A003132 -id:A003132 - cp's OEIS frontend

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

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

A076314 a(n) = floor(n/10) + (n mod 10).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14

Offset: 0

Reinhard Zumkeller, Oct 06 2002

For n<100 this is equal to the digital sum of n (see A007953). - Hieronymus Fischer, Jun 17 2007

			a(15) = floor(15 / 10) + (15 mod 10) = 1 + 5 = 6. - _Indranil Ghosh_, Feb 13 2017

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A076313, A010879, A076309, A076310, A076311, A076312, A055017, A007953, A003132.

a076314 = uncurry (+) . flip divMod 10 -- Reinhard Zumkeller, Jun 01 2013

[n-9*Floor(n/10):n in [0..100]]; // Vincenzo Librandi, Dec 10 2016

Table[n - 9 Floor[n / 10], {n, 0, 100}] (* Vincenzo Librandi Dec 10 2016 *)

a(n)=n\10+n%10 \\ Charles R Greathouse IV, Sep 24 2015

def A076314(n): return (n/10)+(n%10) # Indranil Ghosh, Feb 13 2017

From Hieronymus Fischer, Jun 17 2007: (Start)
a(n) = n - 9*floor(n/10).
a(n) = (n + 9*(n mod 10))/10.
a(n) = n - 9*A002266(A004526(n)) = n - 9*A004526(A002266(n)).
a(n) = (n + 9*A010879(n))/10.
a(n) = (n + 9*A000035(n) + 18*A010874(A004526(n)))/10.
a(n) = (n + 9*A010874(n) + 45*A000035(A002266(n)))/10.
G.f.: x*(8*x^10 - 9*x^9 + 1)/((1 - x^10)*(1 - x)^2). (End)
a(n) = A033930(n) for 1 <= n < 100. - R. J. Mathar, Sep 21 2008
a(n) = +a(n-1) + a(n-10) - a(n-11). - R. J. Mathar, Feb 20 2011

A139566 a(n) is the sum of squares of digits of a(n-1); a(1)=15.

Original entry on oeis.org

15, 26, 40, 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

Robert Gornall (rob(AT)khobbits.net), Jun 11 2008

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A101926, A087965, A074411, A110998, A051861, A048222.

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), A008462 (starting with 8), A008463 (starting with 9), A122065 (starting with 74169). - M. F. Hasler, May 24 2009

a = {15}; Do[AppendTo[a, Plus @@ (IntegerDigits[a[[ -1]]]^2)], {70}]; a (* Stefan Steinerberger, Jun 14 2008 *)
NestList[Total[IntegerDigits[#]^2] &, 15, 70] (* or *) PadRight[ {15,26,40},70,{42,20,4,16,37,58,89,145}](* Harvey P. Dale, Jan 28 2013 *)

/* to check the given terms */
a=[/* paste the terms here */]; a==vector(#a,n,k=if(n>1,A003132(k),15))
/* to check the following code, use: a==vector(99,n,A139566(n)) */
A139566(n)=[15,26,40,16,37,58,89,145,42,20,4][if(n>11,(n-4)%8+4,n)] \\ (End)

Vec(x*(36*x^10+6*x^9-27*x^8-145*x^7-89*x^6-58*x^5-37*x^4-16*x^3 -40*x^2-26*x-15)/((x-1)*(x+1)*(x^2+1)*(x^4+1)) + O(x^70)) \\ Colin Barker, Aug 24 2015

Eventually periodic with period 8.
a(n) = A008463(n) for n > 4. - M. F. Hasler, May 24 2009
a(n) = a(n-8) for n > 11. - Colin Barker, Aug 24 2015
G.f.: x*(36*x^10 + 6*x^9 - 27*x^8 - 145*x^7 - 89*x^6 - 58*x^5 - 37*x^4 - 16*x^3 - 40*x^2 - 26*x - 15) / ((x-1)*(x+1)*(x^2+1)*(x^4+1)). - Colin Barker, Aug 24 2015

More terms from Stefan Steinerberger, Jun 14 2008
Terms checked, using the given PARI code, by M. F. Hasler, May 24 2009
Minor edits and starting value added in name by M. F. Hasler, Apr 27 2018

A008463 Take sum of squares of digits of previous term; start with 9.

Original entry on oeis.org

9, 81, 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, 16, 37, 58, 89, 145

Offset: 1

N. J. A. Sloane

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

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), A008462 (starting with 8), A008463 (starting with 9), A139566 (starting with 15), A122065 (starting with 74169). - M. F. Hasler, May 24 2009

Nest[Append[#, Total[IntegerDigits[Last@ #]^2]] &, {9}, 79] (* Michael De Vlieger, Apr 29 2018 *)
NestList[Total[IntegerDigits[#]^2]&,9,80] (* or *) PadRight[ {9,81,65,61},80,{42,20,4,16,37,58,89,145}] (* Harvey P. Dale, Sep 11 2019 *)

A008463(n)=[9,81,65,61,37, 58,89,145,42,20,4,16][if(n>12,(n-5)%8+5,n)]
/* This code has been checked as follows: */
k=3;vector(99,n,k=A003132(k))==vector(99,n,A008463(n))
/* The given terms have been checked as follows: */
a=[/* paste the terms here */]; apply(A008463,[1..#a])==a \\ (End)

Periodic with period 8.
a(n) = A000218(n+1). - R. J. Mathar, May 24 2008
a(n) = A080709(n-2) for n > 4. - M. F. Hasler, May 24 2009

A034087 Numbers divisible by the sum of the squares of their digits.

Original entry on oeis.org

1, 10, 20, 50, 100, 110, 111, 120, 130, 133, 200, 210, 240, 267, 298, 310, 315, 360, 372, 376, 400, 420, 480, 500, 532, 550, 630, 803, 917, 973, 1000, 1010, 1011, 1020, 1030, 1071, 1100, 1101, 1110, 1134, 1148, 1200, 1211, 1222, 1290, 1300, 1302, 1316

Offset: 1

Erich Friedman

			a(100) = 4131 since 4^2+1^2+3^2+1^2=27 divides 4131. - _Carmine Suriano_, May 04 2013

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3465 from Carmine Suriano)

Cf. A003132, A034088, A169665, A169666.

isA034087 := proc(n) if n mod A003132(n) = 0 then true ; else false ; end if ; end proc:
for n from 1 to 1800 do if isA034087(n) then printf("%d ",n) ; end if ; end do ; # R. J. Mathar, Feb 25 2007

Select[Range[1500], Divisible[#, Plus @@ (IntegerDigits[#]^2)] &] (* Amiram Eldar, Jan 31 2021 *)

isok(m) = !(m % norml2(digits(m))); \\ Michel Marcus, Jan 31 2021

def ok(n): return n and n%sum(di**2 for di in map(int, str(n))) == 0
print([k for k in range(1317) if ok(k)]) # Michael S. Branicky, Jan 10 2025

A003132[a(n)] | a(n). - R. J. Mathar, Feb 25 2007

cp's OEIS Frontend

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

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

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

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

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

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

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A076314 a(n) = floor(n/10) + (n mod 10).