In the OEIS · Yagel Numbers

In the On-Line Encyclopedia of Integer Sequences since October 2026

The first prime in every row of the grid: for each order $k$ , the least exponent $n$ at which $Y_k(n)$ is prime.

The OEIS entry → · All 100 terms (b-file) · Re-prove every term yourself →

The OEIS is the reference database of integer sequences. Sequence A400659 follows the grid $Y_k(n) = p_{k-1}\#\cdot p_k^{\,n} - 1$ across its rows: it records where each row reaches its first prime. The first terms are 2, 1, 1, 3, 1, 1, 2, 5, 5, 4, 16, 4, 1, 12, 9, …

Mind the letters. In an OEIS entry $n$ is always the index of the sequence, so the entry writes the order as $n$ and the exponent as $k$ : $a(n)$ is the least $k \ge 1$ such that $\mathrm{A002110}(n-1)\cdot \mathrm{prime}(n)^{k} - 1$ is prime. This site and the paper use the opposite letters, with $k$ the order and $n$ the exponent. Below, $a(k)$ is the least $n$ with $Y_k(n)$ prime.

$a(1) = 2$ : row 1 is the Mersenne numbers $2^n - 1$ ; $Y_1(1) = 1$ is not prime and $Y_1(2) = 3$ is.
$a(k) = 1$ exactly when the primorial number $p_k\# - 1$ is prime, that is, when $k$ is in A057704.
It is not known whether every row contains a prime. The entry would list $-1$ for a row without one; none is known.

Every term, proved

Each prime in the table is proved, not screened. Every smaller exponent of its row has an explicit compositeness witness, and the prime itself passes a Lucas $N+1$ test, which applies because $N + 1 = p_{k-1}\#\cdot p_k^{\,n}$ is completely factored by construction. The verification page gives two self-contained programs, in PARI/GP and in Python, that re-prove all 100 terms on your own computer without any data from this site. Click a prime to compute its exact value on the datasets page.

