A first digit theorem for powerful integer powers

SpringerPlus

Table 1 GBL fit for first digit of powerful integer powers: MAD vs. WLS criterion

s = 1	Parameters		Δ to LL estimate		MAD GoF measures			WLS GoF measures
m =	WLS	MAD	WLS	MAD	LL	WLS	MAD	LL	WLS	MAD
8	0.50630179	0.50602471	1.819	1.947	282.9	135.2	134.6	90.14	28.44	28.75
9	0.50398677	0.50439385	2.298	1.890	168.3	90.24	86.04	66.89	21.19	22.62
10	0.50258542	0.50277813	2.755	2.340	92.13	45.33	45.02	42.88	12.36	13.05
11	0.50170493	0.50180536	3.116	2.650	44.37	17.57	16.34	22.63	4.505	4.911
12	0.50116789	0.50121344	2.932	2.477	19.77	8.163	7.927	8.998	1.546	1.726
13	0.50080206	0.50083278	2.077	1.415	6.993	4.335	4.077	2.519	0.784	0.960
14	0.50054625	0.50054722	0.289	0.244	1.821	1.802	1.794	0.345	0.330	0.330
15	0.50037082	0.50036094	3.110	2.122	2.042	1.252	1.200	1.154	0.315	0.400

s = 2	Parameters		Δ to LL estimate		MAD GoF measures			WLS GoF measures
m =	WLS	MAD	WLS	MAD	LL	WLS	MAD	LL	WLS	MAD
8	0.25428436	0.25324296	0.383	0.867	238.4	218.8	193.9	163.9	157.8	167.6
9	0.25428436	0.25267279	1.142	0.469	76.66	122.8	59.30	28.98	66.02	28.52
10	0.25166049	0.25155591	0.585	0.811	26.64	19.31	18.58	7.514	4.409	4.870
11	0.25096820	0.25089373	1.021	1.366	22.15	15.50	13.56	10.95	6.568	7.071
12	0.25062939	0.25059921	1.012	1.314	11.19	8.423	7.972	6.168	4.169	4.347
13	0.25042363	0.25041307	0.552	0.779	3.923	3.055	2.805	1.563	1.287	1.334
14	0.25028419	0.25028723	0.369	0.510	1.533	1.302	1.238	0.540	0.482	0.491
15	0.25018970	0.25019163	1.984	2.177	1.344	0.640	0.624	1.027	0.258	0.266

s = 3	Parameters		Δ to LL estimate		MAD GoF measures			WLS GoF measures
m =	WLS	MAD	WLS	MAD	LL	WLS	MAD	LL	WLS	MAD
8	0.16900766	0.16723059	0.495	1.319	274.3	257.7	248.0	575.4	553.1	615.3
9	0.16801302	0.16762481	0.748	1.137	116.4	89.18	81.67	142.8	119.1	125.5
10	0.16769485	0.16753300	0.560	0.909	60.96	58.26	57.25	124.0	117.8	120.2
11	0.16729754	0.16730322	0.748	0.722	18.31	13.37	13.25	24.45	19.34	19.35
12	0.16710133	0.16708812	0.524	0.656	7.100	5.652	5.371	6.058	4.895	4.969
13	0.16695408	0.16695069	0.260	0.333	3.798	3.374	3.255	3.664	3.530	3.541
14	0.16685701	0.16685618	0.287	0.249	2.484	2.358	2.338	3.931	3.856	3.857
15	0.16679314	0.16679442	1.323	1.452	1.321	0.972	0.952	2.021	1.279	1.286

s = 4	Parameters		Δ to LL estimate		MAD GoF measures			WLS GoF measures
m =	WLS	MAD	WLS	MAD	LL	WLS	MAD	LL	WLS	MAD
8	0.12774699	0.12760514	0.089	0.023	336.0	339.0	334.9	1704	1702	1703
9	0.12682400	0.12682674	0.253	0.256	108.7	100.3	100.2	495.4	489.5	489.6
10	0.12592594	0.12589200	0.086	0.160	38.75	37.58	36.59	149.2	148.9	149.1
11	0.12543815	0.12539409	0.724	0.928	21.98	16.83	15.38	52.95	42.62	43.45
12	0.12530767	0.12526505	0.576	1.002	7.884	6.275	5.841	18.01	14.97	16.63
13	0.12521961	0.12520948	0.108	0.326	4.885	4.688	4.291	20.84	20.79	20.99
14	0.12514734	0.12514391	0.428	0.269	1.315	1.279	1.181	4.693	4.331	4.381
15	0.12509739	0.12509771	1.246	1.278	0.940	0.587	0.579	3.039	1.619	1.620

s = 5	Parameters		Δ to LL estimate		MAD GoF measures			WLS GoF measures
m =	WLS	MAD	WLS	MAD	LL	WLS	MAD	LL	WLS	MAD
8	0.10483305	0.10494392	1.295	1.346	551.6	498.4	496.2	8531	7818	7819
9	0.10110871	0.10166842	0.148	0.412	182.7	185.4	175.0	2055	2051	2113
10	0.10041545	0.10050001	0.770	0.588	39.74	30.51	28.02	211.5	157.1	160.2
11	0.10035943	0.10038750	0.538	0.407	21.67	19.45	18.85	124.1	111.8	112.5
12	0.10021129	0.10018888	0.809	1.033	10.05	8.099	7.558	60.76	47.82	48.81
13	0.10015612	0.10013482	0.508	0.967	5.373	4.752	4.238	35.74	33.38	35.31
14	0.10011058	0.10010322	0.004	0.337	1.790	1.792	1.653	9.827	9.827	10.32
15	0.10007403	0.10007147	0.609	0.353	0.794	0.713	0.655	4.477	3.745	3.874