A013584 -id:A013584 - cp's OEIS frontend

A049045 Domain of A049044.

1, 2, 4, 5, 7, 10, 11, 14, 17, 19, 22, 23, 31, 34, 37, 38, 41, 46, 61, 62, 71, 73, 74, 77, 82, 89, 103, 113, 122, 131, 139, 142, 146, 154, 157, 163, 167, 173, 178, 191, 193, 197, 206, 211, 226, 227, 233, 239, 251, 257, 262, 263, 278, 283, 293, 307, 313, 314, 317

Offset: 1

David W. Wilson

nonn

Positive integers that divide some positive element of A003422.
From Robert Israel, Nov 14 2016: (Start)
Numbers n such that A013584(n) > 0.
If n is in the sequence, then so are all divisors of n. (End)

Robert Israel, Table of n, a(n) for n = 1..10000

A049044, A003422, A013584.

Complement of A275608.

filter:= proc(n) local t,r,m;
  r:= 1; t:= 1;
  for m from 1 do
    r:= r*m mod n;
    if r = 0 then return false fi;
    t:= t + r mod n;
    if t = 0 then return true fi;
  od;
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Nov 14 2016

okQ[n_] := Module[{t, r, m}, r = 1; t = 1; For[m = 1, True, m++, r = Mod[r*m, n]; If[r == 0, Return[False]]; t = Mod[t + r, n]; If[t == 0, Return[True]]]];
okQ[1] = True;
Select[Range[1000], okQ] (* Jean-François Alcover, Apr 10 2019, after Robert Israel *)

A275608 Numbers that divide no nonzero terms of A003422.

Original entry on oeis.org

3, 6, 8, 9, 12, 13, 15, 16, 18, 20, 21, 24, 25, 26, 27, 28, 29, 30, 32, 33, 35, 36, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 63, 64, 65, 66, 67, 68, 69, 70, 72, 75, 76, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Robert Israel, Nov 14 2016

nonn

Numbers k such that A013584(k) = 0.

If k is in the sequence, then so is every multiple of k.

			3 is in the sequence because A003422(1)=1 and A003422(2)=2 are not divisible by 3, and A003422(k) == 1 (mod 3) for k >= 3.
4 is not in the sequence because A003422(3) = 4 is divisible by 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003422, A013584.

Complement of A049045.

filter:= proc(n) local t,r,m;
  r:= 1; t:= 1;
  for m from 1 do
    r:= r*m mod n;
    if r = 0 then return true fi;
    t:= t + r mod n;
    if t = 0 then return false fi;
  od;
end proc:
select(filter, [$2..100]);

okQ[n_] := Module[{t, r, m}, r = 1; t = 1; For[m = 1, True, m++, r = Mod[r*m, n]; If[r == 0, Return[True]]; t = Mod[t + r, n]; If[t == 0, Return[False]]]];
Select[Range[2, 100], okQ] (* Jean-François Alcover, Apr 12 2019, after Robert Israel *)

A013585 Smallest m such that 1!+...+m! is divisible by 2n+1, or 0 if no such m exists.

Original entry on oeis.org

1, 2, 0, 0, 3, 4, 0, 0, 5, 0, 0, 12, 0, 7, 19, 0, 4, 0, 24, 0, 32, 19, 0, 0, 0, 5, 20, 0, 0, 0, 0, 0, 0, 20, 12, 0, 7, 0, 0, 57, 7, 0, 0, 19, 0, 0, 0, 0, 6, 8, 83, 0, 0, 15, 33, 24, 0, 0, 0, 0, 12, 32, 0, 38, 19, 9, 0, 0, 0, 23, 0, 0, 0, 0, 70, 71, 5, 0, 57, 20, 0, 17, 0, 0, 0, 0, 26, 0, 0, 0, 0, 0, 0, 0, 0, 28

Offset: 0

Michael R. Mudge (Amsorg(AT)aol.com), additional terms from Allan C. Wechsler

nonn

From Robert Israel, Nov 14 2016: (Start)
a(n) < 2*n for n > 1.
If a(n) = 0, then a((2*k+1)*n + k) = 0 for all k >= 0.
(End)

M. R. Mudge, Smarandache Notions Journal, University of Craiova, Vol. VII, No. 1, 1996.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A003422, A013584.

f:= proc(n) local t,r,m;
  r:= 1; t:= 0;
  for m from 1 do
    r:= r*m mod (2*n+1);
    if r = 0 then return 0 fi;
    t:= t + r mod (2*n+1);
    if t = 0 then return m fi;
  od;
end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Nov 14 2016

a[n_] := Module[{t, r, m}, r = 1; t = 0; For[m = 1, True, m++, r = Mod[r m, 2 n + 1]; If[r == 0, Return[0]]; t = Mod[t + r, 2 n + 1]; If[t == 0, Return[m]]]];
a[0] = 1;
a /@ Range[0, 100] (* Jean-François Alcover, Jul 19 2020, after Maple *)

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A049045 Domain of A049044.

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A275608 Numbers that divide no nonzero terms of A003422.

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A013585 Smallest m such that 1!+...+m! is divisible by 2n+1, or 0 if no such m exists.

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Maple

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