Two sequences with sod: more terms?

Graeme McRae g_m at
Wed Feb 13 09:23:30 CET 2008

Perhaps the following observation means something:  If a unit of measure is chosen at random (with equal probability) from among foot, yard, and mile, there is a roughly 30% probability that any length, X (a random variable), begins with "1" when expressed in base 10 in this unit, as long as the distribution of X is such that its deviation is "large enough" compared to its mean.

--Graeme McRae
This was sent from my BlackBerry® wireless device, which randomly substitutes words for other words.

-----Original Message-----
From: "Jonathan Post" <jvospost3 at>

Date: Mon, 11 Feb 2008 01:56:36 
To:"Richard Guy" <rkg at>
Cc:hv at, "zak seidov" <zakseidov at>, njas at,       seqfan at, "Andrew Post" <andrewpost at>,       "Dr. Geoffrey A. Landis" <geoffrey.landis at>,       "Forrest Bishop" <forrestb at>,       "Chris Phoenix" <cphoenix at>,       "Dr. Philip Vos Fellman" <shirogitsune99 at>,       "T. D. Noe" <noe at>,       "Dr. George Hockney" <george.hockney at>,       "Dr. Tom McDonough" <tom at>
Subject: Re: Two sequences with sod: more terms?

Googling on a phrase that I emailed here about 10 hours ago, led me toa "coincidence" that I just posted on the delightful "Good Math, BadMath" blog run by Mark Chu-Carroll, PhD Computer Scientist, who worksfor Google as a Software Engineer. In one thread, he has a loonobsessed with dull notational artifacts of base 10, which troll ispassionately convinced makes him the most insightful genius alive,suppressed by The Establishment.  We eventually lost patience withhim, and I found myself echoing what I'd said in seqfans in responseto Richard Guy's query on "sod" for "sum of digits."
The coincidence was in multiple hits on "rule of thumb" and "GregoryBenford" -- NOT to be confused with the F. Benford of "Benford's Law:A phenomenological law also called the first digit law, first digitphenomenon, or leading digit phenomenon. Benford's law states that inlistings, tables of statistics, etc., the digit 1 tends to occur withprobability ∼30%, much greater than the expected 11.1% (i.e., onedigit out of 9). Benford's law can be observed, for instance, byexamining tables of logarithms  and noting that the first pages aremuch more worn and smudged than later pages (Newcomb 1881). WhileBenford's law unquestionably applies to many situations in the realworld, a satisfactory explanation has been given only recently throughthe work of Hill (1996)."Weisstein, Eric W. "Benford's Law." From MathWorld--A Wolfram WebResource.
Benford, F. "The Law of Anomalous Numbers." Proc. Amer. Phil. Soc. 78,551-572, 1938.
My blog posting being as follows, and as submitted on
10 being an arbitrary base due to rule-of-thumb with two human thumbsworth of fingers.
By the way, Professor Gregory Benford, whom I've known for very nearly30 years, told me "rules of thumb are well and good, but remember thatother intelligent beings may have different numbers of thumbs."
Which reminds me of the following...
The passion rule of thumb: Gregory Benford, professor of physics andastronomy, a long-time advisor to NASA, and Hugo- and Nebula-winningnovelist, says "passion is inversely proportional to the amount ofreal information available."
Coincidently, I can say more linking "rule of thumb" and Benford.
"If crew were already harvesting, then Killeen knew he had beenrunning a bit too long. He deliberately did not use the time displayin his suit, since the thing was ageold and its symbols were aconfusing scramble of too much data, unreadable to his untutored mind.Instead he checked his inboard system. The timer stuttered out auseless flood of information and then told him he had been runningnearly an hour. He did not know very precisely how long an hour was,but as a rule of thumb it was enough."[Tides of Light, by Gregory Benford, Chapter One, Copyright (c) 1989by Abbenford Associates]
"Languages evolve, just as in biology, by descent and divergence. Wecan read Shakespeare but need notes to fathom some of his archaicwords. Without substantial help, Beowulf is beyond us. As a rule ofthumb, basic vocabularies change about twenty percent in a thousandyears. Even if a language survives (as most don't) for five thousandyears, it will be vastly different."['Deep Time', by Gregory Benford]['Deep Time' fascinating, thought provoking, disturbing'Deep Time'by Gregory BenfordAvon, $20Review by L.D. MeagherMarch 25, 1999Web posted at: 4:22 p.m. EST (2122 GMT)]
"A general rule of thumb is that you should measure metabolic rateunder normal, low-stress conditions."[THE LOST CHAPTER FROM THE LONG TOMORROWThis chapter was left out of MRR's recent book, The Long Tomorrow; HowAdvances in Evolutionary Biology Can Help Us Postpone Aging, OxfordUniversity Press, 2005. Here it is for you to judge if this was theright decision.]
Posted by: Jonathan Vos Post | February 11, 2008 4:33 AM

On Feb 10, 2008 3:00 PM, Jonathan Post <jvospost3 at> wrote:> 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.!
l>> SECONDARY REFERENCES:>> Allouche, J.-P. "Series and Infinite Products Related to Binary> Expansions of Integers." 1992.>>> 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, 200!
4.>> Best,>> prof. Jon!
athan 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> 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 wrote:> >> > > zak seidov <zakseidov at> wro!
te:> > > :%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>

More information about the SeqFan mailing list