[seqfan] Re: A145713 and company

Robert G. Wilson v rgwv at rgwv.com
Sun Aug 29 22:24:40 CEST 2010

```Dear Douglas, Neil and others,

Yes the sequences are inconsistent. A145709 should be the template.
Therefore I suggest that the sequences as follows:

All of the offsets are 1; no need for an offset of 0 since no additional
information would be obtained. Also Douglas' terms are correct (up to 10^8)
but adjusted for the title changes. This should not create a problem because
these sequences are only referenced by themselves.

Sincerely yours, Bob.

A145709: Number of primes among the first 10^n primes that have most
significant and least significant decimal digit both equal to 1.
1,  6,  39,  393,  7737, 103788, 1230898, 13344414, 120133545, 1092391851

A145710: Number of primes among the first 10^n primes that have most
significant and least significant decimal digit both equal to 3.
1,  5,  33,  279,  2251,  18840,  162663,  1434000,  12811209,  115798815

A145711: Number of primes among the first 10^n primes that have most
significant and least significant decimal digit both equal to 7.
1,  1,  33,  265,  2104,  17892,  155957,  1379176,  12374436,  112213302

A145712: Number of primes among the first 10^n primes that have most
significant and least significant decimal digit both equal to 9.
0,  0,   2,  245,  2053,  17518,  153645,  1362970,  12244431,  111153241

A145713: Number of primes among the first 10^n primes that have most
significant and least significant decimal digit are equal.
3, 12, 107, 1182, 14145, 158038, 1703163, 17520560, 157563621, 1431557209

I used the Mathematica coding: f[n_] := Block[{id = IntegerDigits at n},
If[id[[1]] == id[[-1]], id[[1]], 0]];
k = n = 1; p = 2; t = {0, 0, 0, 0}; While[n < 9, While[ k < 10^n + 1,
Switch[ f at p, 1, t[[1]]++, 3, t[[2]]++, 7, t[[3]]++, 9, t[[4]]++]; k++; p =
NextPrime at p]; Print[{t,Plus@@t}]; n++ ]

--------------------------------------------------
From: "Douglas McNeil" <mcneil at hku.hk>
Sent: Sunday, August 29, 2010 4:59 AM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan]  A145713 and company

> --
> %S A145713 1,10,105,1180,14143,150036
> %N A145713 Number of primes p in the range 9 < p <= prime(10^n) that
> begin and end
>               with the same digit.
> %E A145713 Edited by N. J. A. Sloane, Aug 16 2010
> %E A145713 The term 150036 is almost certainly wrong - see A145711.
> --
>
> I find
>
> 1 A145709 [1, 6, 39, 393, 7737, 103788, 1230898, 13344414, 120133545,
> 1092391851]
> 3 A145710 [0, 4, 32, 278, 2250, 18839, 162662, 1433999, 12811208,
> 115798814]
> 7 A145711 [0, 0, 32, 264, 2103, 17891, 155956, 1379175, 12374435,
> 112213301]
> 9 A145712 [0, 0, 2, 245, 2053, 17518, 153645, 1362970, 12244431,
> 111153241]
>   A145713 [1, 10, 105, 1180, 14143, 158036, 1703161, 17520558,
> 157563619, 1431557207]
>
> which agrees with all but the a(6) value on A145713 and extends the
> sequences by several terms.  There's a bit of an inconsistency,
> though, as in the OEIS, A145711 has an extra 0 compared to the above
> because its offset is 0.  All the others use an offset of 1.
>
> Any objections to making the offsets uniform at 1?  If not I'll edit
> when I correct '713.. "almost certainly wrong" sounds like the sort of
> thing we should fix even in summer.  ;-)
>
>
> Doug
>
> --
> Department of Earth Sciences
> University of Hong Kong
>
>
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>

```