A331185 a(n) = n - prime(A067004(n)), where A067004 is the ordinal transform of number of divisors of n (A000005).

-1, 0, 0, 2, 0, 4, 0, 5, 6, 5, 0, 10, 0, 7, 4, 14, 0, 15, 0, 15, 8, 5, 0, 22, 20, 7, 4, 21, 0, 27, 0, 21, 4, 3, -2, 34, 0, -3, -4, 35, 0, 35, 0, 31, 28, -1, 0, 46, 42, 31, -2, 29, 0, 43, -4, 43, -4, -9, 0, 58, 0, -9, 34, 62, -8, 49, 0, 37, -10, 51, 0, 69, 0, -9, 38, 35, -12, 55, 0, 77, 78, -15, 0, 79, -16, -17, -20, 59, 0, 83

Offset: 1

Author

Antti Karttunen, Jan 12 2020

Keywords

• sign
• look

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10201
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A000005, A000040 (positions of zeros), A067004, A331186, A331187.

Programs

Mathematica

b[_] = 0;
c[n_] := c[n] = With[{t = DivisorSigma[0, n]}, b[t] = b[t]+1];
a[n_] := n - Prime[c[n]];
Array[a, 105] (* Jean-François Alcover, Dec 20 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; };
v067004 = ordinal_transform(vector(up_to,n,numdiv(n)));
A067004(n) = v067004[n];
A331185(n) = (n - prime(A067004(n)));

Formula

a(n) = n - A000040(A067004(n)).