# [seqfan] Re: Resurrect A090566?

Neil Sloane njasloane at gmail.com
Wed Dec 2 18:25:08 CET 2015

```Bob, please go ahead and submit that bunch of sequences!

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Email: njasloane at gmail.com

On Sun, Nov 29, 2015 at 1:25 PM, Robert G. Wilson v <rgwv at rgwv.com> wrote:

> Dear David,
>
>         Here are the sequences that Neil asked me to submit. They are in
> internal format and I have not yet asked for any allocation of A#s either.
>
>         Please check the terms and once I receive the OK, I will submit.
> Also can you see a way to extend A000009 below out further easily?
>
>         Also I extended A090566 out to a(1840). a(1841) has to many digits
> to be allowed in the database of the OEIS. And YES the values you gave
> matched mine precisely. I also added the b-text file to A243091.
>
> Sincerely yours, Bob.
>
> %N A000001 a(1) = 2, a(n) = smallest number > a(n-1) such that the
> concatenation of a(n-1) and a(n) is a square.
> %S A000001 2, 5, 29, 241, 1809, 6516, 27729, 70281, 191236, 537636,
> 5052601, 24352064, 50491721, 335176900, 816286736, 1584582656, 5835352241,
> 31064957504, 299026078001, 368254999225, 916181280225, 6283970794161,
> 31966212255489, 247575988078441, 558234718638336, 4773574731628096
> %O A000001 1
> %e A000001 a(3) is 29 since it is the least number greater than 5 which
> concatenated with 5 forms a perfect square, i.e.; 529 = 23^2.
> %t A000001 f[n_] := Block[{x = n, d = 1 + Floor@ Log10@ n}, q = (Floor@
> Sqrt[(10^d + 1) x] + 1)^2; If[q < (10^d) (x + 1), Mod[q, 10^d], Mod[(Floor@
> Sqrt[(10^d)(10x + 1) - 1] + 1)^2, 10^(d + 1)] ]]; NestList[f, 2, 25] (*
> after the algorithm of David W. Wilson in A090566 *)
> %Y A000001 Cf. A090566, A000002, A000003, A000004, A000005, A000006,
> A000007, A000008, A000009.
> %K A000001 nonn
> %A A000001 RGWv
>
> %N A000002 a(1) = 4, a(n) = smallest number > a(n-1) such that the
> concatenation of a(n-1) and a(n) is a square.
> %S A000002 4, 9, 61, 504, 4516, 47504, 382025, 3975209, 33057329,
> 80214016, 454665681, 4507966404, 44168848384, 69005350809, 163894140625,
> 784386132324, 5954843762641, 7954794246144, 53996843222416, 69176076458289,
> 379510987739761, 1641640879622564, 7593632535763529, 31733339799107600
> %O A000002 1
> %e A000002 a(3) is 61 since it is the least number greater than 9 which
> concatenated with 9 forms a perfect square, i.e.; 961 = 31^2.
> %t A000002 f[n_] := Block[{x = n, d = 1 + Floor@ Log10@ n}, q = (Floor@
> Sqrt[(10^d + 1) x] + 1)^2; If[q < (10^d) (x + 1), Mod[q, 10^d], Mod[(Floor@
> Sqrt[(10^d)(10x + 1) - 1] + 1)^2, 10^(d + 1)] ]]; NestList[f, 4, 23] (*
> after the algorithm of David W. Wilson in A090566 *)
> %Y A000002 Cf. A090566, A000001, A000003, A000004, A000005, A000006,
> A000007, A000008, A000009.
>
> %N A000003 a(1) = 8, a(n) = smallest number > a(n-1) such that the
> concatenation of a(n-1) and a(n) is a square.
> %S A000003 8, 41, 209, 764, 5225, 8441, 9344, 63761, 82201, 477264,
> 3191044, 4038489, 34656049, 61233321, 271005625, 3465072801, 36565416324,
> 83511106624, 222222321476, 425286636356, 2743260628100, 9534841632400,
> 33984728488004, 128198574830929, 741089622057984
> %O A000003 1
> %e A000003 a(3) is 209 since it is the least number greater than 41 which
> concatenated with 41 forms a perfect square, i.e.; 41209 = 203^2.
> %t A000003 f[n_] := Block[{x = n, d = 1 + Floor@ Log10@ n}, q = (Floor@
> Sqrt[(10^d + 1) x] + 1)^2; If[q < (10^d) (x + 1), Mod[q, 10^d], Mod[(Floor@
> Sqrt[(10^d)(10x + 1) - 1] + 1)^2, 10^(d + 1)] ]]; NestList[f, 8, 24] (*
> after the algorithm of David W. Wilson in A090566 *)
> %Y A000003 Cf. A090566, A000001, A000002, A000004, A000005, A000006,
> A000007, A000008, A000009.
