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A140172 Powerful happy numbers; if a prime p divides n then p^2 must also divide n and also n must have trajectory under iteration of sum of squares of digits map includes 1.

%I A140172 #8 Aug 20 2012 19:30:11
%S A140172 1,32,49,100,392,784,1000,1125,1152,1323,1444,1521,3200,3364,3456,
%T A140172 4096,4225,4356,4563,4900,7225,7744,8281,8748,9604,10000,10125,10976,
%U A140172 11025,12167,12321,12348,12996,13824,14112,14283,14641,15625,17424,17672,17956
%N A140172 Powerful happy numbers; if a prime p divides n then p^2 must also divide n and also n must have trajectory under iteration of sum of squares of digits map includes 1.
%C A140172 For n=10^x such that x is an integer greater than 1, n is both powerful and happy.
%H A140172 Donovan Johnson, <a href="/A140172/b140172.txt">Table of n, a(n) for n = 1..1000</a>
%H A140172 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>
%e A140172 2 --> 4 --> 16 --> 37 --> ... --> 4, which repeats with period 8, so never reaches 1, so 2 (and 4, 16 etc.) are unhappy.
%e A140172 7 --> 7^2=49 --> 4^2+9^2=97 --> 9^2+7^2=130 --> 1^2+3^2+0^2=10 --> 1^2+0^2=1
%e A140172 However, 7 divides by 7 and not 7^2.
%e A140172 32 --> 3^2+2^2=13 --> 1^2+3^2=10 --> 1^2+0^2=1
%e A140172 and 32 divides by 2 (and no other prime number) and by 2^2. So 32 is powerful and happy
%Y A140172 Cf. A001694, A007770.
%K A140172 base,nonn
%O A140172 1,2
%A A140172 Robin James Kerrison (rjk1994(AT)googlemail.com), Jun 22 2008