A000216 Take sum of squares of digits of previous term, starting with 2.

2, 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

Keywords

• nonn
• base
• easy

Comments

As the orbit of 2 under A003132, this could also have offset 0. Merges into A080709 right after the first term: a(n+1) = A080709(n) for all n >= 1. - M. F. Hasler, Apr 27 2018

References

• R. Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 83.
• P. Kiss, A generalization of a problem in number theory, Math. Sem. Notes Kobe Univ., 5 (1977), no. 3, 313-317. MR 0472667 (57 #12362).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• H. G. Grundmann, Semihappy Numbers, J. Int. Seq. 13 (2010), 10.4.8, Theorem 1.
• P. Kiss, A generalization of a problem in number theory, [Hungarian], Mat. Lapok, 25 (No. 1-2, 1974), 145-149.
• Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
• A. Porges, A set of eight numbers, Amer. Math. Monthly, 52 (1945), 379-382. [Annotated scanned copy]
• H. J. J. te Riele, Iteration of number-theoretic functions, Nieuw Archief v. Wiskunde, (4) 1 (1983), 345-360.
• 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, 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

Programs

Haskell

a000216 n = a000216_list !! (n-1)
a000216_list = iterate a003132 2 -- Reinhard Zumkeller, Aug 24 2011

Magma

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

Mathematica

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

PARI

A000216(n)=[42, 20, 4, 16, 37, 58, 89, 145, 2][n%8+8^(n<2)] \\ M. F. Hasler, May 24 2009, edited Apr 27 2018

Formula

Periodic with period 8.