order k	a(k)	first prime of the row	digits
1	2	2² − 1	1
2	1	p₁#·3¹ − 1	1
3	1	p₂#·5¹ − 1	2
4	3	p₃#·7³ − 1	5
5	1	p₄#·11¹ − 1	4
6	1	p₅#·13¹ − 1	5
7	2	p₆#·17² − 1	7
8	5	p₇#·19⁵ − 1	13
9	5	p₈#·23⁵ − 1	14
10	4	p₉#·29⁴ − 1	15
11	16	p₁₀#·31¹⁶ − 1	34
12	4	p₁₁#·37⁴ − 1	18
13	1	p₁₂#·41¹ − 1	15
14	12	p₁₃#·43¹² − 1	35
15	9	p₁₄#·47⁹ − 1	32
16	2	p₁₅#·53² − 1	22
17	5	p₁₆#·59⁵ − 1	29
18	3	p₁₇#·61³ − 1	27
19	6	p₁₈#·67⁶ − 1	35
20	2	p₁₉#·71² − 1	29
21	34	p₂₀#·73³⁴ − 1	91
22	3	p₂₁#·79³ − 1	35
23	2	p₂₂#·83² − 1	35
24	1	p₂₃#·89¹ − 1	35
25	6	p₂₄#·97⁶ − 1	47
26	65	p₂₅#·101⁶⁵ − 1	167
27	3	p₂₆#·103³ − 1	45
28	5	p₂₇#·107⁵ − 1	51
29	6	p₂₈#·109⁶ − 1	55
30	3	p₂₉#·113³ − 1	51
31	229	p₃₀#·127²²⁹ − 1	529
32	11	p₃₁#·131¹¹ − 1	72
33	17	p₃₂#·137¹⁷ − 1	88
34	5	p₃₃#·139⁵ − 1	64
35	7	p₃₄#·149⁷ − 1	71
36	16	p₃₅#·151¹⁶ − 1	93
37	19	p₃₆#·157¹⁹ − 1	102
38	15	p₃₇#·163¹⁵ − 1	95
39	46	p₃₈#·167⁴⁶ − 1	167
40	11	p₃₉#·173¹¹ − 1	91
41	14	p₄₀#·179¹⁴ − 1	100
42	20	p₄₁#·181²⁰ − 1	116
43	10	p₄₂#·191¹⁰ − 1	96
44	207	p₄₃#·193²⁰⁷ − 1	549
45	12	p₄₄#·197¹² − 1	105
46	12	p₄₅#·199¹² − 1	108
47	32	p₄₆#·211³² − 1	157
48	63	p₄₇#·223⁶³ − 1	233
49	6	p₄₈#·227⁶ − 1	101
50	93	p₄₉#·229⁹³ − 1	309
51	4	p₅₀#·233⁴ − 1	101
52	16	p₅₁#·239¹⁶ − 1	132
53	96	p₅₂#·241⁹⁶ − 1	325
54	5	p₅₃#·251⁵ − 1	111
55	5	p₅₄#·257⁵ − 1	113
56	329	p₅₅#·263³²⁹ − 1	900
57	172	p₅₆#·269¹⁷² − 1	524
58	34	p₅₇#·271³⁴ − 1	191
59	68	p₅₈#·277⁶⁸ − 1	277
60	4	p₅₉#·281⁴ − 1	123
61	395	p₆₀#·283³⁹⁵ − 1	1,084
62	150	p₆₁#·293¹⁵⁰ − 1	488
63	20	p₆₂#·307²⁰ − 1	171
64	2	p₆₃#·311² − 1	128
65	20	p₆₄#·313²⁰ − 1	176
66	1	p₆₅#·317¹ − 1	131
67	54	p₆₆#·331⁵⁴ − 1	267
68	1	p₆₇#·337¹ − 1	136
69	106	p₆₈#·347¹⁰⁶ − 1	405
70	5	p₆₉#·349⁵ − 1	151
71	103	p₇₀#·353¹⁰³ − 1	403
72	11	p₇₁#·359¹¹ − 1	172
73	20	p₇₂#·367²⁰ − 1	197
74	34	p₇₃#·373³⁴ − 1	236
75	22	p₇₄#·379²² − 1	208
76	63	p₇₅#·383⁶³ − 1	316
77	139	p₇₆#·389¹³⁹ − 1	516
78	48	p₇₇#·397⁴⁸ − 1	284
79	71	p₇₈#·401⁷¹ − 1	346
80	25	p₇₉#·409²⁵ − 1	229
81	55	p₈₀#·419⁵⁵ − 1	311
82	45	p₈₁#·421⁴⁵ − 1	287
83	27	p₈₂#·431²⁷ − 1	243
84	22	p₈₃#·433²² − 1	233
85	119	p₈₄#·439¹¹⁹ − 1	492
86	9	p₈₅#·443⁹ − 1	204
87	124	p₈₆#·449¹²⁴ − 1	511
88	470	p₈₇#·457⁴⁷⁰ − 1	1,435
89	20	p₈₈#·461²⁰ − 1	241
90	51	p₈₉#·463⁵¹ − 1	326
91	6	p₉₀#·467⁶ − 1	209
92	71	p₉₁#·479⁷¹ − 1	386
93	74	p₉₂#·487⁷⁴ − 1	397
94	95	p₉₃#·491⁹⁵ − 1	457
95	83	p₉₄#·499⁸³ − 1	428
96	589	p₉₅#·503⁵⁸⁹ − 1	1,798
97	171	p₉₆#·509¹⁷¹ − 1	672
98	30	p₉₇#·521³⁰ − 1	294
99	118	p₉₈#·523¹¹⁸ − 1	536
100	1279	p₉₉#·541¹²⁷⁹ − 1	3,713

The grid's edges in the OEIS

The first row and the first column of the grid, on both faces, were in the OEIS before this project, as were rows 2 and 3. Each sequence below is written in this site's letters; the OEIS entries use their own.

edge of the grid	numbers	OEIS
row $k = 1$ , minus face	Mersenne numbers $2^n - 1$ : exponents giving primes	A000043
row $k = 2$ , both faces	$2\cdot 3^n - 1$ and $2\cdot 3^n + 1$ : exponents giving primes	A003307, A003306
row $k = 3$ , both faces	$6\cdot 5^n - 1$ and $6\cdot 5^n + 1$ : exponents giving primes	A257790, A143279
column $n = 1$ , minus face	primorial numbers $p_k\# - 1$ : the orders $k$ , and the primes $p_k$ , at which it is prime	A057704, A006794
column $n = 1$ , plus face	Euclid numbers $p_k\# + 1$ : the orders $k$ , and the primes $p_k$ , at which it is prime	A014545, A005234
the coefficients	primorials $p_k\#$ , and the primes	A002110, A000040

Checked 9 October 2026: on every edge listed, the OEIS terms agree with this site's catalogue of proven primes as far as both reach.