A048377 -id:A048377 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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

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.

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

[0] cat [&+[d^2: d in Intseq(n)]: n in [1..80]]; // Bruno Berselli, Feb 01 2013

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

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 *)

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

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

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

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

A048376 Replace each 1 in decimal expansion of n with 1 1's, each 2 with 2 2's, etc. (0 vanishes).

Original entry on oeis.org

1, 22, 333, 4444, 55555, 666666, 7777777, 88888888, 999999999, 1, 11, 122, 1333, 14444, 155555, 1666666, 17777777, 188888888, 1999999999, 22, 221, 2222, 22333, 224444, 2255555, 22666666, 227777777, 2288888888, 22999999999, 333, 3331

Offset: 1

Patrick De Geest, Mar 15 1999

			12 -> 122, 123->122333.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A048377, A057628, A057630.

With[{rules=Table[n->Table[n,{n}],{n,0,9}]},Table[FromDigits[ Flatten[ IntegerDigits[x]/.rules]],{x,40}]] (* Harvey P. Dale, Oct 09 2011 *)

A048376(n)={sum(i=1,#n=concat( apply( t->vector(t,i,t),digits(n) )),n[i]*10^(#n-i))} \\ M. F. Hasler, Jan 23 2013

A067043 Nondecreasing sums of digits: a(0) = 0 and for n>0: a(n) = Min{m>n|SumOfDigits(m)>= SumOfDigits(a(n-1))}, where SumOfDigits = A007953.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 28, 29, 38, 39, 48, 49, 58, 59, 68, 69, 78, 79, 88, 89, 98, 99, 189, 198, 199, 289, 298, 299, 389, 398, 399, 489, 498, 499, 589, 598, 599, 689, 698, 699, 789, 798, 799, 889, 898, 899, 989, 998

Offset: 0

Reinhard Zumkeller, Feb 17 2002

A138472(a(n)) = 0. - Reinhard Zumkeller, Mar 19 2008

Positions of records in A048377: A192686(n) = A048377(a(n)). [Reinhard Zumkeller, Jul 10 2011]

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007953, A051885.

a067043 n = a067043_list !! n
a067043_list = 0 : f 1 1 0 1 where
   f k x y z
     | y > 0     = (x-y) : f k x (y `div` 10) z
     | k < 9     = x : f (k+1) (2*x-k*z+1) (z `div` 10) z
     | otherwise = x : f 1 (20*z-1) z (10*z)
-- Reinhard Zumkeller, Jul 10 2011

cp's OEIS Frontend

A192686 Record values in A048377.

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A003132 Sum of squares of digits of n.

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A048376 Replace each 1 in decimal expansion of n with 1 1's, each 2 with 2 2's, etc. (0 vanishes).

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Mathematica

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A067043 Nondecreasing sums of digits: a(0) = 0 and for n>0: a(n) = Min{m>n|SumOfDigits(m)>= SumOfDigits(a(n-1))}, where SumOfDigits = A007953.

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Haskell