A393434 a(0) = 0; thereafter a(n) is the least integer (in absolute value) not yet in the sequence such that the absolute difference between a(n-1) and a(n) is a factorial number k!, k >= 1; in case of a tie, preference is given to the positive value.

0, 1, -1, -2, -3, 3, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, -12, -6, -4, -5, -7, -8, -9, -10, -11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, -60, -36, -30, -24, -18, -16, -14, -13, -15, -17, -19, -20, -21, -22, -23

Offset

0,4

Comments

This sequence could be called the Heraclitus transform of the factorials. For other examples of Heraclitus transforms, see A377090, A377091, A377092, A379719.

It is conjectured that every integer appears in this sequence. (It appears that the proof in A379719 can be modified to apply here. The start is somewhat irregular, but beginning at n = 25, the positive terms appear in blocks of consecutive numbers, where block B(k), k >= 4, has length k*k!/2, starts at n = k!/2 + 1 and ends at n = (k+1)!/2. The structure of the negative blocks is less clear.)

Links

Paolo Xausa, Table of n, a(n) for n = 0..9000 (terms 0..4000 from N. J. A. Sloane).

Mathematica

A393434list[nmax_] := Module[{s, a, u = 1, factQ},
  factQ[n_] := factQ[n] = Module[{d = 2, x = n}, While[Divisible[x, d], x /= d++]; x == 1];
  s[_] := False; s[0] = True;
  NestList[(While[s[u] && s[-u], u++]; a = u; While[s[a] || !factQ[Abs[# - a]], a = Boole[a < 0] - a]; s[a] = True; a) &, 0, nmax]];
A393434list[100] (* Paolo Xausa, Feb 17 2026 *)
Module[{F=Table[k!, {k, 1, 10}]}, FoldList[Function[a, Append[a, First@SortBy[Select[Flatten[{Last[a]+F, Last[a]-F}], FreeQ[a, #]&], {Abs[#]&, -#&}]]], {0}, Range[200]][[-1]]] (* Vincenzo Librandi, Feb 18 2026 *)

Prog

(Magma) Nmax := 200; a := [ Integers() | 0 ]; used := { 0 }; F := [ Factorial(n) : n in [1..10] ]; for n in [1..Nmax] do S := { a[#a] + f : f in F } join { a[#a] - f : f in F }; S := { x : x in S | not x in used }; t := Minimum({ <Abs(x), -x, x> : x in S }); m := t[3]; Append(~a, m); Include(~used, m); end for; a; // Vincenzo Librandi, Feb 18 2026

Crossrefs

Cf. A000142, A377090, A377091, A377092, A379719.

Cf. A392317 (partial sums).

Sequence in context: A182865 A131469 A309076 * A379777 A343326 A073078

Adjacent sequences: A393431 A393432 A393433 * A393435 A393436 A393437

Keyword

sign

Author

N. J. A. Sloane, Feb 17 2026

Status

approved