cp's OEIS Frontend

A295865 Numbers k, the smallest of at least 4 consecutive numbers x, for which phi(x) <= phi(x+1).

%I A295865 #45 Mar 29 2018 10:10:14
%S A295865 1,2,14,104,164,254,494,584,1484,2204,2534,2834,3002,3674,3926,4454,
%T A295865 4484,4784,4844,5186,5264,5312,5984,6104,7994,8294,8414,8774,8834,
%U A295865 9074,9164,9944,10004,10604,10724,11024,11684,11894,12254,13034,13064,13166,13364,13454,13754,14234,15344,15554,16184,16214
%N A295865 Numbers k, the smallest of at least 4 consecutive numbers x, for which phi(x) <= phi(x+1).
%C A295865 There are 3988536 terms below 2*10^9.
%C A295865 Up to a(3988356):
%C A295865 - a(1) is the only odd term.
%C A295865 - a(1) is the only term with 5 consecutive numbers where phi(k) <= phi(k+1).
%C A295865 - the only powers of 2 are a(1)=1 and a(2) = 2.
%C A295865 - of the residues of a(n) mod 10, 4 accounts for greater than 91%.
%C A295865 - if a(n) is divisible by 4, then phi(a(n)) is divisible by 4.
%C A295865 Numbers k such that A057000(k) >= 0 for 3 consecutive terms. - _Michel Marcus_, Mar 21 2018
%e A295865 14 is a term because phi(14) <= phi(15) <= phi(16) <= phi(17).
%e A295865 15 is not a term because phi(15) <= phi(16) <= phi(17) > phi(18).
%p A295865 Phi:= map(numtheory:-phi, [$1..20001]):
%p A295865 DPhi:= Phi[2..-1]-Phi[1..-2]:
%p A295865 C:= select(t -> DPhi[t]>=0, [$1..20000]):
%p A295865 C[select(t -> C[t+2]=C[t]+2, [$1..nops(C)-3])]; # _Robert Israel_, Mar 26 2018
%t A295865 Drop[#, -2] & /@ Select[SplitBy[#, Last@ # >= 0 &], Length@ # > 2 && #[[1, -1]] >= 0 &][[All, All, 1]] &@ MapIndexed[{First@ #2, #1} &, Differences@ Array[EulerPhi, 2^14]] // Flatten (* _Michael De Vlieger_, Mar 26 2018 *)
%o A295865 (PARI) isok(n) = {my(v = vector(4, k, eulerphi(n+k-1))); (v[1] <= v[2]) && (v[2] <= v[3]) && (v[3] <= v[4]);} \\ _Michel Marcus_, Mar 21 2018
%Y A295865 Cf. A000010, A057000.
%K A295865 nonn
%O A295865 1,2
%A A295865 _Torlach Rush_, Feb 13 2018