A303759 Number of times the largest prime power factor of n (A034699) is largest prime power factor for numbers <= n; a(1) = 1.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 2, 1, 2, 1, 4, 1, 5, 1, 1, 3, 2, 5, 3, 1, 2, 3, 3, 1, 6, 1, 4, 4, 2, 1, 2, 1, 2, 3, 4, 1, 2, 5, 4, 3, 2, 1, 6, 1, 2, 5, 1, 5, 6, 1, 4, 3, 7, 1, 6, 1, 2, 3, 4, 7, 6, 1, 3, 1, 2, 1, 8, 5, 2, 3, 8, 1, 7, 7, 4, 3, 2, 5, 2, 1, 2, 9, 4, 1, 6, 1, 8, 9

Offset: 1

Author

Antti Karttunen, Apr 30 2018

Keywords

• nonn

Comments

Ordinal transform of A034699.

Links

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

Crossrefs

Cf. A000961 (positions of ones), A034699.

Cf. also A078899, A284600, A302789.

Programs

Maple

b:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= max(1, seq(i[1]^i[2], i=ifactors(n)[2]));
      b(t):= b(t)+1
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Apr 30 2018

Mathematica

f[n_] := Max[Power @@@ FactorInteger[n]];
b[_] = 0;
a[n_] := With[{t = f[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Jan 03 2022 *)

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; };
A034699(n) = if(1==n,n,fordiv(n, d, if(isprimepower(n/d), return(n/d))));
v303759 = ordinal_transform(vector(up_to,n,A034699(n)));
A303759(n) = v303759[n];