cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A367871 a(n) is the least prime q such that A367798(n)^2 is the sum of q and its reversal.

Original entry on oeis.org

2, 29, 20147, 261104399, 20870609999, 2059108419947, 2080783998959, 200000140570007, 206841324099959, 261359249999999, 20401390509044927, 20421109564999967, 20000105691609287, 27180442947919997, 20105039549690939, 22040085159209699, 24000605788991999, 2008220921060899607, 2008804724799599927
Offset: 1

Views

Author

Robert Israel, Dec 03 2023

Keywords

Comments

a(n) is the first term q of A367796 such that A056964(q) = A367798(n)^2.

Examples

			a(4) = 261104399 because A367798(3) = 35419 and 35419^2 = 1254505561 = 261104399 + 993401162 and 261104399 is the first prime that works.
		

Crossrefs

Programs

  • Maple
    f:= proc(n) local y, c, d, dp, i, delta, m;
     y:= convert(n^2, base, 10);
     d:= nops(y);
     if d::even then
        if y[-1] <> 1 then return false fi;
        dp:= d-1;
        y:= y[1..-2];
        c[dp]:= 1;
     else
        dp:= d;
        c[dp]:= 0;
     fi;
     c[0]:= 0;
     for i from 1 to floor(dp/2) do
        delta:= y[i] - y[dp+1-i] - c[i-1] - 10*c[dp+1-i];
        if delta = 0 then c[dp-i]:= 0; c[i]:= 0;
        elif delta = -1 then c[dp-i]:= 1; c[i]:= 0;
        elif delta = -10 then c[dp-i]:= 0 ; c[i]:= 1;
        elif delta = -11 then c[dp-i]:= 1; c[i]:= 1;
        else return false
        fi;
        if y[i] + 10*c[i] - c[i-1] < 0  or (i=1 and y[i]+10*c[i]-c[i-1]=1) then return false fi;
      od;
      m:= (dp+1)/2;
      delta:= y[m] + 10*c[m] - c[m-1];
      if not member(delta, [seq(i, i=0..18, 2)]) then return false fi;
      [seq(y[i]+ 10*c[i]-c[i-1], i=1..m)]
    end proc:
    g:= proc(L) local T, d, t, p, x, i; uses combinat;
      d:= nops(L);
      T:= cartprod([select(t -> t[1]::odd, [seq([L[1]-x, x], x=max(1, L[1]-9)..min(L[1], 9))]),
        seq([seq([L[i]-x, x], x=max(0, L[i]-9)..min(9, L[i]))], i=2..d-1)]);
      while not T[finished] do
        t:= T[nextvalue]();
        p:= add(t[i][1]*10^(i-1), i=1..d-1) + L[-1]/2 * 10^(d-1) +
          add(t[i][2]*10^(2*d-i-1), i=1..d-1);
        if isprime(p) then return p fi;
      od;
    -1
    end proc:
    p:= 11: Q:=29:
    while p < 10^8 do
      p:= nextprime(p);
      d:= 1+ilog10(p^2);
      if d::even and p^2 >= 2*10^(d-1) then p:= nextprime(floor(10^(d/2)));  fi;
      v:= f(p);
      if v = false then next fi;
      q:= g(v);
      if q = -1 then next fi;
      Q:= Q, q;
    od:
    Q;

Formula

A056964(a(n)) = A367798(n)^2.

A367900 a(n) is the greatest prime q such that A367798(n)^2 is the sum of q and its reversal.

Original entry on oeis.org

2, 83, 81131, 894500063, 88990607813, 8499228209501, 8597793891803, 800072140300001, 859981720058603, 899969843983163, 82943190509220401, 86999838571212401, 88290616680100001, 89991996902408171, 83909667566050103, 89690298128004023, 89919974791600043, 8069990701280128001, 8299959944574088001
Offset: 1

Views

Author

Robert Israel, Dec 04 2023

Keywords

Comments

a(n) is the last term q of A367796 such that A056964(q) = A367798(n)^2.

Examples

			a(4) = 894500063 because A367798(3) = 35419 and 35419^2 = 1254505561 = 894500063 + 360005498 and 894500063 is the greatest prime that works.
		

Crossrefs

Programs

  • Maple
    f:= proc(n) local y, c, d, dp, i, delta, m;
     y:= convert(n^2, base, 10);
     d:= nops(y);
     if d::even then
        if y[-1] <> 1 then return false fi;
        dp:= d-1;
        y:= y[1..-2];
        c[dp]:= 1;
     else
        dp:= d;
        c[dp]:= 0;
     fi;
     c[0]:= 0;
     for i from 1 to floor(dp/2) do
        delta:= y[i] - y[dp+1-i] - c[i-1] - 10*c[dp+1-i];
        if delta = 0 then c[dp-i]:= 0; c[i]:= 0;
        elif delta = -1 then c[dp-i]:= 1; c[i]:= 0;
        elif delta = -10 then c[dp-i]:= 0 ; c[i]:= 1;
        elif delta = -11 then c[dp-i]:= 1; c[i]:= 1;
        else return false
        fi;
        if y[i] + 10*c[i] - c[i-1] < 0  or (i=1 and y[i]+10*c[i]-c[i-1]=1) then return false fi;
      od;
      m:= (dp+1)/2;
      delta:= y[m] + 10*c[m] - c[m-1];
      if not member(delta, [seq(i, i=0..18, 2)]) then return false fi;
      [seq(y[i]+ 10*c[i]-c[i-1], i=1..m)]
    end proc:
    g:= proc(L) local T, d, t, p,  x, i; uses combinat;
      d:= nops(L);
      T:= cartprod([select(t -> t[1]::odd, [seq([L[1]-x, x], x=min(L[1], 9)..max(1, L[1]-9),-1)]),
        seq([seq([L[i]-x, x], x=min(9, L[i])..max(0, L[i]-9),-1)], i=2..d-1)]);
      while not T[finished] do
        t:= T[nextvalue]();
        p:= add(t[i][1]*10^(i-1), i=1..d-1) + L[-1]/2 * 10^(d-1) +
          add(t[i][2]*10^(2*d-i-1), i=1..d-1);
        if isprime(p) then return p fi;
      od;
    -1
    end proc:
    p:= 2, 11: Q:= 83:
     while p < 10^10 do
      p:= nextprime(p);
      d:= 1+ilog10(p^2);
      if d::even and p^2 >= 2*10^(d-1) then p:= nextprime(floor(10^(d/2)));  fi;
      v:= f(p);
      if v = false then next fi;
      q:= g(v);
      if q = -1 then next fi;
      Q:= Q, q;
    od:
    Q;

Formula

A056964(a(n)) = A367798(n)^2.
Showing 1-2 of 2 results.