Generalizing Benford's Law Using Power Laws: Application to Integer Sequences

Table 3

Fitting integer sequences to the Benford and generalized Benford distributions

Name of sequence	Sample size	Benford		Twosided Power Benford		Pareto Benford
		chi-square	𝑃-value	chi-square	𝑃-value	chi-square	𝑃-value
Square	100	9.096	33.43	7.837	34.72	0.362	99.91
Cube	500	9.696	28.70	5.808	56.23	0.286	99.96
Cube	1000	46.459	0.00	43.725	0.00	0.48	99.81
Cube	10000	443.745	0.00	472.011	0.00	3.138	79.13
Square root	99	8.612	37.61	7.002	42.86	2.778	83.61
Prime < 100	25	7.741	45.91	7.299	39.84	1.849	93.30
Prime < 1000	168	45.016	0.00	36.651	0.00	0.333	99.93
Prime < 10000	1229	387.194	0.00	307.322	0.00	3.297	77.07
Princeton number	25	3.452	90.29	2.762	89.72	1.302	97.16
Mixing sequence	618	15.550	4.93	9.014	25.17	1.819	93.55
Pentagonal number	100	5.277	72.76	2.127	95.24	1.968	92.26
Keith number	71	9.215	32.45	7.688	36.09	7.402	28.53
Bell number	100	3.069	93.00	3.014	88.37	2.607	85.63
Catalan number	100	2.404	96.61	2.304	94.11	1.934	92.57
Lucky number	45	7.693	46.40	5.165	63.98	5.564	47.37
Ulam number	44	6.350	60.81	2.520	92.56	2.526	86.56
Numeri ideoni	65	2.594	95.72	2.522	92.54	2.584	85.89
Fibonacci number	100	1.029	99.81	1.021	99.45	1.027	98.46
Partition number	94	1.394	99.43	1.132	99.24	1.513	95.86