A303759 -id:A303759 - cp's OEIS frontend

A330753 Number of values of k, 1 <= k <= n, with A309639(k) = A309639(n), where A309639 gives the index of the least harmonic number whose denominator is divisible by n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 3, 2, 4, 5, 1, 2, 3, 2, 1, 6, 1, 4, 7, 2, 1, 2, 1, 2, 3, 4, 1, 2, 5, 3, 3, 2, 1, 6, 1, 2, 8, 1, 5, 6, 1, 4, 1, 5, 1, 9, 1, 2, 3, 4, 7, 6, 1, 3, 1, 2, 1, 10, 5, 2, 3, 8, 1, 11, 7, 3, 3, 2, 5, 2, 1, 2, 9, 4, 1, 6, 1, 8, 12

Offset: 1

Antti Karttunen, Dec 30 2019

nonn

Ordinal transform of A309639.

For all n, a(A000961(n)) = 1, but the sequence obtains value 1 also on other n that are not prime powers. In range 1..65537 these extra 1's occur at n = 69, 201, 407, 505, 576, 869, 1791, 5157, 9383, 9691, 10219, 10571, 10991, 12575, 12731, 13343, 13739, 14179, 14483, 14693, 16173, 16723, 23347, 24209, 26233, 26377, 37393, 44407, 46089, 53707, 62063.

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

Cf. A000961, A309639.

Cf. also A303759, A330754.

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; };
v330753 = ordinal_transform(vector(up_to, n, A309639(n)));
A330753(n) = v330753[n];

A330755 Number of values of k, 1 <= k <= n, with A034684(k) = A034684(n), where A034684 gives the smallest unitary divisor of n that is larger than 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 3, 1, 1, 5, 1, 2, 4, 6, 1, 5, 1, 7, 1, 3, 1, 8, 1, 1, 6, 9, 2, 4, 1, 10, 7, 3, 1, 11, 1, 5, 4, 12, 1, 8, 1, 13, 9, 6, 1, 14, 5, 2, 10, 15, 1, 11, 1, 16, 3, 1, 6, 17, 1, 7, 12, 18, 1, 2, 1, 19, 13, 8, 4, 20, 1, 7, 1, 21, 1, 14, 8, 22, 15, 3, 1, 23, 5, 9, 16, 24, 9, 17, 1, 25, 2, 10, 1, 26, 1, 4, 18

Offset: 1

Antti Karttunen, Dec 30 2019

nonn

Ordinal transform of A034684.

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

Cf. A034684, A078898, A303759.

A034684[n_] := With[{f = FactorInteger[n]}, Min[f[[All, 1]]^f[[All, 2]]]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A034684[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 10 2022 *)

up_to = 65537;
A034684(n) = if(1==n,n,my(f=factor(n)); vecmin(vector(#f[, 1], i, f[i, 1]^f[i, 2]))); \\ After code in A034699
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; };
v330755 = ordinal_transform(vector(up_to, n, A034684(n)));
A330755(n) = v330755[n];

cp's OEIS Frontend

A330753 Number of values of k, 1 <= k <= n, with A309639(k) = A309639(n), where A309639 gives the index of the least harmonic number whose denominator is divisible by n.

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