A000218 -id:A000218 - 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

A039943 Every integer eventually goes to one of these under the "x goes to sum of squares of digits of x" map.

Original entry on oeis.org

0, 1, 4, 16, 20, 37, 42, 58, 89, 145

Offset: 0

N. J. A. Sloane

The subset of the first three terms also satisfies the current definition. An alternate definition would be: Periodic points of A003132. - M. F. Hasler, May 24 2009

Following I. Ja. Tanatar (Moscow), one can easily prove that, for a given x, there exists an iteration of the map f(x) given in the definition which reaches 1 or 89. Indeed, it is easy to see that if x has at least 3 digits, then f(x) < x. Therefore there exists an iteration of f with not more than 2 digits. For two-digit numbers the property is verified directly. See Kordemsky. - Vladimir Shevelev, May 06 2013

B. A. Kordemsky, Matematicheskaja Smekalka, Moscow, 1955, pp. 305 and 557 (in Russian).

Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
Arthur Porges, A set of eight numbers, Amer. Math. Monthly, 52 (1945), 379-382. [Annotated scanned copy]
Eric W. Weisstein, MathWorld: Happy Number
Wikipedia, Periodic point

Cf. A000216, A003621.

Cf. A003132 (the iterated map), A003621, A039943, A031176, A007770, A000216 (orbit of 2), A000218 (orbit of 3), A080709 (orbit of 4, the only nontrivial limit cycle), A000221 (orbit of 5), A008460 (orbit of 6), A008462 (orbit of 8), A008463 (orbit of 9), A139566 (orbit of 15), A122065 (orbit of 74169).

a039943 n = a039943_list !! n
a039943_list = [0,1,4,16,20,37,42,58,89,145]
-- Reinhard Zumkeller, Mar 16 2013

lst = {}; Do[a = NestWhile[Plus @@ (IntegerDigits@#^2) &, n, Unequal, All]; If[FreeQ[lst, a], AppendTo[lst, a]], {n, 10^4}] (* Robert G. Wilson v, Jan 19 2006 *)
Union[Table[NestWhile[Total[IntegerDigits[#]^2]&,n,Unequal,All],{n,0,100}]] (* Harvey P. Dale, Nov 26 2013 *)

A000216 Take sum of squares of digits of previous term, starting with 2.

Original entry on oeis.org

2, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37

Offset: 1

N. J. A. Sloane

As the orbit of 2 under A003132, this could also have offset 0. Merges into A080709 right after the first term: a(n+1) = A080709(n) for all n >= 1. - M. F. Hasler, Apr 27 2018

R. Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 83.
P. Kiss, A generalization of a problem in number theory, Math. Sem. Notes Kobe Univ., 5 (1977), no. 3, 313-317. MR 0472667 (57 #12362).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
H. G. Grundmann, Semihappy Numbers, J. Int. Seq. 13 (2010), 10.4.8, Theorem 1.
P. Kiss, A generalization of a problem in number theory, [Hungarian], Mat. Lapok, 25 (No. 1-2, 1974), 145-149.
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
A. Porges, A set of eight numbers, Amer. Math. Monthly, 52 (1945), 379-382. [Annotated scanned copy]
H. J. J. te Riele, Iteration of number-theoretic functions, Nieuw Archief v. Wiskunde, (4) 1 (1983), 345-360.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A003132 (the iterated map), A003621, A039943, A099645, A031176, A007770, A000218 (starting with 3), A080709 (starting with 4), 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

a000216 n = a000216_list !! (n-1)
a000216_list = iterate a003132 2 -- Reinhard Zumkeller, Aug 24 2011

[2] cat &cat[[4, 16, 37, 58, 89, 145, 42, 20]: n in [0..17]]; // Vincenzo Librandi, Jan 29 2013

NestList[Total[IntegerDigits[#]^2]&, 2, 80] (* Vincenzo Librandi, Jan 29 2013 *)

A000216(n)=[42, 20, 4, 16, 37, 58, 89, 145, 2][n%8+8^(n<2)] \\ M. F. Hasler, May 24 2009, edited Apr 27 2018

Periodic with period 8.

A099645 Number of iterations until n reaches a number in A039943 under "x goes to sum of squares of digits of x" map.

