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a(1) = 2, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
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8
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
EXAMPLE
a(3) is 29 since it is the least number greater than a(2)=5 which concatenated with 5 forms a perfect square, i.e., 529 = 23^2.
MATHEMATICA
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 *)
a(1) = 4, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
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8
4, 9, 61, 504, 4516, 47504, 382025, 3975209, 33057329, 80214016, 454665681, 4507966404, 44168848384, 69005350809, 163894140625, 784386132324, 5954843762641, 7954794246144, 53996843222416, 69176076458289, 379510987739761, 1641640879622564, 7593632535763529, 31733339799107600
EXAMPLE
a(3) is 61 since it is the least number greater than a(2)=9 which concatenated with 9 forms a perfect square, i.e., 961 = 31^2.
MATHEMATICA
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 *)
a(1) = 8, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
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8
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, 5579432351776489
EXAMPLE
a(3) is 209 since it is the least number greater than a(2)=41 which concatenated with 41 forms a perfect square, i.e., 41209 = 203^2.
MATHEMATICA
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, 25] (* after the algorithm of David W. Wilson in A090566 *)
a(1) = 10, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
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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, 180798422222500
EXAMPLE
a(3) is 336 since it is the least number greater than a(2)=24 which concatenated with 24 forms a perfect square, i.e., 24336 = 156^2.
MATHEMATICA
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, 25] (* after the algorithm of David W. Wilson in A090566 *)
a(1) = 11, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
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8
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
EXAMPLE
a(3) is 169 since it is the least number greater than a(2)=56 which concatenated with 56 forms a perfect square, i.e., 56169 = 237^2.
MATHEMATICA
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 *)
a(1) = 15, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
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8
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
EXAMPLE
a(3) is 316 since it is the least number greater than a(2)=21 which concatenated with 21 forms a perfect square, i.e., 21316 = 146^2.
MATHEMATICA
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 *)
a(1) = 16, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
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8
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, 67835332918689
EXAMPLE
a(3) is 225 since it is the least number greater than a(2)=81 which concatenated with 81 forms a perfect square, i.e., 81225 = 285^2.
MATHEMATICA
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 *)
Integers which are unique starting points for the algorithm described in A090566.
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8
1, 2, 4, 8, 10, 11, 14, 15, 16, 17, 18, 19, 21, 22, 23
COMMENTS
Consider the family of integer sequences generated from a starting value b(1) and the rule that each subsequent term is the smallest number greater than the previous term such that the concatenation of the two is a square. Then using
b(1) = a(1) = 1 yields {1, 6, 25, 281, 961, ...} ( A090566), b(1) = a(2) = 2 yields {2, 5, 29, 241, 1809, ...} ( A265147), b(1) = a(3) = 4 yields {4, 9, 61, 504, 4516, ...} ( A265148), b(1) = a(4) = 8 yields {8, 41, 209, 764, 5225, ...} ( A265149), b(1) = a(5) = 10 yields {10, 24, 336, 400, 689, ...} ( A265150), b(1) = a(6) = 11 yields {11, 56, 169, 744, 769, ...} ( A265151), ...
EXAMPLE
The complement of {a(n)} is {3, 5, 6, 7, 9, 12, 13, 20, ...}; using any of these values as b(1) yields a sequence that quickly merges into one of the sequences obtained using a value from {a(n)} as b(1):
b(1) = 3 -> {3, 6, 25, 281, 961, ...}, which quickly merges into A090566 (as does the result of using b(1) = 6 or 12 or 20 ...);
b(1) = 5 -> {5, 29, 241, 1809, ...}, which quickly merges into A265147 (as does the result of using b(1) = 7 ...);
b(1) = 9 -> {9, 61, 504, 4516, ...}, which quickly merges into A265148; b(1) = 13 -> {13, 69, 169, 744, 769, ...}, which quickly merges into A265151.
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