A062775 again

[Gottfried Helms]

T. D. Noe schrieb:
> Because this sequence is multiplicative, we need only give the formula for
> powers of primes.  I have computed the sequence for all prime powers <
> 2500, and found the following formulas empirically:
> 
> There are 4 cases: for even and odd primes, and for even and odd powers.
> 
> For p=2,
> power 2k-1:	2^(3k-1) (2^k     - 1)
> power 2k:	2^(3k)   (2^(k+1) - 1)
> 
> For odd primes p,
> power 2k-1:	p^(3k-2) (p^k     + p^(k-1) - 1)
> power 2k:	p^(3k-1) (p^(k+1) + p^k     - 1)
> 
> Regards,
> 
> Tony
> 
> 
Hmmm. I think, there is a serious flaw in the formula.
Think of exponent j=3, prime p=7

Then we have
   k=2 , j=2k-1 thus formula 3:

>    power 2k-1:	p^(3k-2) (p^k     + p^(k-1) - 1)

      7^(3k-2) (7^k + 7^(k-1) -1)
      7^4     * ( 7^2 + 7 - 1)
      7^4     * 55

The number, that I computed empirically is 55 - just the rightmost
parentheses alone. So may be a typo is in the first part?


For p=11 I got empirically a_3(11) = 121 , but this formula
gives

     A_(11) = 11^4 * (121 + 11 -1 ) = 11^4 * 131

where even the right factor 131 is different to my result of counting
and could not be corrected by any correction of the p^(3k-2)-term.

Gottfried Helms

my empirical results for A_3(i)  i=1 to 64

i       A_3(i)
-----------------
1	1
2	4
3	9
4	20
5	25
6	36
7	55
8	112
9	189
10	100
11	121
12	180
13	109
14	220
15	225
16	448
17	289
18	756
19	487
20	500
21	495
22	484
23	529
24	1008
25	725
26	436
27	2187
28	1100
29	841
30	900
31	1081
32	2048
33	1089
34	1156
35	1375
36	3780
37	973
38	1948
39	981
40	2800
41	1681
42	1980
43	1513
44	2420
45	4725
46	2116
47	2209
48	4032
49	2989
50	2900
51	2601
52	2180
53	2809
54	8748
55	3025
56	6160
57	4383
58	3364
59	3481
60	4500
61	3781
62	4324
63	10395
64	10240