A361088 Irregular table, read by rows, where row n holds the tau signature of n, i.e., the shortest sequence (tau(n+k), 0 <= k <= m) that uniquely identifies n; tau = A000005.

1, 2, 2, 2, 3, 3, 2, 2, 4, 2, 4, 2, 4, 2, 4, 3, 4, 3, 3, 4, 2, 4, 2, 6, 2, 6, 2, 4, 6, 2, 4, 2, 4, 4, 5, 4, 4, 5, 4, 5, 5, 2, 2, 6, 2, 6, 6, 2, 6, 2, 6, 4, 4, 2, 6, 4, 4, 2, 8, 4, 4, 2, 8, 4, 2, 8, 2, 8, 3, 8, 3, 3, 4, 4, 6, 2, 4, 4, 6, 2, 8, 4, 6, 2, 8, 2, 6, 2, 8, 2, 6, 2, 8

Offset

1,2

Comments

Row lengths are given by A309981(n) + 1; see there (and the OEIS wiki page) for examples.

Links

Table of n, a(n) for n=1..93.

M. F. Hasler et al, tau signature, OEIS Wiki, April 2023.

Example

The first 20 rows read as follows:
   n | row n: tau-signature of n
  ---+--------------------------
   1 | [1]
   2 | [2, 2]
   3 | [2, 3]
   4 | [3, 2]
   5 | [2, 4, 2]
   6 | [4, 2, 4]
   7 | [2, 4, 3]
   8 | [4, 3]
   9 | [3, 4, 2]
  10 | [4, 2, 6]
  11 | [2, 6, 2, 4]
  12 | [6, 2, 4]
  13 | [2, 4, 4, 5]
  14 | [4, 4, 5]
  15 | [4, 5]
  16 | [5, 2]
  17 | [2, 6, 2, 6]
  18 | [6, 2, 6]
  19 | [2, 6, 4, 4, 2]
  20 | [6, 4, 4, 2, 8]
See the wiki page for proofs.

Prog

(PARI) signatures=Map(); LIMIT=10^5 /* This search limit should (possibly dynamically, or by hand) be increased as n grows beyond 100. As of today, the value for n=49 is not yet proven. */
A361088_row(n, s=0)={if(!s, s=iferr(mapget(signatures, n), E, []); #s|| for(L=1, oo, s=concat(s, numdiv(n+L-1)); A361088_row(n, s)|| [mapput(signatures, n, [s, LIMIT]); return(s)]); s[2]>=LIMIT&& return(s[1]); s=s[1]; while(A361088_row(n, s), s=concat(s, numdiv(n+#r))); mapput(signatures, n, [s, LIMIT]); return(s)); my(r=iferr(mapget(signatures, s), E, [])); if(!r, r=[n, n], r[2]<n, r=[r[1], n, n]; mapput(signatures, s, r); return(n), #r>2, return(r[#r-1]), r[#r]>=LIMIT, return); for(j=max(r[2], n)+1, LIMIT, for(k=1, #s, numdiv(j+k-1)!=s[k]&& next(2)); mapput(signatures, s, [n, j, j]); return(j)); mapput(signatures, s, [n, LIMIT])

Crossrefs

Cf. A309981, A327265, A161460, A000005 (tau = numdiv).

Sequence in context: A096916 A098014 A059957 * A165924 A212628 A272851

Adjacent sequences: A361085 A361086 A361087 * A361089 A361090 A361091

Keyword

nonn,tabf

Author

M. F. Hasler, Apr 07 2023

Status

approved