please expand

[Tanya Khovanova]

sequences.)
 Thanks,
Return-Path: <israel at math.ubc.ca>
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Date: Sun, 14 Oct 2007 11:46:22 -0700 (PDT)
From: Robert Israel <israel at math.ubc.ca>
To: Leroy Quet <qq-quet at mindspring.com>
cc: seqfan at ext.jussieu.fr
Subject: Re: Sequence Of Primes, a(n)+a(n-1) is divisible by n
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The first 1000 members of this sequence are

2, 3, 5, 7, 13, 17, 19, 23, 41, 31, 29, 37, 11, 67, 59, 61, 83, 53, 73, 
79, 101, 109, 89, 233, 103, 47, 239, 139, 113, 293, 97, 151, 137, 127, 43, 
167, 157, 509, 251, 373, 107, 467, 163, 181, 347, 193, 313, 439, 281, 307, 
443, 271, 197, 227, 367, 733, 331, 353, 401, 71, 229, 503, 179, 199, 569, 
211, 317, 487, 397, 431, 269, 1009, 359, 809, 449, 601, 311, 613, 479, 
1259, 421, 389, 349, 149, 523, 2027, 811, 929, 743, 859, 131, 1871, 337, 
593, 911, 419, 541, 1399, 463, 1319, 881, 937, 491, 2393, 727, 953, 1061, 
223, 641, 1103, 547, 563, 557, 2381, 241, 2749, 383, 787, 983, 683, 277, 
691, 773, 457, 659, 1091, 673, 1613, 947, 1117, 1093, 1789, 191, 607, 599, 
751, 1153, 1039, 617, 1051, 769, 1487, 643, 1931, 661, 499, 2129, 1693, 
379, 1409, 991, 821, 1307, 1447, 709, 1151, 409, 2417, 1217, 1327, 433, 
1499, 283, 1021, 619, 701, 461, 1543, 977, 2741, 829, 1223, 1013, 1063, 
677, 1423, 2273, 1621, 1049, 2531, 1069, 2551, 907, 1289, 919, 1301, 1303, 
941, 1879, 1523, 757, 4591, 1361, 2113, 797, 1933, 1987, 1559, 1213, 2767, 
1033, 173, 1847, 2213, 1459, 3461, 1277, 3691, 2341, 3511, 1109, 2267, 
1549, 1433, 1777, 1663, 2441, 1031, 1367, 823, 1597, 6359, 967, 1709, 
1427, 1723, 1667, 1511, 997, 3583, 2857, 839, 1249, 3877, 1973, 1787, 
1753, 2039, 1531, 2293, 587, 1823, 3259, 1601, 1571, 3329, 853, 1123, 
5077, 3389, 1861, 2657, 1627, 2927, 883, 647, 1657, 2969, 2707, 2473, 
2207, 4057, 3541, 1193, 2239, 1471, 1721, 1483, 1733, 3109, 1481, 1229, 
2579, 4519, 1783, 2617, 971, 8447, 1283, 3181, 739, 2633, 2161, 1801, 
1607, 2383, 1907, 2111, 3361, 4153, 3677, 2143, 2237, 1279, 2543, 3947, 
1381, 2777, 1097, 3089, 2011, 2203, 4139, 3739, 1429, 5281, 1451, 1619, 
2693, 2251, 2089, 2887, 857, 6029, 3391, 1019, 877, 9901, 2819, 6113, 
1567, 4211, 3517, 2297, 4507, 1993, 2897, 3643, 2917, 2347, 3593, 6337, 
3623, 2371, 1637, 11093, 2683, 6079, 5413, 4079, 3061, 4441, 1373, 4801, 
2423, 5167, 2099, 6229, 3167, 5209, 2141, 4177, 3919, 3847, 2879, 2801, 
2539, 2459, 5059, 3557, 4003, 3217, 1489, 7949, 2243, 2137, 1889, 8387, 
1181, 4723, 827, 5851, 2333, 6619, 1609, 4391, 2377, 4409, 2017, 257, 
3163, 4457, 5857, 9463, 521, 4099, 4007, 3733, 1699, 5303, 4447, 2591, 
4073, 3001, 2909, 1831, 5297, 4231, 3331, 3851, 2549, 5471, 2971, 2671, 
3793, 1877, 5431, 4337, 3823, 3539, 6301, 7673, 3863, 3571, 1811, 2339, 
4733, 3607, 2663, 3203, 577, 1949, 5647, 3659, 4397, 1553, 2281, 2843, 
4861, 2003, 1867, 719, 1873, 3323, 2753, 2467, 3637, 4229, 4093, 2053, 
4547, 4273, 2357, 1187, 3253, 2087, 4157, 3889, 3727, 3457, 1493, 8429, 
4679, 9817, 3803, 12577, 4751, 6217, 653, 3019, 3881, 9949, 2063, 6271, 
2081, 3499, 3491, 5849, 1171, 8209, 2131, 4463, 1201, 3529, 263, 5437, 
2179, 10223, 8419, 4993, 2687, 5009, 2221, 2609, 1747, 1163, 3697, 3121, 
