A051853 Table of solutions to all possible Chinese Remainder Equations x = a1 mod p1, x = a2 mod p2, ..., x = an mod pn, where p1 - pn are the first n primes and each a1 - an varies between 1 and (its respective) p-1, with the leftmost a varying fastest.
1, 1, 5, 1, 11, 7, 17, 13, 23, 19, 29, 1, 71, 127, 197, 43, 113, 169, 29, 121, 191, 37, 107, 163, 23, 79, 149, 31, 101, 157, 17, 73, 143, 199, 59, 151, 11, 67, 137, 193, 53, 109, 179, 61, 131, 187, 47, 103, 173, 19, 89, 181, 41, 97, 167, 13, 83, 139, 209, 1, 1541
Offset: 1
Examples
Rows have lengths 1,2,8,48,480,5760,92160,... (A005867(n)) and terms 1; 1,5; 1,11,7,17,13,23,19,29; 1,71,127,197,43,113,169,29,121,191,37,107,163,23,79,149,31,101,157,17,73,143,199,59,151,11,67,137,193,53,109,179,61,131,187,47,103,173,19,89,181,41,97,167,13,83,139,209;
Crossrefs
Cf. A051854.
Programs
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Maple
with(numtheory); incr_plist_from_left := proc(aa) local i,n,a; a := aa; n := nops(a); for i from 1 to n do if(a[i] < (ithprime(i)-1)) then a[i] := a[i]+1; RETURN(a); else a[i] := 1; fi; od; RETURN([op(a),1]); end; incr_plist_from_left_n_times := proc(aa,n) local a,i; a := aa; for i from 1 to n do a := incr_plist_from_left(a); od; RETURN(a); end; form_modlist := proc(a) local b,i; b := []; for i from 1 to nops(a) do b := [op(b),ithprime(i)]; od; RETURN(b); end; prim_chrem_left := proc(n) local r,m; r := incr_plist_from_left_n_times([],n); m := form_modlist(r); RETURN(chrem(r,m)); end;
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Mathematica
row[n_] := Module[{i}, pp = Prime[Range[n]]; iter = Sequence @@ Table[{i[k], 1, pp[[k]] - 1}, {k, n, 1, -1}]; Table[ChineseRemainder[Array[i, n], pp], iter // Evaluate] // Flatten]; Table[row[n], {n, 1, 5}] // Flatten (* Jean-François Alcover, Mar 06 2016 *)
Formula
a(n) = prim_chrem_left(n) (see Maple code)