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

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

Author

N. J. A. Sloane, Mar 04 2003

Keywords

• nonn
• base
• easy

Comments

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)

References

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

Links

• 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).

Crossrefs

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

Programs

Haskell

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

Magma

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

Mathematica

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

PARI

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

Formula

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