1759, 2153, 4217, 4621, 3251, 4637, 1291, 4649, 2791, 9137, 1321, 13649, 
2351, 2659, 4871, 5189, 6907, 8243, 5419, 4721, 3407, 3719, 3931, 11399, 
1913, 4243, 2953, 6317, 6067, 10477, 4027, 6353, 3527, 3767, 1453, 2731, 
5653, 1697, 3037, 6449, 1999, 4349, 6781, 8087, 3617, 7043, 6841, 2789, 
6323, 2269, 5801, 10369, 3671, 8231, 5861, 1741, 4787, 5023, 2621, 3943, 
6469, 3413, 3187, 10037, 8179, 6199, 4327, 7883, 7129, 8467, 5483, 7933, 
5507, 3469, 2713, 10799, 3301, 6869, 3319, 10289, 3343, 6899, 5641, 5779, 
3373, 12671, 4549, 4651, 2837, 7549, 3433, 4673, 2287, 8171, 4051, 4111, 
5233, 2957, 2903, 8837, 571, 2963, 10607, 4759, 5897, 9521, 3547, 4783, 
8329, 7193, 7159, 9613, 5387, 16249, 4219, 7841, 10883, 12107, 2437, 
15773, 7331, 3631, 3079, 4253, 6151, 9787, 5563, 6737, 2503, 7369, 1901, 
4289, 5011, 12377, 5039, 11159, 5689, 9311, 3209, 5569, 5107, 4957, 4493, 
10651, 4517, 11941, 7079, 8161, 3923, 2447, 10313, 19081, 5879, 5659, 761, 
5669, 3347, 5683, 2069, 14753, 7927, 7649, 5351, 6367, 6673, 5081, 4729, 
12301, 11971, 3797, 6073, 9743, 2797, 14389, 6133, 7127, 2833, 5147, 7507, 
8501, 5527, 7853, 4877, 8543, 8929, 11261, 4241, 11959, 11701, 8609, 4951, 
8629, 5651, 2521, 7027, 12097, 1583, 6637, 4339, 5279, 7793, 5987, 3673, 
12911, 15461, 2557, 9241, 19949, 9283, 4657, 5813, 8167, 10733, 6091, 
5843, 2593, 10079, 4021, 4451, 9689, 5179, 7583, 3067, 13997, 5227, 7607, 
2389, 7621, 3119, 8353, 3853, 20593, 6047, 11257, 6793, 6221, 3191, 6959, 
7561, 9887, 6857, 7723, 5417, 4817, 3967, 9227, 8389, 3371, 3989, 6329, 
5479, 7823, 6977, 6361, 9221, 5639, 5521, 4909, 4789, 10151, 3313, 10169, 
4831, 26711, 10889, 8689, 4129, 4931, 8677, 9491, 6427, 5717, 7963, 8779, 
4937, 16427, 4201, 3449, 5743, 6529, 8831, 21929, 2711, 4999, 6581, 7333, 
2729, 5021, 10499, 15919, 5087, 4261, 1979, 7393, 5119, 18371, 3581, 1129, 
4373, 8219, 1237, 6653, 5197, 7459, 4421, 17783, 2861, 6679, 5261, 23431, 
3701, 13877, 4523, 6691, 9349, 8317, 2939, 9941, 3761, 5923, 6197, 13219, 
9461, 14869, 7867, 14897, 3011, 31219, 5501, 12473, 19429, 6779, 13721, 
2699, 6343, 3533, 5531, 2719, 12149, 7699, 8861, 1087, 7213, 11069, 3907, 
4423, 3083, 15287, 16481, 10303, 8971, 6131, 7309, 7829, 9011, 6163, 9029, 
11251, 1439, 12113, 12479, 2803, 6547, 7069, 5711, 7937, 2311, 11369, 
6607, 8819, 9199, 6263, 10937, 4561, 5783, 6299, 4933, 7177, 9277, 20201, 
631, 4583, 4987, 7207, 3257, 7219, 7639, 8111, 11161, 6379, 6791, 10789, 
2411, 9923, 3307, 46141, 9551, 11689, 4259, 17029, 4283, 11719, 6971, 
5503, 7877, 18913, 7013, 7307, 5237, 7321, 10639, 14533, 3467, 10949, 
7993, 10067, 7109, 9181, 11657, 8297, 5323, 16493, 5347, 20161, 6287, 
13799, 4481, 10159, 10909, 14767, 13691, 2851, 9109, 18521, 6373, 4703, 
2689, 36161, 8287, 4691, 12941, 11213, 4597, 12161, 887, 6577, 7433, 5657, 
4639, 12227, 13099, 7559, 7481, 5693, 6553, 21737, 17911, 6659, 14153, 
25621, 11351, 3833, 9467, 5749, 13291, 15299, 10459, 8641, 12391, 8663, 
8581, 6763, 8597, 18311, 6701, 8707, 7681, 7759, 4799, 22277, 8699, 12619, 
12601, 8761, 17483, 5869, 8741, 20509, 10723, 6863, 6829, 4919, 9781, 
9839, 13729, 15761, 1951, 11839, 8867, 14821, 5927, 7919, 10891, 6947, 
12893, 6967, 6949, 16931, 10957, 20947, 6997, 12983, 8017

