[seqfan] Re: Resurrect A090566?

David Wilson davidwwilson at comcast.net
Thu Dec 3 12:51:58 CET 2015


They all need the 'base' and 'nonn' keywords

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Robert
> G. Wilson v
> Sent: Wednesday, December 02, 2015 8:47 PM
> To: 'Sequence Fanatics Discussion list'
> Subject: [seqfan] Re: Resurrect A090566?
> 
> On their way. thanx.
> 
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Neil
> Sloane
> Sent: Wednesday, December 02, 2015 12:25 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: Resurrect A090566?
> 
> 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.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> 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]
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
> 
> _______________________________________________
> 
> Seqfan Mailing list - http://list.seqfan.eu/
> 
> 
> _______________________________________________
> 
> Seqfan Mailing list - http://list.seqfan.eu/



More information about the SeqFan mailing list