OFFSET
0,3
COMMENTS
It is easy to show that a(n) < 81*(log_10(n)+1). - Stefan Steinerberger, Mar 25 2006
It is known that a(0)=0 and a(1)=1 are the only fixed points of this map. For more information about iterations of this map, see A007770, A099645 and A000216 ff. - M. F. Hasler, May 24 2009
Also known as the "Happy number map", since happy numbers A007770 are those whose trajectory under iterations of this map ends at 1. - M. F. Hasler, Jun 03 2025
The number of zeroless preimages that map to n under the A003132 map is given by the number of compositions of n into positive squares of single-digit numbers, i.e., the number of compositions of n into parts of 1, 4, 9, ..., 81. For n <100 this is the same as A006465(n), but for n>=100 less because A006465 also admits parts of 100, 121, 144 etc. The full set of preimages is infinite for each n>0, because one can insert or append zeros anywhere in a preimage and still obtain the same sum-of-squared-digits. - R. J. Mathar, Sep 27 2025
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Hugo Steinhaus, One Hundred Problems in Elementary Mathematics, Dover New York, 1979, republication of English translation of Sto Zadań, Basic Books, New York, 1964. Chapter I.2, An interesting property of numbers, pp. 11-12 (available on Google Books).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..10000
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29; [preprint from author's web page, PostScript format].
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
B. M. Stewart, Sums of functions of digits, Canad. J. Math., 12 (1960), 374-389.
Robert Walker, Self Similar Sloth Canon Number Sequences
FORMULA
a(n) = n^2 - 20*n*floor(n/10) + 81*(Sum_{k>0} floor(n/10^k)^2) + 20*Sum_{k>0} floor(n/10^k)*(floor(n/10^k) - floor(n/10^(k+1))). - Hieronymus Fischer, Jun 17 2007
a(10n+k) = a(n)+k^2, 0 <= k < 10. - Hieronymus Fischer, Jun 17 2007
MAPLE
A003132 := proc(n) local d; add(d^2, d=convert(n, base, 10)) ; end proc: # R. J. Mathar, Oct 16 2010
MATHEMATICA
Table[Sum[DigitCount[n][[i]]*i^2, {i, 1, 9}], {n, 0, 40}] (* Stefan Steinerberger, Mar 25 2006 *)
Total/@(IntegerDigits[Range[0, 80]]^2) (* Harvey P. Dale, Jun 20 2011 *)
PROG
(PARI) A003132(n)=norml2(digits(n)) \\ M. F. Hasler, May 24 2009, updated Apr 12 2015
(Haskell)
a003132 0 = 0
a003132 x = d ^ 2 + a003132 x' where (x', d) = divMod x 10
-- Reinhard Zumkeller, May 10 2015, Aug 07 2012, Jul 10 2011
(Magma) [0] cat [&+[d^2: d in Intseq(n)]: n in [1..80]]; // Bruno Berselli, Feb 01 2013
(Python)
def A003132(n): return sum(int(d)**2 for d in str(n)) # Chai Wah Wu, Apr 02 2021
CROSSREFS
Concerning iterations of this map, see A003621, A039943, A099645, A031176, A007770, A000216 (starting with 2), A000218 (starting with 3), A080709 (starting with 4, this is the only nontrivial limit cycle), A000221 (starting with 5), A008460 (starting with 6), A008462 (starting with 8), A008463 (starting with 9), A139566 (starting with 15), A122065 (starting with 74169). - M. F. Hasler, May 24 2009
KEYWORD
AUTHOR
EXTENSIONS
More terms from Stefan Steinerberger, Mar 25 2006
Terms checked using the given PARI code, M. F. Hasler, May 24 2009
Replaced the Maple program with a version which works also for arguments with >2 digits, R. J. Mathar, Oct 16 2010
Added ref to Porges. Steinhaus also treated iterations of this function in his Polish book Sto zadań, but I don't have access to it. - Don Knuth, Sep 07 2015
STATUS
approved
