A008462 -id:A008462 - cp's OEIS frontend

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

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)]

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

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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.

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

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