>
> %N A000004 a(1) = 10, a(n) = smallest number > a(n-1) such that the
> concatenation of a(n-1) and a(n) is a square.
> %S A000004 10, 24, 336, 400, 689, 5876, 7556, 8249, 53284, 335556,
> 4512400, 25092921, 165947209, 496186596, 3891489129, 6897736129,
> 10128495225, 18547234816, 81770476100, 203672467856, 909690622025,
> 6063906517681, 14045408555225, 50912872680100, 145763131189824
> %O A000004 1
> %e A000004 a(3) is 336 since it is the least number greater than 24 which
> concatenated with 24 forms a perfect square, i.e.; 24336 = 156^2.
> %t A000004 f[n_] := Block[{x = n, d = 1 + Floor@ Log10@ n}, q = (Floor@
> Sqrt[(10^d + 1) x] + 1)^2; If[q < (10^d) (x + 1), Mod[q, 10^d], Mod[(Floor@
> Sqrt[(10^d)(10x + 1) - 1] + 1)^2, 10^(d + 1)] ]]; NestList[f, 10, 24] (*
> after the algorithm of David W. Wilson in A090566 *)
> %Y A000004 Cf. A090566, A000001, A000002, A000003, A000005, A000006,
> A000007, A000008, A000009.
>
> %N A000005 a(1) = 11, a(n) = smallest number > a(n-1) such that the
> concatenation of a(n-1) and a(n) is a square.
> %S A000005 11, 56, 169, 744, 769, 5076, 5625, 43524, 390625, 1827776,
> 2562500, 8273225, 37136225, 38371001, 43037561, 258421444, 792669636,
> 2928667041, 38512058944, 260125180889, 405701529401, 688085041025,
> 5890084946609, 22508111494025, 64017148660004, 537387232526336
> %O A000005 1
> %e A000005 a(3) is 169 since it is the least number greater than 56 which
> concatenated with 56 forms a perfect square, i.e.; 56169 = 237^2.
> %t A000005 f[n_] := Block[{x = n, d = 1 + Floor@ Log10@ n}, q = (Floor@
> Sqrt[(10^d + 1) x] + 1)^2; If[q < (10^d) (x + 1), Mod[q, 10^d], Mod[(Floor@
> Sqrt[(10^d)(10x + 1) - 1] + 1)^2, 10^(d + 1)] ]]; NestList[f, 11, 25] (*
> after the algorithm of David W. Wilson in A090566 *)
> %Y A000005 Cf. A090566, A000001, A000002, A000003, A000005, A000006,
> A000007, A000008, A000009.
>
> %N A000006 a(1) = 14, a(n) = smallest number > a(n-1) such that the
> concatenation of a(n-1) and a(n) is a square.
> %S A000006 14, 44, 89, 401, 956, 6649, 17796, 58596, 432489, 4211044,
> 22847241, 34268944, 85740489, 530152900, 718608036, 3266783209,
> 33250749225, 96733442161, 617288020224, 5959324297569, 20015258667081,
> 123104551223296, 420105398760804, 552382701059344, 967075372931216
> %O A000006 1
> %e A000006 a(3) is 89 since it is the least number greater than 44 which
> concatenated with 44 forms a perfect square, i.e.; 4489 = 67^2.
> %t A000006 f[n_] := Block[{x = n, d = 1 + Floor@ Log10@ n}, q = (Floor@
> Sqrt[(10^d + 1) x] + 1)^2; If[q < (10^d) (x + 1), Mod[q, 10^d], Mod[(Floor@
> Sqrt[(10^d)(10x + 1) - 1] + 1)^2, 10^(d + 1)] ]]; NestList[f, 14, 24] (*
> after the algorithm of David W. Wilson in A090566 *)
> %Y A000006 Cf. A090566, A000001, A000002, A000003, A000004, A000005,
> A000007, A000008, A000009.
>
> %N A000007 a(1) = 15, a(n) = smallest number > a(n-1) such that the
> concatenation of a(n-1) and a(n) is a square.
