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

Author

M. F. Hasler, Apr 07 2023

Keywords

• nonn
• tabf

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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]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])}