A071128
Squares which repeat with at least two full periods when written in base 2.
Original entry on oeis.org
36, 1849, 6241, 29929, 418609, 672400, 1320201, 7458361, 28121809, 119355625, 302794801, 386554921, 1211179204, 1277991001, 1967188609, 5111964004, 7681646025, 30542106169, 73073443041, 79698371481, 91424593225, 120297291921, 292293772164, 318793485924
Offset: 1
173^2 = 29929 = 1110100|1110100|1 in base 2.
-
psq[b_, nd_] := Block[{sq={}, r, x}, Do[r = Mod[nd, p]; Do[x = (b^( Floor[nd/p] p) - 1)/(b^p - 1) t b^r + Floor[t/b^(p - r)]; If[ IntegerQ@ Sqrt@ x, AppendTo[ sq, x]], {t, b^(p - 1), b^p - 1}], {p, nd/2}]; sq]; Union @@ (psq[2, #] & /@ Range[32]) (* Giovanni Resta, Aug 01 2018 *)
A071135
Squares which repeat with at least two full periods when written in base 9.
Original entry on oeis.org
557904400, 1255284900, 2231617600, 508239242281, 110723679691024, 267200559990920521, 4624943568803015273104, 89426738951269414045129, 106343953870710063227329, 141828621071731373745001, 153228095100765507880225, 155946118585671902125225
Offset: 1
23620^2 = 557904400 = 13857|13857 in base 9.
712909^2 = 508239242281 = 171676|171676|1 in base 9.
-
psq[b_, nd_] := Block[{sq={}, r, x}, Do[r = Mod[nd, p]; Do[x = (b^( Floor[nd/p] p) - 1)/(b^p - 1) t b^r + Floor[t/b^(p - r)]; If[ IntegerQ@ Sqrt@ x, AppendTo[ sq, x]], {t, b^(p - 1), b^p - 1}], {p, nd/2}]; sq]; Union @@ (psq[9, #] & /@ Range[10]) (* Giovanni Resta, Aug 01 2018 *)
A116502
Squares with structure dmdmd, where d is a single digit and m a string of digits.
Original entry on oeis.org
69696, 56722567225, 95540955409, 1108033241108033241, 1220096161220096161, 4312080784312080784, 4432132964432132964, 4880384644880384644, 9024307889024307889, 9972299169972299169, 946308825294630882529, 11542927396115429273961, 46171709584461717095844
Offset: 1
309097^2 = a(3) = 9[5540]9[5540]9.
-
w={}; Do[s = Reduce[(1 + 10^(1+e) + 100^(1+e)) d + 10 (1+10^(1+e)) x == y^2 && 0 <= x < 10^e && y>0, {x,y}, Integers]; If[s =!= False, w = Union[w, y^2 /. List@ ToRules@ s]], {e, 12}, {d, 9}]; w (* Giovanni Resta, Aug 01 2018 *)
A071129
Squares which repeat with at least two full periods when written in base 3.
Original entry on oeis.org
4, 121, 841, 6889, 60025, 62001, 68644, 534361, 540225, 1042441, 4791721, 4809249, 20811844, 43072969, 43125489, 43283241, 387499225, 387656721, 388129401, 567725929, 713584369, 1255284900, 2231617600, 3487020601, 3487493025, 3488910489, 31381768201
Offset: 1
11^2 = 121 = 1|1|1|1|1 in base 3.
249^2 = 62001 = 1001|1001|100 in base 3.
-
psq[b_, nd_] := Block[{sq={}, r, x}, Do[r = Mod[nd, p]; Do[x = (b^( Floor[nd/p] p) - 1)/(b^p - 1) t b^r + Floor[t/b^(p - r)]; If[ IntegerQ@ Sqrt@ x, AppendTo[ sq, x]], {t, b^(p - 1), b^p - 1}], {p, nd/2}]; sq]; Union @@ (psq[3, #] & /@ Range[20]) (* Giovanni Resta, Aug 01 2018 *)
A071134
Squares which repeat with at least two full periods when written in base 8.
Original entry on oeis.org
9, 36, 6241, 116964, 418609, 28121809, 477247716, 1277991001, 1967188609, 5111964004, 7681646025, 73073443041, 79698371481, 91424593225, 120297291921, 292293772164, 318793485924, 1954689202404, 5429696889241, 5480313774049, 21718787556964, 21921255096196
Offset: 1
647^2 = 418609 = 146|146|1 in base 8.
21846^2 = 477247716 = 34344|34344 in base 8.
