A331186 -id:A331186 - cp's OEIS frontend

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

Antti Karttunen, Jan 12 2020

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

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

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 *)

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)));

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

A331187 a(n) is n divided by the highest power of prime(A067004(n)) which divides it, where A067004 is the ordinal transform of number of divisors of n (A000005).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 8, 1, 2, 1, 3, 1, 2, 15, 1, 1, 2, 1, 4, 21, 22, 1, 3, 1, 26, 27, 4, 1, 10, 1, 32, 33, 34, 35, 9, 1, 38, 39, 8, 1, 6, 1, 44, 45, 46, 1, 3, 1, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 15, 1, 62, 63, 1, 65, 66, 1, 68, 69, 70, 1, 8, 1, 74, 75, 76, 77, 78, 1, 80, 1, 82, 1, 84, 85, 86, 87, 88, 1, 90

Offset: 1

Antti Karttunen, Jan 12 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000040, A067004, A331186.

Cf. also A331185.

b[_] = 0;
A067004[n_] := A067004[n] = With[{t = DivisorSigma[0, n]}, b[t] = b[t]+1];
A331186[n_] := IntegerExponent[n, Prime[A067004[n]]];
a[n_] := n/(Prime[A067004[n]]^A331186[n]);
Array[a, 105] (* Jean-François Alcover, Dec 21 2021 *)

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];
A331187(n) = { my(p=prime(A067004(n))); n/(p^valuation(n,p)); };

a(n) = n / (A000040(A067004(n))^A331186(n)).

cp's OEIS Frontend

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

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