> %S A000007 15, 21, 316, 969, 6996, 55401, 390625, 1827776, 2562500,
> 8273225, 37136225, 38371001, 43037561, 258421444, 792669636, 2928667041,
> 38512058944, 260125180889, 405701529401, 688085041025, 5890084946609,
> 22508111494025, 64017148660004, 537387232526336, 4166255964768676
> %O A000007 1
> %e A000007 a(3) is 316 since it is the least number greater than 21 which
> concatenated with 21 forms a perfect square, i.e.; 21316 = 146^2.
> %t A000007 f[n_] := Block[{x = n, d = 1 + Floor@ Log10@ n}, q = (Floor@
> Sqrt[(10^d + 1) x] + 1)^2; If[q < (10^d) (x + 1), Mod[q, 10^d], Mod[(Floor@
> Sqrt[(10^d)(10x + 1) - 1] + 1)^2, 10^(d + 1)] ]]; NestList[f, 15, 24] (*
> after the algorithm of David W. Wilson in A090566 *)
> %Y A000007 Cf. A090566, A000001, A000002, A000003, A000004, A000005,
> A000006, A000008, A000009.
>
> %N A000008 a(1) = 16, a(n) = smallest number > a(n-1) such that the
> concatenation of a(n-1) and a(n) is a square.
> %S A000008 16, 81, 225, 625, 681, 2100, 3889, 17841, 33121, 452049,
> 2561025, 9392964, 9776361, 69946276, 104857889, 232947041, 619807376,
> 729085444, 5435467076, 8236728484, 52686818481, 370961353041,
> 3290130736249, 4333224368201, 44310474545225, 67348431045184
> %O A000008 1
> %e A000008 a(3) is 225 since it is the least number greater than 81 which
> concatenated with 81 forms a perfect square, i.e.; 81225 = 285^2.
> %t A000008 f[n_] := Block[{x = n, d = 1 + Floor@ Log10@ n}, q = (Floor@
> Sqrt[(10^d + 1) x] + 1)^2; If[q < (10^d) (x + 1), Mod[q, 10^d], Mod[(Floor@
> Sqrt[(10^d)(10x + 1) - 1] + 1)^2, 10^(d + 1)] ]]; NestList[f, 16, 25] (*
> after the algorithm of David W. Wilson in A090566 *)
> %Y A000008 Cf. A090566, A000001, A000002, A000003, A000004, A000005,
> A000006, A000007, A000009.
>
> %N A000009 Integers, n, which are unique starting points for the algorithm
> described in A090566.
> %S A000009 1, 2, 4, 8, 10, 11, 14, 15, 16, 17, 18, 19, 21, 22, 23
> %O A000009 1
> %C A000009 The algorithm of A090566, i.e.; a(n) is the initial term and
> subsequent terms are the smallest number greater that the previous term
> such that the concatenation of the two is a perfect square.
> %C A000009 Complement 3, 5, 6, 7, 9, 12, 13, 20, …, .
> %C A000009 An initial value of 3, 6, 12, 20, …,  quickly merges into
> A090566.
> %e A000009 An initial value of 5, 7, 9, …, quickly merges into A000002.
> %e A000009 An initial value of 9, …, quickly merges into A000003.
> %e A000009 An initial value of 13, …, quickly merges into A000005.
> %e A000009 An initial value of 15, …, quickly merges into A000006.
> %t A000009 f[n_] := Block[{x = n, d = 1 + Floor@ Log10@ n}, q = (Floor@
> Sqrt[(10^d + 1) x] + 1)^2; If[q < (10^d) (x + 1), Mod[q, 10^d], Mod[(Floor@
> Sqrt[(10^d)(10x + 1) - 1] + 1)^2, 10^(d + 1)]
> %Y A000009 Cf. A090566, A243091, A000001, A000002, A000003, A000004,
> A000005, A000006, A000007, A000008.
>
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of David
> Wilson
> Sent: Sunday, November 22, 2015 5:46 PM
> To: 'Sequence Fanatics Discussion list'
> Subject: [seqfan] Re: Resurrect A090566?
>
> RGW:
>
> My interest was just to salvage A090566.
> If you want to submit variant sequences, feel free.
> I could check your computations.
>
> Does your Mma program agree with my entire A090566 b-file?
> Does Mma agree that concatenating any two b-file values produces a square
> number?
>
> > -----Original Message-----
> > From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of
> > Robert G. Wilson v
> > Sent: Sunday, November 22, 2015 5:21 PM
> > To: 'Sequence Fanatics Discussion list'
> > Subject: [seqfan] Re: Resurrect A090566?
> >
> > Dear David,
> >
> >       Now that we know that our algorithm is working properly in
> > Mathematica, how about the following sequences:
> >
> > [variants on A090566 elided]
>
>
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```