Original entry on oeis.org

0, 1, 5, 0, 4, 9, 5, 5, 4, 1, 2, 5, 2, 6, 3, 0, 5, 3, 4, 0, 5, 6, 3, 1, 3, 2, 6, 3, 2, 5, 2, 3, 4, 4, 5, 8, 0, 2, 5, 1, 6, 0, 4, 4, 7, 4, 3, 6, 4, 4, 3, 3, 5, 7, 5, 2, 4, 0, 2, 9, 1, 2, 8, 4, 2, 7, 2, 2, 5, 5, 5, 6, 1, 3, 4, 2, 2, 4, 3, 5, 3, 3, 2, 6, 1, 2, 4, 7, 0, 4, 4, 2, 5, 4, 2, 5, 3, 1, 8, 1, 2, 5, 2, 6, 3

Offset: 1

Labos Elemer, Nov 08 2004

Length of transient when the f[n]=Sum[digit^2 of n] function is iterated.

In A031176 including cycle lengths[=c] of this iteration only c=1 and c=8 occur. A007770 lists cases of c=1, the happy numbers.

			n=99999999999: iteration-list={99999999999,891,146,53,34,25,29,85,89,145,42,20,[4,16,37,58,89,145,42,20],4,...}. Lengths of transient=12, of cycle=8.

Hugo Steinhaus: "Sto zadan" (1958), "One Hundred Problems in Elementary Mathematics" (1964), problem 2. - M. F. Hasler, May 24 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382. - _M. F. Hasler_, May 24 2009

Cf. A007700, A031176.
Cf. A000216, A003132, A003621
Cf. A039943, A031176, A007770, A000216 (orbit of 2), A000218 (orbit of 3), A080709 (orbit of 4), A000221 (orbit of 5), A008460 (orbit of 6), A008462 (orbit of 8), A008463 (orbit of 9), A139566 (orbit of 15), A122065 (orbit of 74169). - M. F. Hasler, May 24 2009

a099645 = length . takeWhile (`notElem` a039943_list) . iterate a003132
a099645_list = map a099645 [1..]
-- Reinhard Zumkeller, Aug 24 2011

fu[x_] :=Apply[Plus, IntegerDigits[x]^2];hs=20; (* transient lengths are obtained by: *) a[n_] :=-1+Min[Flatten[Position[NestList[fu, n, Length[Union[NestList[fu, n, hs]]]] -Last[NestList[fu, n, Length[Union[NestList[fu, n, hs]]]]], 0]]]; Table[a[n], {n, 1, 256}]

A099645(n)={ local( c=0, S=Set([1,4,16,37,58,89,145,42,20])); while( !setsearch(S,n), n=A003132(n); c++); c} \\ M. F. Hasler, May 24 2009

Terms checked using the given PARI code. However, according to the domain of A003132 and the definition of A039943 (which both include 0), an initial a(0)=0 should be added here, too. - M. F. Hasler, May 24 2009

A000221 Take sum of squares of digits of previous term; start with 5.

Original entry on oeis.org

5, 25, 29, 85, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145

Offset: 1

N. J. A. Sloane

Essentially the same as A080709, cf. formula. - M. F. Hasler, May 24 2009

As the orbit of 5 under A003132, this could as well start with index 0. - M. F. Hasler, Apr 27 2018

R. Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 83.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 25.

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A003132 (the iterated map), A003621, A039943, A099645, A031176, A007770, A000216 (starting with 2), A000218 (starting with 3), A080709 (starting with 4), 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

a000221 n = a000221_list !! (n-1)
a000221_list = iterate a003132 5
-- Reinhard Zumkeller, Mar 04 2013

[5, 25, 29, 85] cat &cat[[89, 145, 42, 20, 4, 16, 37, 58]: n in [0..17]]; // Vincenzo Librandi, Jan 29 2013

NestList[Plus @@ IntegerDigits[ # ]^2 &, 5, 50]
PadRight[{5,25,29,85},120,{4,16,37,58,89,145,42,20}] (* Harvey P. Dale, Jan 14 2022 *)

A000221(n)=[20,4,16,37,58,89,145,42,5,25,29,85][n%8+8^(n<5)] \\ M. F. Hasler, May 24 2009, edited Apr 27 2018

Ultimately periodic with period 8.
a(n) = A080709(n) for n >= 5. - M. F. Hasler, May 24 2009
a(n+1) = A003132(a(n)). - Reinhard Zumkeller, Dec 19 2011

A008460 Take sum of squares of digits of previous term; start with 6.

Original entry on oeis.org

6, 36, 45, 41, 17, 50, 25, 29, 85, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4

Offset: 1

N. J. A. Sloane

R. Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 83.

