A115872 Square array where row n gives all solutions k > 0 to the cross-domain congruence n*k = A048720(A065621(n),k), zero sequence (A000004) if no such solutions exist.
1, 2, 1, 3, 2, 3, 4, 3, 6, 1, 5, 4, 7, 2, 7, 6, 5, 12, 3, 14, 3, 7, 6, 14, 4, 15, 6, 7, 8, 7, 15, 5, 28, 7, 14, 1, 9, 8, 24, 6, 30, 12, 15, 2, 15, 10, 9, 28, 7, 31, 14, 28, 3, 30, 7, 11, 10, 30, 8, 56, 15, 30, 4, 31, 14, 3, 12, 11, 31, 9, 60, 24, 31, 5, 60, 15, 6, 3, 13, 12, 48, 10, 62, 28, 56, 6, 62, 28, 12, 6, 5, 14, 13, 51, 11, 63, 30, 60, 7, 63, 30, 15, 7, 10, 7
Offset: 1
Examples
Fifteen initial terms of rows 1 - 19 are listed below: 1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... 2: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... 3: 3, 6, 7, 12, 14, 15, 24, 28, 30, 31, 48, 51, 56, 60, 62, ... 4: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... 5: 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, ... 6: 3, 6, 7, 12, 14, 15, 24, 28, 30, 31, 48, 51, 56, 60, 62, ... 7: 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, ... 8: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... 9: 15, 30, 31, 60, 62, 63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ... 10: 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, ... 11: 3, 6, 12, 15, 24, 27, 30, 31, 48, 51, 54, 60, 62, 63, 96, ... 12: 3, 6, 7, 12, 14, 15, 24, 28, 30, 31, 48, 51, 56, 60, 62, ... 13: 5, 10, 15, 20, 21, 30, 31, 40, 42, 45, 47, 60, 61, 62, 63, ... 14: 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, ... 15: 15, 30, 31, 60, 62, 63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ... 16: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... 17: 31, 62, 63, 124, 126, 127, 248, 252, 254, 255, 496, 504, 508, 510, 511, ... 18: 15, 30, 31, 60, 62, 63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ... 19: 7, 14, 28, 31, 56, 62, 63, 112, 119, 124, 126, 127, 224, 238, 248, ...
Links
Crossrefs
A few odd-positioned rows: row 1: A000027, Row 3: A048717, Row 5: A115770 (? Checked for all values less than 2^20), Row 7: A115770, Row 9: A115801, Row 11: A115803, Row 13: A115772, Row 15: A115801 (? Checked for all values less than 2^20), Row 17: A115809, Row 19: A115874, Row 49: A114384, Row 57: A114386.
Programs
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Mathematica
X[a_, b_] := Module[{A, B, C, x}, A = Reverse@IntegerDigits[a, 2]; B = Reverse@IntegerDigits[b, 2]; C = Expand[ Sum[A[[i]]*x^(i-1), {i, 1, Length[A]}]* Sum[B[[i]]*x^(i-1), {i, 1, Length[B]}]]; PolynomialMod[C, 2] /. x -> 2]; T[n_, k_] := Module[{x = BitXor[n-1, 2n-1], k0 = k}, For[i = 1, True, i++, If[n*i == X[x, i], If[k0 == 1, Return[i], k0--]]]]; Table[T[n-k+1, k], {n, 1, 14}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 04 2022 *) -
PARI
up_to = 120; A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2); A065621(n) = bitxor(n-1,n+n-1); A115872sq(n, k) = { my(x = A065621(n)); for(i=1,oo,if((n*i)==A048720(x,i),if(1==k,return(i),k--))); }; A115872list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A115872sq(col,(a-(col-1))))); (v); }; v115872 = A115872list(up_to); A115872(n) = v115872[n]; \\ (Slow) - Antti Karttunen, May 08 2019
Extensions
Example section added and the data section extended up to n=105 by Antti Karttunen, May 08 2019
Comments