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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.

%I A361088 #23 Mar 25 2025 08:56:15
%S A361088 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,
%T A361088 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,
%U A361088 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
%N 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.
%C A361088 Row lengths are given by A309981(n) + 1; see there (and the OEIS wiki page) for examples.
%H A361088 M. F. Hasler et al, <a href="/wiki/A309981">tau signature</a>, OEIS Wiki, April 2023.
%e A361088 The first 20 rows read as follows:
%e A361088    n | row n: tau-signature of n
%e A361088   ---+--------------------------
%e A361088    1 | [1]
%e A361088    2 | [2, 2]
%e A361088    3 | [2, 3]
%e A361088    4 | [3, 2]
%e A361088    5 | [2, 4, 2]
%e A361088    6 | [4, 2, 4]
%e A361088    7 | [2, 4, 3]
%e A361088    8 | [4, 3]
%e A361088    9 | [3, 4, 2]
%e A361088   10 | [4, 2, 6]
%e A361088   11 | [2, 6, 2, 4]
%e A361088   12 | [6, 2, 4]
%e A361088   13 | [2, 4, 4, 5]
%e A361088   14 | [4, 4, 5]
%e A361088   15 | [4, 5]
%e A361088   16 | [5, 2]
%e A361088   17 | [2, 6, 2, 6]
%e A361088   18 | [6, 2, 6]
%e A361088   19 | [2, 6, 4, 4, 2]
%e A361088   20 | [6, 4, 4, 2, 8]
%e A361088 See the wiki page for proofs.
%o A361088 (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. */
%o A361088 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])}
%Y A361088 Cf. A309981, A327265, A161460, A000005 (tau = numdiv).
%K A361088 nonn,tabf
%O A361088 1,2
%A A361088 _M. F. Hasler_, Apr 07 2023