A082123 Smallest difference > 1 between d and p/d for any divisor d of the partial product p of the sequence, starting with 17.

17, 16, 26, 36, 76, 94, 432, 37220, 996768, 158267352, 973348166592, 8429202561226344, 419324164827901536306744, 339991740461303603766175692597227316, 12025891484499365294341150949542442100059557280661504

Offset: 1

Author

Ralf Stephan, Apr 04 2003

Keywords

• nonn
• hard
• more

Crossrefs

Cf. A082120, A003681 (starts with 2, 3), A082124.

Programs

Maple

branch:= proc(m,dm, bestyet)
    local t,x, nby,r;
    nby:= bestyet;
    for t from F[m][2] by -1 to 0 do
      x:= dm*F[m][1]^t;
      if x >= nby then next
      elif x >= c then nby:= x
      elif (x*R[m] < c) or (m=nF) then break
      else nby:= branch(m+1,x,nby);
      fi
    od;
    nby
end proc:
P:= 17: A[1]:= 17:
for n from 2 to 15 do
  c:= ceil(1/2+1/2*sqrt(5+4*P));
  while not type(c,integer) do Digits:= 2*Digits; c:= eval(c) od:
  F:= ifactors(P)[2]; nF:= nops(F);
  F:= sort(F,(s,t)->s[1]>t[1]);
  R:= [seq(mul(F[i][1]^F[i][2],i=j+1..nF),j=1..nF)];
  d:= branch(1,1,P);
  A[n]:= d - P/d;
  P:= P*A[n]
od:
seq(A[n],n=1..15); # Robert Israel, May 20 2015

PARI

p=17; print1(p,","); for(n=1,13,r=floor(sqrt(p)); d1=1; d2=1; nE=omega(p); P=factor(p); E=P[,2]; P=P[,1]; forvec(v=vector(nE,i,[0,E[i]]),x=prod(k=1,nE,P[k]^v[k]); if(x<=r && x>=d2,d1=d2; d2=x,if(x<=d2 && x>=d1,d1=x))); difer=p/d2-d2; if(difer<=1,difer=p/d1-d1); print1(difer","); p*=difer)

Extensions

a(12) from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 02 2006

a(13) from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 14 2006

a(14) and a(15) from Robert Israel, May 20 2015