where the first prime that hasn't occured so far is 863.  Up to
this point the only n for which a(n) <= n-1 are
a(12) = 11, a(201) = 173, a(379) = 257, a(588) = 571, a(868) = 631,
a(932) = 887.

I don't have a proof, but it seems likely that the answer to
both of Leroy's questions is yes: the sequence is infinite, and
is a permutation of the primes.

Robert Israel                                israel at math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada


On Sun, 14 Oct 2007, Leroy Quet wrote:

> I just submitted this sequence. (I also submitted a few related
> sequences.)
>
>
>> %I A134204
>> %S A134204 2,3,5,7,13,17,19,23,41,31,29,37,11,67
>> %N A134204 a(0)=2. a(n) = the smallest prime not occurring earlier in the
>> sequence such that a(n-1)+a(n) is a multiple of n.
>> %C A134204 Is this sequence infinite, and, if so, is it a permutation of
>> the primes?
>> This sequence is infinite if and only if a(n-1) never divides n for any n.
>> %e A134204 The primes that don't occur among terms a(0) through a(6) form
>> the sequence 11,23,29,31,... Of these, 23 is the smallest that when added
>> to a(6)=19 gets a multiple of 7 --  19+23 = 42 = 6*7. (19+11 = 30, which
>> is not a multiple of 7.) So a(7) = 23.
>> %Y A134204 A134205,A134206,A134207
>> %O A134204 0
>> %K A134204 ,more,nonn,
>
> (Hopefully I did not make an error.)
>
> Is this sequence infinite? In other words, does a(n-1) not divide n for
> every positive integer n?
>
> And if it is infinite, is the sequence a permutation of the primes? (ie
> Does every prime occur somewhere in the sequence?)
>
> Thanks,
> Leroy Quet
>