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

1,2

Comments

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

Links

Donovan Johnson, Table of n, a(n) for n = 1..1000

Index entries for sequences related to powerful numbers

Example

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

Crossrefs

Cf. A001694, A007770.

Sequence in context: A222300 A259770 A066472 * A259765 A256521 A037008

Adjacent sequences: A140169 A140170 A140171 * A140173 A140174 A140175

Keyword

base,nonn

Author

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

Status

approved