A139566 -id:A139566 - 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

cp's OEIS Frontend

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

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