An array alternating non-primes

and primes

[Sent: Monday, April 8, 2013 0:00 AM]

Hello SeqFans,

http://oeis.org/A083197... has given me the idea of a variation. Here is A083197:

> Triangular array, read by rows, where

if n is odd the n-th row consists of n least unused non-primes, while

if n is even the n-th row consists of the n least unused primes.

Triangle begins:

2 3

4 6 8

5 7 11 13

9 10 12 14 15

17 19 23 29 31 37

etc.

-----------------

... In my variation the size of each row is given by the successive integers of the sequence itself, not by n. Thus the array would begin:

row size & content

1 1 (non-prime)

2 3 2 (primes)

4 6 8 3 (non-primes)

5 7 11 13 4 (primes)

9 10 12 14 15 16 6 (non-primes)

17 19 23 29 31 37 41 43 8 (primes)

18 20 21 22 24 5 (non-primes)

47 51 53 59 61 67 71 7 (primes)

25 26 27 28 30 32 33 34 35 36 38 11 (non-primes)

...

As A083197, this new sequence is of course a permutation of the natural numbers.

Best,

É.

__________

[Maximilian Hasler]:

Dear Eric & SeqFans,

First (yet somehow least important), let me just correct a smallerror, "51" is not prime and so the row starting with 47 should read 47,53,59,61,67,71,73.

Your post inspired me some more ideas:

First, I noticed that your construction can be iterated. The first lines remain the same, but then, due to variing row lengths, primes and non-primes get mixed in different ways.

But the changes appear later and later: In the next step, the sequence would differ due to the row with length 16 instead of 17, and the index of that row is the sum of all preceding numbers (way over 100).

Nonetheless, there is the "limiting" sequence to which the construction converges (which coincides for the abovementioned reason with the (corrected) terms of your example,

2, 3,

4, 6, 8,

5, 7, 11, 13,

9, 10, 12, 14, 15, 16,

17, 19, 23, 29, 31, 37, 41, 43,

18, 20, 21, 22, 24,

47, 53, 59, 61, 67, 71, 73,

25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38,

79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139,

...

The same definition as A083197, but filling with nonnegative instead of positive integers, yields another variation not in the OEIS:

2 3

1 4 6

5 7 11 13

8 9 10 12 14

(The odd rows are just "shifted" by 1 element wrt A083197, due to the initial 0.)

To apply your idea here, we could say that there’d be an initial row 0 with 0 primes, so this empty row zero would be followed by:

row 1 with 2 non-primes : 0, 1,

row 2 with 3 primes : 2, 3, 5

row 3 with 1 non-prime : 4

row 4 with 4 primes : 7, 11, 13, 17

row 5 with 6 composites : 6, 8, 9, 10, 12, 14

etc.

Iterating this construction once more gives:

row 0 with 0 primes,

row 1 with 1 non-primes : 0,

row 2 with 2 primes : 2, 3,

row 3 with 3 non-primes : 1,4,6

row 4 with 5 primes : 5, 7, 11, 13, 17

row 5 with 4 composites : 8, 9, 10, 12

etc.

I think that even and odd sequences of this sequence (of sequences) converge respectively to two distinct limits

2, 3,

1, 4, 6,

5, 7, 11, 13, 17,

8, 9, 10, 12,

19, 23, 29, 31, 37, 41, 43,

14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27,

47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103,

28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50,

...

And

0, 1,

2, 3, 5,

7, 11, 13, 17,

6, 8, 9, 10, 12, 14,

19, 23, 29, 31, 37,

15, 16, 18, 20, 21, 22, 24,

41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83,

25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40,

...

PS: For my records, I get this by iterating pnp(%,,1) with

{pnp(a,nnp=-1,f=0,na=[],maxrow=10)=np=1;for(n=1,min(maxrow,#a),for(j=1,a[n],

na=concat(na,if(bittest(n,0)==f,np=nextprime(np+1),until(!isprime(nnp++),);nnp));print1(na[#na]","));print);na}

While for your sequences (starting with 1) I have to set the 2^nd parameter nnp to 0 and the 3^rd (hack to exchange n even<=>odd) to 0

__________

[Hans Havermann]:

Maximilian Hasler:

> 1,

> 2, 3,

> 4, 6, 8,

> 5, 7, 11, 13,

> 9, 10, 12, 14, 15, 16,

> 17, 19, 23, 29, 31, 37, 41, 43,

> 18, 20, 21, 22, 24,

> 47, 53, 59, 61, 67, 71, 73,

> 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38,

> 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139,

> ...

If you just want to see more rows, I’ve put another 150 here:

http://chesswanks.com/num/NotTriangular/

1
2	3
4	6	8
5	7	11	13
9	10	12	14	15	16
17	19	23	29	31	37	41	43
18	20	21	22	24
47	53	59	61	67	71	73
25	26	27	28	30	32	33	34	35	36	38
79	83	89	97	101	103	107	109	113	127	131	137	139
39	40	42	44	45	46	48	49	50
149	151	157	163	167	173	179	181	191	193
51	52	54	55	56	57	58	60	62	63	64	65
197	199	211	223	227	229	233	239	241	251	257	263	269	271
66	68	69	70	72	74	75	76	77	78	80	81	82	84	85
277	281	283	293	307	311	313	317	331	337	347	349	353	359	367	373
86	87	88	90	91	92	93	94	95	96	98	99	100	102	104	105	106
379	383	389	397	401	409	419	421	431	433	439	443	449	457	461	463	467	479	487
108	110	111	112	114	115	116	117	118	119	120	121	122	123	124	125	126	128	129	130	132	133	134
491	499	503	509	521	523	541	547	557	563	569	571	577	587	593	599	601	607	613	617	619	631	641	643	647	653	659	661	673
135	136	138	140	141	142	143	144	145	146	147	148	150	152	153	154	155	156	158	159	160	161	162	164	165	166	168	169	170	171	172
677	683	691	701	709	719	727	733	739	743	751	757	761	769	773	787	797	809	811	821	823	827	829	839	853	857	859	863	877	881	883	887	907	911	919	929	937
174	175	176	177	178	180	182	183	184	185	186	187	188	189	190	192	194	195	196	198	200	201	202	203	204	205	206	207	208	209	210	212	213	214	215	216	217	218	219	220	221
941	947	953	967	971	977	983	991	997	1009	1013	1019	1021	1031	1033	1039	1049	1051	1061	1063	1069	1087	1091	1093	1097	1103	1109	1117	1123	1129	1151	1153	1163	1171	1181	1187	1193	1201	1213	1217	1223	1229	1231
222	224	225	226	228	230	231	232	234	235	236	237	238	240	242	243	244	245
1237	1249	1259	1277	1279	1283	1289	1291	1297	1301	1303	1307	1319	1321	1327	1361	1367	1373	1381	1399
246	247	248	249	250	252	253	254	255	256	258	259	260	261	262	264	265	266	267	268	270
1409	1423	1427	1429	1433	1439	1447	1451	1453	1459	1471	1481	1483	1487	1489	1493	1499	1511	1523	1531	1543	1549
272	273	274	275	276	278	279	280	282	284	285	286	287	288	289	290	291	292	294	295	296	297	298	299

__________

Many thanks, Maximilian and Hans!

Best,

É.