A265147
a(1) = 2, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
Original entry on oeis.org
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
Offset: 1
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.
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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 *)
A265148
a(1) = 4, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
Original entry on oeis.org
4, 9, 61, 504, 4516, 47504, 382025, 3975209, 33057329, 80214016, 454665681, 4507966404, 44168848384, 69005350809, 163894140625, 784386132324, 5954843762641, 7954794246144, 53996843222416, 69176076458289, 379510987739761, 1641640879622564, 7593632535763529, 31733339799107600
Offset: 1
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.
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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 *)
A265149
a(1) = 8, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
Original entry on oeis.org
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
Offset: 1
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.
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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 *)
A265150
a(1) = 10, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
Original entry on oeis.org
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
Offset: 1
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.
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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 *)
A265152
a(1) = 14, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
Original entry on oeis.org
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
Offset: 1
a(3) is 89 since it is the least number greater than a(2)=44 which concatenated with 44 forms a perfect square, i.e., 4489 = 67^2.
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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 *)
A265153
a(1) = 15, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
Original entry on oeis.org
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
Offset: 1
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.
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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 *)
A265154
a(1) = 16, a(n) = smallest number > a(n-1) such that the concatenation of a(n-1) and a(n) is a square.
Original entry on oeis.org
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
Offset: 1
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.
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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 *)
A265155
Integers which are unique starting points for the algorithm described in A090566.
Original entry on oeis.org
1, 2, 4, 8, 10, 11, 14, 15, 16, 17, 18, 19, 21, 22, 23
Offset: 1
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.
Showing 1-8 of 8 results.
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