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A003132 (the iterated map), A003621, A039943, A099645, A031176, A007770, A000216 (starting with 2), A000218 (starting with 3), A080709 (starting with 4), A000221 (starting with 5), A008462 (starting with 8), A008463 (starting with 9), A139566 (starting with 15), A122065 (starting with 74169). - M. F. Hasler, May 24 2009

[6, 36, 45, 41, 17, 50, 25, 29, 85] cat &cat[[89, 145, 42, 20, 4, 16, 37, 58]: n in [0..17]]; // Vincenzo Librandi, Jan 29 2013

NestList[Total[IntegerDigits[#]^2]&, 6, 80] (* Vincenzo Librandi, Jan 29 2013 *)

A008460(n)=[6,36,45,41,17, 50,25,29,85,89, 145,42,20,4,16, 37,58][if(n<18,n,(n-10)%8+10)] \\ M. F. Hasler, May 24 2009

Periodic with period 8.

a(13+n) = A080709(n). - M. F. Hasler, May 24 2009

An erroneous (duplicate) term deleted by M. F. Hasler, May 24 2009

A008462 Take sum of squares of digits of previous term; start with 8.

Original entry on oeis.org

8, 64, 52, 29, 85, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145

Offset: 1

N. J. A. Sloane

R. Honsberger, Ingenuity in Math., Random House, 1970, p. 83.

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A003132 (the iterated map), A003621, A039943, A099645, A031176, A007770, A000216 (starting with 2), A000218 (starting with 3), A080709 (starting with 4), A000221 (starting with 5), A008460 (starting with 6), A008463 (starting with 9), A139566 (starting with 15), A122065 (starting with 74169). - M. F. Hasler, May 24 2009

[8, 64, 52, 29, 85] cat &cat[[89, 145, 42, 20, 4, 16, 37, 58]: n in [0..17]]; // Vincenzo Librandi, Jan 29 2013

NestList[Total[IntegerDigits[#]^2]&, 8, 80] (* Vincenzo Librandi, Jan 29 2013 *)
PadRight[{8,64,52,29,85},80,{20,4,16,37,58,89,145,42}] (* Harvey P. Dale, Dec 27 2019 *)

A008462(n)=[8, 64, 52, 29, 85, 89, 145, 42, 20, 4, 16, 37, 58][if(n>13,(n-6)%8+6,n)] \\ M. F. Hasler, May 24 2009

Vec(x*(8 + 64*x + 52*x^2 + 29*x^3 + 85*x^4 + 89*x^5 + 145*x^6 + 42*x^7 + 12*x^8 - 60*x^9 - 36*x^10 + 8*x^11 - 27*x^12) / ((1 - x)*(1 + x)*(1 + x^2)*(1 + x^4)) + O(x^60)) \\ Colin Barker, Apr 28 2018

Periodic with period 8.
a(n) = A080709(n-1) for n >= 5 and a(n) = A000221(n-1) = A008460(n+4) for all n >= 4. - M. F. Hasler, May 24 2009; edited and extended Apr 27 2018
From Colin Barker, Apr 28 2018: (Start)
G.f.: x*(8 + 64*x + 52*x^2 + 29*x^3 + 85*x^4 + 89*x^5 + 145*x^6 + 42*x^7 + 12*x^8 - 60*x^9 - 36*x^10 + 8*x^11 - 27*x^12) / ((1 - x)*(1 + x)*(1 + x^2)*(1 + x^4)).
a(n) = a(n-8) for n>13.
(End)

A080709 Take sum of squares of digits of previous term, starting with 4.

Original entry on oeis.org

4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37

Offset: 1

N. J. A. Sloane, Mar 04 2003

Occurs as puzzle in the Nintendo DS game "Professor Layton and the Diabolical Box". - M. F. Hasler, Dec 18 2009
From M. F. Hasler, Apr 27 2018: (Start)
As the orbit of 4 under A003132, this could rather have offset 0. Merges with the orbit of 5 at the 5th term of both sequences, and with other orbits as given in the formula section.
Porges gave his "set of eight numbers" as a(1)..a(8) in this order, rather than that of the set A039943. (End)

R. Honsberger, Ingenuity in Math., Random House, 1970, p. 83.

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A003132 (the iterated map), A003621, A039943, A099645, A031176, A007770, A000216 (starting with 2), A000218 (starting with 3), 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

a080709 n = a080709_list !! (n-1)
a080709_list = iterate a003132 4
-- Reinhard Zumkeller, Aug 24 2011

