# [seqfan] Re: not all prime

Douglas McNeil mcneil at hku.hk
Wed Aug 18 03:34:41 CEST 2010

```Andrew Weinholm wrote:

> There's another misplaced comma in here which happens to split a prime into
> two primes: 283 has been split into 2 and 83.

This suggests another quick search for comma problems, namely
sequences with runs of (a,b,c,d) with b < a where a < concat(b,c) < d.
Applying some restrictive cuts to weed out false positives, and also
looking only for cases with one error (which wouldn't have caught the
2,83 one):

--
http://oeis.org/classic/?q=id%3aA158189
%N A158189 Prime numbers p such that sum of nonprime digits of p + sum
of even digits  of p = prime.
[2, 11, 23, 29, 61, 101, 113, 127, 131, 151, 163, 167, 191, 199, 229,
233, 239, 241, 251, 257, 269, 271, 277, 281, 293, 311, 349, 409, 421,
439, 479, 521, 523, 601, 61, 3, 617, 631, 727, 761, 821, 911, 919,
947, 991, 1013, 1019, 1031, 1051, 1063, 1087, 1091]
[[601, 61, 3, 617]]

[which survived my prime test because 61 and 3 are both prime, just
like the 2,83 case]

http://oeis.org/classic/?q=id%3aA091856
%N A091856 Beginning with 1, minimum value such that
gcd(a(2n-1),a(2n)) = 1, gcd(a(2n), a(2n+1))>1 and a(n) > a(n-1).
[1, 2, 4, 5, 10, 11, 22, 23, 46, 47, 94, 95, 100, 101, 202, 203, 210,
211, 422, 423, 426, 427, 4, 34, 435, 438, 439, 878, 879, 882, 883,
1766, 1767, 1770, 1771, 1778, 1779, 1782, 1783, 3566, 3567, 3570,
3571, 7142, 7143, 7146, 7147, 7154, 7155, 7158, 7159]
[[427, 4, 34, 435]]

http://oeis.org/classic/?q=id%3aA157134
%N A157134 G.f. satisfies: A(x) = Sum_{n>=0} x^(n^2) * A(x)^n.
[1, 1, 1, 1, 2, 4, 7, 11, 18, 33, 63, 117, 211, 383, 713, 1348, 2547,
4793, 9039, 17165, 32785, 62761, 120243, 230768, 444119, 857015,
1656931, 3207990, 6219994, 12079544, 23496417, 45767352, 8, 9256038,
174269488, 340646238, 666604642]
[[45767352, 8, 9256038, 174269488]]

http://oeis.org/classic/?q=id%3aA098533
%N A098533 Sum of seventh powers of first n Fibonacci numbers.
[0, 1, 2, 130, 2317, 80442, 2177594, 64926111, 1866014652,
54389364796, 15768245991, 71, 45808159494700, 1329726624043564,
38611060907763141, 1121015217730946894, 32548443577940946894,
945021540449512861377]
[[54389364796, 15768245991, 71, 45808159494700]]

http://oeis.org/classic/?q=id%3aA138252
%N A138252 Beatty sequence of the number t satisfying 1/s + 1/t = 1,
where s is  the positive root of x^3 - x^2 - 1.
[3, 6, 9, 12, 15, 18, 22, 25, 28, 31, 34, 37, 40, 44, 47, 50, 53, 56,
59, 62, 66, 69, 72, 75, 78, 81, 84, 88, 91, 94, 97, 100, 103, 107,
110, 113, 116, 119, 122, 125, 129, 132, 135, 138, 141, 1, 44, 147,
151, 154, 157, 160, 163, 166, 169, 173, 176, 179, 182, 185, 188]
[[141, 1, 44, 147]]

http://oeis.org/classic/?q=id%3aA166068
%N A166068 a(n) = a(n-1)+ [least square > a(n-1)]
[1, 5, 14, 30, 66, 147, 316, 640, 1316, 2685, 5389, 10865, 21890,
43794, 87894, 176103, 352503, 705339, 1410939, 2822283, 5644683,
11290059, 22586380, 45177389, 90362673, 180726709, 361467845,
722962014, 1445926, 558, 2891903234]
[[722962014, 1445926, 558, 2891903234]]

http://oeis.org/classic/?q=id%3aA141361
%N A141361 E.g.f.: A(x) =
exp(x*A(x)*exp(x*A(x)^2*exp(x*A(x)^3*exp(x*A(x)^4*exp(...))))),  an
infinite power tower.
[1, 1, 5, 55, 981, 24621, 803143, 32390247, 1560845289, 87688371385,
5637912173451, 408922311037659, 33077570245035517,
2956347175261764597, 2897160705852956894, 55, 30931475430329804121871]
[[2956347175261764597, 2897160705852956894, 55, 30931475430329804121871]]

http://oeis.org/classic/?q=id%3aA141363
%N A141363 E.g.f.: A(x) =
exp(x*A(x)^3*exp(x*A(x)^4*exp(x*A(x)^5*exp(x*A(x)^6*exp(...))))),  an
infinite power tower.
[1, 1, 9, 175, 5357, 225461, 12112675, 792855043, 61249418585,
5456747990665, 550924441708031, 62176714054787135,
7758184127489208517, 1060631125759075562, 797,
157674821045525700085499]
[[7758184127489208517, 1060631125759075562, 797, 157674821045525700085499]]

http://oeis.org/classic/?q=id%3aA047723
%N A047723 Sum of 12 but no fewer nonzero fourth powers.
[12, 27, 42, 57, 72, 92, 107, 122, 137, 152, 172, 187, 192, 202, 217,
232, 25, 2, 267, 282, 297, 312, 332, 347, 362, 377, 392, 412, 427,
432, 442, 457, 472, 492, 497, 507, 522, 537, 552, 572, 587, 602, 617,
636, 652, 667, 672, 682, 697, 716, 732, 747, 762]
[[232, 25, 2, 267]]

http://oeis.org/classic/?q=id%3aA175260
%N A175260 Numbers n with property that n+41, n^2+41 and n^3+41 are all primes.
[0, 66, 546, 1860, 2490, 6720, 6930, 7290, 12360, 14010, 15960, 17076,
1875, 6, 21726, 23016, 28710, 30096, 30840, 32520, 33150, 36846,
37476, 43410, 51306, 57246, 61710, 64536, 68736, 78750, 79560, 80106,
81660, 82026, 96540, 100866, 100896]
[[17076, 1875, 6, 21726]]

http://oeis.org/classic/?q=id%3aA137819
%N A137819 Year numbers, i.e. phi(n) = 2 phi(sigma(n)), divisible by a
5th prime  power.
[1811079, 4473387, 67009923, 77242167, 88605819, 110475819, 120781449,
132208767, 134082297, 165515319, 183408867, 225548469, 232275681,
272876607, 284339403, 32, 6557251, 349538247, 402371793, 425844621,
501668883, 566867727]
[[284339403, 32, 6557251, 349538247]]

--

Doug

--
Department of Earth Sciences
University of Hong Kong

```