A273279
Least perfect power that is the sum of two nonzero squares in exactly n ways.
Original entry on oeis.org
8, 125, 3125, 4225, 1953125, 48828125, 105625, 274625, 762939453125, 2640625, 476837158203125, 17850625, 1221025, 34328125, 186264514923095703125, 1650390625, 446265625, 1160290625, 41259765625, 4291015625, 45474735088646411895751953125, 30525625
Offset: 1
8 is a term because 8 = 2^3 = 2^2 + 2^2.
125 is a term because 125 = 5^3 = 2^2 + 11^2 = 5^2 + 10^2.
3125 is a term because 3125 = 5^5 = 10^2 + 55^2 = 25^2 + 50^2 = 38^2 + 41^2.
-
p = Select[Prime@ Range@ 90, Mod[#, 4] == 1 &]; f[w_] := Times @@ (Take[p, Length@w]^Reverse[w]); c[w_] := Floor[(1/2) Times @@ (w+1)];r[w_] := Block[{v, k = If[Length@w == 1, 1,2]}, While[(v = cn[k w]) < trg, k++]; If[v == trg, b = Min[b, f[k*w]]]; If[cn[w] < trg, r[Append[w, 1]]; v=w; v[[-1]]++; r[v]]]; a[1]=8; a[n_] := (b=Infinity; trg = n; r[{2}]; r[{1, 1}]; b); Array[a, 50] (* Giovanni Resta, May 19 2016 *)
A274686
Least number k such that k-th triangular number is the sum of two nonzero squares in exactly n ways.
Original entry on oeis.org
4, 40, 25, 145, 625, 169, 31249, 985, 2600, 2500, 87890625, 3649, 384199200, 15625, 33124, 6409
Offset: 1
a(2) = 40 because 40*41 / 2 = 820 = 6^2 + 28^2 = 12^2 + 26^2.
A273545
Least number k such that k*n is the sum of two nonzero squares in exactly n ways.
Original entry on oeis.org
2, 25, 975, 1105, 1625, 16575, 739375, 27625, 71825, 27625, 58093750, 480675, 93925, 8547175, 1077375, 801125, 76765625, 2082925, 783935546875, 801125, 942703125, 23740234375, 1404178750, 17784975, 12138425, 8300781250, 106229175, 700984375, 221252441406250
Offset: 1
a(2) = 25 because 25*2 = 50 is the least even number that is the sum of two nonzero squares in exactly 2 ways; 50 = 1^2 + 7^2 = 5^2 + 5^2.
-
nR[n_] := (SquaresR[2, n] + Plus @@ Pick[{-4, 4}, IntegerQ /@ Sqrt[{n, n/2}]])/8; a[n_] := Block[{k=1}, While[nR[n * k] != n, k++]; k]; Array[a, 10] (* Giovanni Resta, May 27 2016 *)
A273787
Least number k such that A001844(k) (sums of two consecutive squares) is the sum of two nonzero squares in exactly n ways.
Original entry on oeis.org
1, 6, 21, 23, 221, 78, 7278, 153, 703, 1653, 6695846, 496, 670758346, 8346, 1471, 1081
Offset: 1
a(2) = 6 from 6^2 + 7^2 = 2^2 + 9^2.
a(3) = 21 from 21^2 + 22^2 = 5^2 + 30^2 = 14^2 + 27^2.
a(4) = 23 form 23^2 + 24^2 = 4^2 + 33^2 = 9^2 + 32^2 = 12^2 + 31^2.
-
A025426(n)=my(v=valuation(n, 2), f=factor(n>>v), t=1); for(i=1, #f~, if(f[i, 1]%4>1, if(f[i, 2]%2, return(0)), t*=f[i, 2]+1)); if(t%2, t-(-1)^v, t)/2
a(n)=my(k=1); while(A025426(2*k*(k+1)+1)!=n, k++); k \\ Charles R Greathouse IV, Jun 03 2016
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