matches for "sod", of which some have a comment
> > explaining the term, but not all. I didn't check whether they all
> > use it to mean the same thing.
> >
> > Note also that in British and Australian English "sod" has a pejorative
> > sense (though far more mildly so than its origins might suggest). I would
> > not for a moment suggest replacing it for that reason, but others might.
> >
> > Hugo

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