A344770 Ordinal transform of A011772.

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

Offset: 1

Author

Antti Karttunen, May 30 2021

Keywords

• nonn
• look

Comments

Number of values of k, 1 <= k <= n, with A011772(k) = A011772(n).

a(n) = 1 for all powers of primes, A000961. Ones occur on some other positions as well: 15, 22, 35, 38, 42, 44, 45, 46, 51, 52, 54, 65, ...

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000961, A011772, A344590.

Programs

Mathematica

A011772[n_] := For[m = 1, True, m++, If[Divisible[m(m+1)/2, n], Return[m]]];
b[_] = 0;
a[n_] := a[n] = With[{t = A011772[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 21 2021 *)

PARI

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
v344770 = ordinal_transform(vector(up_to,n,A011772(n)));
A344770(n) = v344770[n];

Formula

a(n) >= A344590(n).