

A003132


Sum of squares of digits of n.
(Formerly M3355)


109



0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 4, 5, 8, 13, 20, 29, 40, 53, 68, 85, 9, 10, 13, 18, 25, 34, 45, 58, 73, 90, 16, 17, 20, 25, 32, 41, 52, 65, 80, 97, 25, 26, 29, 34, 41, 50, 61, 74, 89, 106, 36, 37, 40, 45, 52, 61, 72, 85, 100, 117, 49
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OFFSET

0,3


COMMENTS

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


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. 1112 (available on Google Books).


LINKS



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


MAPLE

A003132 := proc(n) local d; add(d^2, d=convert(n, base, 10)) ; end proc: # R. J. Mathar, Oct 16 2010


MATHEMATICA

Total/@(IntegerDigits[Range[0, 80]]^2) (* Harvey P. Dale, Jun 20 2011 *)


PROG

(Haskell)
a003132 0 = 0
a003132 x = d ^ 2 + a003132 x' where (x', d) = divMod x 10
(Magma) [0] cat [&+[d^2: d in Intseq(n)]: n in [1..80]]; // Bruno Berselli, Feb 01 2013
(Python)


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

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



