A393634 Numbers m such that there is no solution to the equation phi(k+m) = sigma(k).

0, 3, 11, 19, 51, 801, 2833

Offset

1,2

Comments

0, and m such that A094466(m) = -1.

If m is odd and k is an odd prime with phi(m+k) = sigma(k), then phi(m+k) <= (m+k)/2 while sigma(k) = k+1, so k <= m-2.

If k is composite and phi(m+k) = sigma(k), then phi(m+k) <= m+k-2 while sigma(k) >= 1 + k + sqrt(k), so k <= (m-3)^2.

This means that when m is odd, a finite search can determine that m is a term.

a(8) > 100000 if it exists.

Links

Table of n, a(n) for n=1..7.

Example

a(3) = 11 is a term because phi(11+k) <> sigma(k) for 1 <= k <= (11-3)^2 = 64.

Maple

f:= proc(n) local km, k; uses NumberTheory;
  km:= max(2, (n-3)^2);
  for k from 1 to km do
      if phi(n+k) = sigma(k) then return k fi;
  od;
  if n::odd then return -1 fi;
  k:= km;
  do
    k:= nextprime(k);
    if phi(n+k) = k+1 then return k fi
  od;
end proc:
f(0):= -1:
select(n -> f(n) = -1, [$0..3000]);

Mathematica

f[n_]:=Module[{km, k}, km=Max[2, (n-3)^2]; For[k=1, k<=km, k++, If[EulerPhi[n+k]==DivisorSigma[1, k], Return[k]]]; If[OddQ[n], Return[-1]]; k=km; While[True, k=NextPrime[k]; If[EulerPhi[n+k]==k+1, Return[k]]]; ]; f[0]=-1; Select[Range[0, 20000], f[#]==-1&] (* Vincenzo Librandi, Mar 30 2026 *)

Prog

(Magma) f := function(n) if n eq 0 then return -1; end if; km := Max(2, (n-3)^2); for k in [1..km] do if EulerPhi(n+k) eq SumOfDivisors(k) then return k; end if; end for; if IsOdd(n) then return -1; end if; k := km; repeat k := NextPrime(k); until EulerPhi(n+k) eq k+1; return k; end function; [n : n in [0..1000] | f(n) eq -1]; // Vincenzo Librandi, Mar 30 2026

Crossrefs

Cf. A094466.

Sequence in context: A309027 A213891 A336790 * A163851 A213051 A238362

Adjacent sequences: A393631 A393632 A393633 * A393635 A393636 A393637

Keyword

nonn,more

Author

Robert Israel, Mar 29 2026

Status

approved