[seqfan] Re: A139414

[Harry J. Smith]

Tony:

I know I cannot change your mind, but the 4 polynomials of A139414 look
remarkable to me.
Here are the first 100 (0..100) values. The non prime values have been
zeroed out

n  p1    p2      p3     p4     pE(Euler's polynomial)
0 1373 1459 1301 1877 41
1 1231 1319 1163 1741 43
2 1097 1187 1033 1613 47
3 971 1063 911 1493 53
4 853 947 797 1381 61
5 743 839 691 1277 71
6 641 739 593 1181 83
7 547 647 503 1093 97
8 461 563 421 1013 113
9 383 487 347 941 131
10 313 419 281 877 151
11 251 359 223 821 173
12 197 307 173 773 197
13 151 263 131 733 223
14 113 227 97 701 251
15 83 199 71 677 281
16 61 179 53 661 313
17 47 167 43 653 347
18 41 163 41 653 383
19 43 167 47 661 421
20 53 179 61 677 461
21 71 199 83 701 503
22 97 227 113 733 547
23 131 263 151 773 593
24 173 307 197 821 641
25 223 359 251 877 691
26 281 419 313 941 743
27 347 487 383 1013 797
28 421 563 461 1093 853
29 503 647 547 1181 911
30 593 739 641 1277 971
31 691 839 743 1381 1033
32 797 947 853 1493 1097
33 911 1063 971 1613 1163
34 1033 1187 1097 1741 1231
35 1163 1319 1231 1877 1301
36 1301 1459 1373 0 1373
37 1447 1607 1523 0 1447
38 1601 0 0 2333 1523
39 0 0 1847 0 1601
40 1933 2099 0 2677 0
41 2111 0 2203 2861 0
42 2297 2467 2393 0 1847
43 0 2663 2591 3253 1933
44 2693 0 2797 3461 0
45 2903 3079 3011 3677 2111
46 3121 3299 0 0 2203
47 3347 3527 3463 4133 2297
48 3581 0 3701 4373 2393
49 3823 4007 3947 4621 0
50 4073 4259 4201 4877 2591
51 0 4519 4463 0 2693
52 4597 4787 4733 5413 2797
53 4871 0 5011 5693 2903
54 5153 5347 5297 5981 3011
55 5443 5639 5591 6277 3121
56 5741 5939 0 6581 0
57 6047 6247 6203 0 3347
58 6361 6563 6521 7213 3463
59 0 0 0 7541 3581
60 7013 7219 0 7877 3701
61 7351 7559 7523 8221 3823
62 0 7907 7873 8573 3947
63 0 8263 8231 8933 4073
64 0 8627 8597 0 4201
65 8783 8999 8971 9677 0
66 9161 0 0 10061 4463
67 9547 9767 9743 10453 4597
68 9941 10163 10141 10853 4733
69 10343 10567 0 11261 4871
70 10753 10979 0 11677 5011
71 11171 11399 11383 12101 5153
72 11597 11827 11813 0 5297
73 0 12263 12251 12973 5443
74 12473 0 12697 13421 5591
75 12923 13159 13151 13877 5741
76 13381 13619 13613 14341 0
77 0 14087 14083 14813 6047
78 14321 14563 14561 0 6203
79 0 0 0 0 6361
80 0 0 15541 0 6521
81 15791 0 0 0 0
82 0 16547 16553 17293 0
83 16811 0 0 0 7013
84 17333 0 17597 18341 0
85 17863 18119 18131 0 7351
86 18401 0 0 19421 7523
87 18947 19207 0 19973 0
88 19501 19763 0 20533 7873
89 20063 20327 20347 21101 0
90 0 20899 20921 0 8231
91 21211 0 21503 0 0
92 0 22067 22093 22853 8597
93 22391 0 22691 0 8783
94 22993 0 23297 24061 8971
95 23603 23879 23911 24677 9161
96 0 24499 24533 25301 0
97 24847 25127 25163 25933 9547
98 0 25763 25801 26573 9743
99 0 26407 0 0 9941
100 0 27059 0 0 10141

-Harry


> -----Original Message-----
> From: T. D. Noe [mailto:noe at sspectra.com]
> Sent: Thursday, January 29, 2009 3:19 PM
> To: Harry J. Smith
> Cc: 'David Wilson'; 'zak seidov'; 'Olivier Gerard'; 'Maximilian Hasler'
> Subject: RE: [seqfan] Re: A139414
> 
> >A long time ago the amazing mathematician Leonard Euler showed that the
> >polynomial
> >
> >	n^2  + n + 41         (6)
> >
> >generates distinct prime values for all integral inputs between 0 and 39.
> >This was an amazing discovery, and even today the record for producing
> >consecutive primes by a polynomial sits at just 57 (in 2005, the record
was
> >just 43!). So one might ask: is there a polynomial out there which only
> >takes prime values? Sadly, the answer is know. However..
> >
> >
> >I think the 4 polynomials in sequence A139414 are unique in how many
primes
> >they generate Sequence A155814 is just a nice companion.
> >
> >p1(n) =4*x^2 - 146*x + 1373,
> >p2(n)= 4*x^2 - 144*x + 1459,
> >p3(n)= 4*x^2 - 142*x + 1301,
> >p4(n)= 4*x^2 - 140*x + 1877.
> >
> >I know you cannot please everybody, but I will miss the sequence on
> >polynomial primes.
> 
> 
> I know Euler's polynomial.  I think the set of 4 quadratics does not come
> close in importance.  They seem random.  If something distinguishes these
4
> from some other set of 4, that might change my mind.
> 
> Tony