&cat[[4, 16, 37, 58, 89, 145, 42, 20]: n in [0..17]]; // Vincenzo Librandi, Jan 29 2013

NestList[Total[IntegerDigits[#]^2]&, 4, 80] (* Vincenzo Librandi, Jan 29 2013 *)

A080709(n)=[4, 16, 37, 58, 89, 145, 42, 20][(n-1)%8+1] \\ M. F. Hasler, May 24 2009

Periodic with period 8.
a(n) = A000216(n+1). - R. J. Mathar, Sep 19 2008
By definition, a(n+1) = A003132(a(n)) for n >= 1. a(n) = A000221(n) = A000218(n+3) = A008460(n+6) = A008462(n+1) = A008463(n+2) = A122065(n+3) = A139566(n+2) for n >= 8 or earlier. - M. F. Hasler, May 24 2009, edited Apr 27 2018

A122065 Take sum of squares of digits of previous term; start with 74169.

Original entry on oeis.org

74169, 183, 74, 65, 61, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4

Offset: 1

Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 14 2006

From a quiz, cf. Russel & Carter reference.

K. Russell and P. Carter, Number Puzzles, W. Foulsham and Co. Ltd. (1993).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A000216 (main entry for related sequences), A003132 (the iterated "sum digits squared" map).

[74169, 183, 74, 65, 61] cat &cat[[37, 58, 89, 145, 42, 20, 4, 16]: n in [0..17]]; // Vincenzo Librandi, Jan 29 2013

NestList[Total[IntegerDigits[#]^2]&, 74169, 80] (* Vincenzo Librandi, Jan 29 2013 *)

A122065_vec=vector(50,n,t=if(n>1,norml2(digits(t)),74169));
A122065(n)=A122065_vec[n%8+(n>7)*8] \\ M. F. Hasler, Apr 27 2018

a(n) = A000218(n) for n >= 4, a(n) = A000216(n-2) for n >= 6. - M. F. Hasler, Apr 27 2018

A003621 Number of iterations until n reaches 1 or 4 under x goes to sum of squares of digits map.

Original entry on oeis.org

0, 1, 11, 0, 8, 13, 5, 9, 10, 1, 2, 9, 2, 10, 10, 7, 9, 9, 4, 1, 9, 10, 3, 2, 7, 9, 10, 3, 6, 11, 2, 3, 10, 8, 9, 12, 6, 7, 11, 8, 10, 2, 8, 4, 11, 8, 9, 10, 4, 8, 10, 7, 9, 11, 9, 8, 10, 5, 8, 13, 7, 9, 12, 8, 8, 11, 6, 2, 12, 5, 9, 10, 6, 9, 10, 6, 5, 4, 3, 9

Offset: 1

N. J. A. Sloane, Mira Bernstein

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 13.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
A. Porges, A set of eight numbers, Amer. Math. Monthly, 52 (1945), 379-382. [Annotated scanned copy]
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.

Cf. A000216, A000218, A300081, etc.

Cf. A003132.

f:= n -> convert(map(t -> t^2, convert(n,base,10)),`+`):
g:= proc(n) option remember;
  if n = 1 or n = 4 then 0 else 1 + procname(f(n)) fi
end proc:
map(g, [$1..100]); # Robert Israel, Apr 11 2019

Table[Length[NestWhileList[Total[IntegerDigits[#]^2]&,n,#!=1&&#!=4&]],{n,80}]-1 (* Harvey P. Dale, Dec 31 2016 *)

a(n) = 0 if n = 1 or 4, otherwise a(n) = 1 + a(A003132(n)). - Robert Israel, Apr 11 2019

cp's OEIS Frontend

A003132 Sum of squares of digits of n.

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A039943 Every integer eventually goes to one of these under the "x goes to sum of squares of digits of x" map.

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A000216 Take sum of squares of digits of previous term, starting with 2.

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Haskell

Magma

Mathematica

PARI

Formula

A099645 Number of iterations until n reaches a number in A039943 under "x goes to sum of squares of digits of x" map.

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Haskell

Mathematica

PARI

Extensions

A000221 Take sum of squares of digits of previous term; start with 5.

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Haskell

Magma

Mathematica

PARI

Formula

A008460 Take sum of squares of digits of previous term; start with 6.

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Magma