A000218 -id:A000218 - cp's OEIS frontend

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

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

A171250 Row lengths of A082381: number of iterations of "sum of digits squared" until 1 or 4 is reached.

Original entry on oeis.org

1, 1, 11, 1, 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, 9, 3, 7, 10, 5, 2, 4, 14, 4, 10, 4, 6, 11, 4, 8, 12, 3, 4, 12, 1

Offset: 1

M. F. Hasler, Dec 18 2009

In the spirit of A082381, the map A003132 ("sum of digits squared") is applied at least once to the initial value n; the sequence gives the number of iterations until 1 or 4 is reached.

Cf. A082381, A003132, A082382, A000216 (n=2), A000218 (n=3), A080709 (n=4), A000221 (n=5), A008460 (n=6), A008461 (n=7: ends in 1), A008462 (n=8), A008462 (n=9), A139566 (n=15), A122065 (n=74169), A000012 (n=1).

Table[Length[NestWhileList[Total[IntegerDigits[#]^2]&,n,!MemberQ[{1,4},#]&]]-1,{n,100}]/.(0->1) (* Harvey P. Dale, Jun 08 2017 *)

A171250(n)=my(c=0); until( n==4 || n==1, c++; n=norml2(eval(Vec(Str(n))))); c

a(n) = O(log* n).

Formula from Charles R Greathouse IV, Aug 02 2010

Corrected and extended by Harvey P. Dale, Jun 08 2017

A082381 Sequence of the squared digital root of a number until 1 or 4 is reached. The initial numbers 1,2,..n are not output.

Original entry on oeis.org

1, 4, 9, 81, 65, 61, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, 89, 145, 42, 20, 4, 25, 29, 85, 89, 145, 42, 20, 4, 36, 45, 41, 17, 50, 25, 29, 85, 89, 145, 42, 20, 4, 49, 97, 130, 10, 1, 64, 52, 29, 85, 89, 145, 42, 20, 4, 81, 65, 61, 37, 58, 89, 145, 42, 20, 4, 1, 2, 4, 5, 25, 29

Offset: 1

Cino Hilliard, Apr 13 2003

Conjecture: The sequence always terminates with 1 or the 4 16 37 58 89 145 42 20 4... loop (cf. A080709).
From M. F. Hasler, Dec 18 2009: (Start)
This sequence should be read as fuzzy table, where the n-th row contains the successive results under the map "sum of digits squared", when starting with n, until either 1 or 4 is reached. So either of these two marks the end of a row: See example.
Row lengths (i.e. "stopping times") are given in A171250. (End)

			From _M. F. Hasler_, Dec 18 2009: (Start)
The table reads:
[n=1] 1 (n=1 -> 1^2=1 -> STOP)
[n=2] 4 (n=2 -> 2^2=4 -> STOP)
[n=3] 9,81,65,61,37,58,89,145,42,20,4 (n=3 -> 3^2=9 -> 9^2=81 -> 8^2+1^2=65 -> ...)
[n=4] 16,37,58,89,145,42,20,4 (n=4 -> 4^2=16 -> 1^2+6^2=37 -> 3^2+7^2=58 -> ...)
...
[n=7] 49,97,130,10,1 (n=7 -> 7^2=49 -> 4^2+9^2=97 -> 130 -> 10 -> 1 -> STOP)
etc. (End)

C. Stanley Ogilvy, Tomorrow's Math, 1972

Cf. A082382 (list also the initial value); sequences ending in the 4-loop: A000216 (n=2), A000218 (n=3), A080709 (n=4), A000221 (n=5), A008460 (n=6), A008462 (n=8), A008462 (n=9), A139566 (n=15), A122065 (n=74169); sequences ending in 1: A000012 (n=1), A008461 (n=7). [From M. F. Hasler, Dec 18 2009]

digitsq2(m) = {y=0; for(x=1,m, digitsq(x) ) }
/* The squared digital root of a number */ digitsq(n) = { while(1, s=0; while(n > 0, d=n%10; s = s+d*d; n=floor(n/10); ); print1(s" "); if(s==1 || s==4,break); n=s; ) }

Corrected and edited, added explanations M. F. Hasler, Dec 18 2009

A250202 The "sum of squares of digits" problem in base 12, start with 6 (written in base 10).

Original entry on oeis.org

6, 36, 9, 81, 117, 162, 38, 13, 2, 4, 16, 17, 26, 8, 64, 41, 34, 104, 128, 164, 66, 61, 26, 8, 64, 41, 34, 104, 128, 164, 66, 61, 26, 8, 64, 41, 34, 104, 128, 164, 66, 61, 26, 8, 64, 41, 34, 104, 128, 164, 66, 61, 26, 8, 64, 41, 34, 104, 128, 164, 66, 61, 26, 8, 64, 41

Offset: 1

Eric Chen, Mar 13 2015

Periodic with period 10.
In base 12, there are 3 fixed points and 4 cycles (only 1 fixed point and 1 cycle in base 10, see A161772):
1 -> 1 (length 1);
5 -> 21 -> 5 (length 2);
8 -> 54 -> 35 -> 2a -> 88 -> a8 -> 118 -> 56 -> 51 -> 22 -> 8 (length 10);
18 -> 55 -> 42 -> 18 (length 3);
25 -> 25 (length 1);
68 -> 84 -> 68 (length 2);
a5 -> a5 (length 1);
Notice 25 (decimal 29) and a5 (decimal 125) are Armstrong numbers in base 12 (A161949), there are no 2-digit Armstrong numbers in base 10.
In base 12, there are only few happy numbers (no such between 10 (decimal 12) and 100 (decimal 144)), but in base 10, there are 20 happy numbers less than or equal to 100 (see A007770).

Eric Chen, Table of n, a(n) for n = 1..1000
Eric Chen, List of A250202 starts with n for n up to 40 (decimal 48)
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A000216, A000218, A080709, A000221, A008460, A008461, A008462, A008463, A161772.

NestList[Total[IntegerDigits[#, 12]^2]&, 6, 144]
Join[{6, 36, 9, 81, 117, 162, 38, 13, 2, 4, 16, 17},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{26, 8, 64, 41, 34, 104, 128, 164, 66, 61},54]] (* Ray Chandler, Aug 26 2015 *)
PadRight[{6,36,9,81,117,162,38,13,2,4,16,17},80,{66,61,26,8,64,41,34,104,128,164}] (* Harvey P. Dale, Aug 06 2017 *)

a(n) = [6, 36, 9, 81, 117, 162, 38, 13, 2, 4, 16, 17, 26, 8, 64, 41, 34, 104, 128, 164, 66, 61][n%10+10*(n>=10)+10*(n%10<3 & n>=20)]

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A171250 Row lengths of A082381: number of iterations of "sum of digits squared" until 1 or 4 is reached.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A082381 Sequence of the squared digital root of a number until 1 or 4 is reached. The initial numbers 1,2,..n are not output.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

Extensions

A250202 The "sum of squares of digits" problem in base 12, start with 6 (written in base 10).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI