cp's OEIS Frontend

A257795 Bihappy numbers: numbers that reach 1 under iteration of the sum-of-squares-of-two-digits map s_2.

%I A257795 #35 Dec 20 2024 01:03:59
%S A257795 1,10,100,103,301,367,608,806,1000,1030,1826,2363,2618,2896,3010,3056,
%T A257795 3640,4036,4498,4596,5294,5630,6080,6323,6703,6791,8060,8484,9167,
%U A257795 9452,9628,9645,9844,10000,10003,10275,10300,10451,10979,11241,11540,12336,12770,12939,13623,13929,14015,14112,15104,15161,16151,16286,17027
%N A257795 Bihappy numbers: numbers that reach 1 under iteration of the sum-of-squares-of-two-digits map s_2.
%C A257795 If n has an even number of digits, say n = abcdef, the map is n -> s_2(n) := (ab)^2+(cd)^2+(ef)^2. If n has an odd number of digits, say n = abcde, the map is n -> s_2(n) = a^2+(bc)^2+(de)^2. The sequence {s_2(n), n >= 0} does not have its own entry in the OEIS because it begins {0, 1, ..., 9801, 1, 2, 5, ...} and agrees with the squares (A000290) for the first 100 terms. - _N. J. A. Sloane_, May 10 2015
%C A257795 This sequence is infinite because it contains several infinite subsequences (powers of 10, for example).
%H A257795 Pieter Post and Giovanni Resta, <a href="/A257795/b257795.txt">Table of n, a(n) for n = 1..10000</a> (first 244 terms from Pieter Post)
%F A257795 All 10^k are members of this sequence.
%F A257795 If n is a member each permutation of a set of pairs of digits gives another member.
%F A257795 Placing two zeros between the sets of two digits gives another member.
%F A257795 All other numbers have loops of lengths 1, 2, 4, 5, 6, 10, 14, 35 or 56.
%F A257795 The first number with a loop of length 2 is 51, which reaches the loop (5965, 7706) after 3 iterations.
%F A257795 The first number with a loop of length 4 is 342, loop of 5 is 57, loop of 6 is 389, loop of 10 is 21, loop of 14 is 28, loop of 35 is 2 and the first number with a loop of 56 is 5.
%F A257795 And there are some numbers which end up in a loop of length 1. The first such number is 1233 (= 12^2 + 33^2)
%F A257795 All numbers appear to end up in one of these loops.
%e A257795 367 is in the sequence since 3^2+67^2 = 4498 => 44^2+98^2= 11540 => 1^2+15^2+40^2 = 1826 => 18^2+26^2 = 1000 => 10^2+0^2 = 100 => 1^2+0^2 = 1, so in 6 iterations 367 reaches 1.
%o A257795 (PARI) select( {is_A257795(n,S=[])=!while(1<n=norml2(digits(n,100)), setsearch(S,n)&&return; S=setunion(S,[n]))}, [1..10^3]) \\ _M. F. Hasler_, Dec 20 2024
%Y A257795 Cf. A257810 (minimum of the limiting cycle), A007770 (happy numbers), A055616, A000290 (the squares).
%K A257795 nonn,base
%O A257795 1,2
%A A257795 _Pieter Post_, May 09 2015