A003132 Sum of squares of digits of n.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• look
• base
• nice

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

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
• Index entries for Colombian or self numbers and related sequences

Crossrefs

Cf. A052034, A052035.

Cf. A007953, A055017, A076313, A076314.

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

Cf. A080151, A051885 (record values and where they occur).

Cf. A257588, A332919.

Programs

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

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 *)

PARI

A003132(n)=norml2(digits(n)) \\ M. F. Hasler, May 24 2009, updated Apr 12 2015

Python

def A003132(n): return sum(int(d)**2 for d in str(n)) # Chai Wah Wu, Apr 02 2021

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

a(n) = A007953(A048377(n)) - A007953(n). - Reinhard Zumkeller, Jul 10 2011

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