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.

1, 32, 49, 100, 392, 784, 1000, 1125, 1152, 1323, 1444, 1521, 3200, 3364, 3456, 4096, 4225, 4356, 4563, 4900, 7225, 7744, 8281, 8748, 9604, 10000, 10125, 10976, 11025, 12167, 12321, 12348, 12996, 13824, 14112, 14283, 14641, 15625, 17424, 17672, 17956

Offset: 1

Author

Robin James Kerrison (rjk1994(AT)googlemail.com), Jun 22 2008

Keywords

• base
• nonn

Comments

For n=10^x such that x is an integer greater than 1, n is both powerful and happy.

Examples

			2 --> 4 --> 16 --> 37 --> ... --> 4, which repeats with period 8, so never reaches 1, so 2 (and 4, 16 etc.) are unhappy.
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
However, 7 divides by 7 and not 7^2.
32 --> 3^2+2^2=13 --> 1^2+3^2=10 --> 1^2+0^2=1
and 32 divides by 2 (and no other prime number) and by 2^2. So 32 is powerful and happy
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to powerful numbers

Crossrefs

Cf. A001694, A007770.