[seqfan] Re: Resurrect A090566?

Sun Nov 29 19:25:42 CET 2015

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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