-
psq[b_, nd_] := Block[{sq={}, r, x}, Do[r = Mod[nd, p]; Do[x = (b^( Floor[nd/p] p) - 1)/(b^p - 1) t b^r + Floor[t/b^(p - r)]; If[ IntegerQ@ Sqrt@ x, AppendTo[ sq, x]], {t, b^(p - 1), b^p - 1}], {p, nd/2}]; sq]; Union @@ (psq[8, #] & /@ Range[11]) (* Giovanni Resta, Aug 01 2018 *)
A071130
Squares which repeat with at least two full periods when written in base 4.
Original entry on oeis.org
6241, 378225, 672400, 1320201, 28121809, 302794801, 386554921, 1211179204, 7681646025, 73073443041, 79698371481, 91424593225, 120297291921, 292293772164, 318793485924, 657660987369, 1521820771641, 18145972716481, 72583890865924, 163313754448329
Offset: 1
79^2 = 6241 = 120|120|1 in base 4.
615^2 = 378225 = 11301|11301 in base 4.
-
psq[b_, nd_] := Block[{sq={}, r, x}, Do[r = Mod[nd, p]; Do[x = (b^( Floor[nd/p] p) - 1)/(b^p - 1) t b^r + Floor[t/b^(p - r)]; If[ IntegerQ@ Sqrt@ x, AppendTo[ sq, x]], {t, b^(p - 1), b^p - 1}], {p, nd/2}]; sq]; Union @@ (psq[4, #] & /@ Range[16]) (* Giovanni Resta, Aug 01 2018 *)
A071131
Squares which repeat with at least two full periods when written in base 5.
Original entry on oeis.org
7056, 71824, 17380561, 10479821641, 1695422743056, 3046710023334121, 3484661204178361, 4238552821180561, 16390460540754756, 2341118082098985289, 49597060188790044841, 413921144300672743056, 931719854156507158681, 1149780956458574459881, 1254278391764733006841
Offset: 1
268^2 = 71824 = 424|424|4 in base 5.
1302084^2 = 1695422743056 = 210234211|210234211 in base 5.
-
psq[b_, nd_] := Block[{sq={}, r, x}, Do[r = Mod[nd, p]; Do[x = (b^( Floor[nd/p] p) - 1)/(b^p - 1) t b^r + Floor[t/b^(p - r)]; If[ IntegerQ@ Sqrt@ x, AppendTo[ sq, x]], {t, b^(p - 1), b^p - 1}], {p, nd/2}]; sq]; Union @@ (psq[5, #] & /@ Range[15]) (* Giovanni Resta, Aug 01 2018 *)
A071132
Squares which repeat with at least two full periods when written in base 6.
Original entry on oeis.org
5776, 190096, 6739216, 97990201, 241989136, 242611776, 3975176401, 8707875856, 8711608896, 14393520729, 25588481296, 39982002025, 57574082916, 95174101009, 149392299169, 313461135376, 313483530816, 380696404036, 4724628251641, 11284466503696, 11284600873536
Offset: 1
76^2 = 5776 = 42|42|4 in base 6.
119973^2 = 14393520729 = 1034013|1034013 in base 6.
-
psq[b_, nd_] := Block[{sq={}, r, x}, Do[r = Mod[nd, p]; Do[x = (b^( Floor[nd/p] p) - 1)/(b^p - 1) t b^r + Floor[t/b^(p - r)]; If[ IntegerQ@ Sqrt@ x, AppendTo[ sq, x]], {t, b^(p - 1), b^p - 1}], {p, nd/2}]; sq]; Union @@ (psq[6, #] & /@ Range[13]) (* Giovanni Resta, Aug 01 2018 *)
Missing a(12), a(13), a(16) and more terms from
Giovanni Resta, Aug 01 2018
A071133
Squares which repeat with at least two full periods when written in base 7.
Original entry on oeis.org
16, 400, 900, 1600, 29584, 70627216, 394618225, 2214643600, 4982948100, 8858574400, 16052636601, 64210546404, 169556179984, 727092173809, 1120597967889, 2908368695236, 6543829564281, 6703226505481, 79822717134736, 407103419654416, 12766762698010000
Offset: 1
19865^2 = 394618225 = 12531|12531|1 in base 7.
411772^2 = 169556179984 = 1515152|1515152 in base 7.
-
psq[b_, nd_] := Block[{sq={}, r, x}, Do[r = Mod[nd, p]; Do[x = (b^( Floor[nd/p] p) - 1)/(b^p - 1) t b^r + Floor[t/b^(p - r)]; If[ IntegerQ@ Sqrt@ x, AppendTo[ sq, x]], {t, b^(p - 1), b^p - 1}], {p, nd/2}]; sq]; Union @@ (psq[7, #] & /@ Range[12]) (* Giovanni Resta, Aug 01 2018 *)
Showing 1-9 of 9 results.