A371415 -id:A371415 - cp's OEIS frontend

A371413 Dedekind psi function applied to the cubefull numbers (A036966).

1, 12, 24, 36, 48, 96, 108, 150, 192, 432, 324, 384, 392, 864, 768, 750, 1296, 972, 1728, 1800, 1536, 2592, 1452, 3456, 3888, 3600, 3072, 2916, 2366, 2744, 5184, 4704, 3750, 5400, 6912, 7776, 7200, 6144, 5202, 9000, 10368, 9408, 11664, 8748, 7220, 13824, 15552

Offset: 1

Amiram Eldar, Mar 22 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001615, A036966, A065464.

Similar sequences: A323332, A371412, A371415.

psi[n_] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); psi[1] = 1; Join[{1}, psi /@ Select[Range[20000], AllTrue[Last /@ FactorInteger[#], #1 > 2 &] &]]
(* or *)
f[n_] := Module[{f = FactorInteger[n], p, e}, If[n == 1, 1, p = f[[;;, 1]]; e = f[[;;, 2]]; If[Min[e] > 2, Times @@ ((p+1) * p^(e-1)), Nothing]]]; Array[f, 20000]

dedpsi(f) = prod(i = 1, #f~, (f[i, 1] + 1) * f[i, 1]^(f[i, 2]-1));
lista(max) = {my(f); print1(1, ", "); for(k = 2, max, f = factor(k); if(vecmin(f[, 2]) > 2, print1(dedpsi(f), ", "))); }

a(n) = A001615(A036966(n)).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/((p^2-1)*p)) = 1.231291... (A065487).

A371414 Euler phi function applied to the cubefull exponentially odd numbers (A335988).

Original entry on oeis.org

1, 4, 18, 16, 100, 64, 72, 162, 294, 256, 288, 400, 1210, 648, 1024, 1458, 2028, 1176, 2500, 1800, 1152, 1600, 4624, 6498, 2592, 4096, 5292, 4840, 4704, 11638, 4608, 6400, 14406, 5832, 8112, 13122, 23548, 10000, 7200, 28830, 16200, 10368, 16384, 21780, 18496, 19360

Offset: 1

Amiram Eldar, Mar 22 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A098198, A335988.

Similar sequences: A323333, A371414, A371415.

Join[{1}, EulerPhi /@ Select[Range[20000], AllTrue[Last /@ FactorInteger[#], #1 > 1 && OddQ[#1] &] &]]

lista(max) = {my(f, ans); print1(1, ", "); for(k = 2, max, f = factor(k); ans = 1; for (i = 1, #f~, if (f[i, 2] == 1 || !(f[i, 2] % 2), ans = 0; break)); if(ans, print1(eulerphi(f), ", ")));}

a(n) = A000010(A335988(n)).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/((p-1)^2*(p+1))) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 1/p^3 + 2/p^4) = 1.43921640806700099050... .

A374457 The Dedekind psi function values of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 4, 6, 12, 8, 12, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 48, 42, 36, 30, 72, 32, 48, 48, 54, 48, 38, 60, 56, 72, 42, 96, 44, 72, 48, 72, 54, 108, 72, 96, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 144, 90

Offset: 1

Amiram Eldar, Jul 09 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001615, A065463, A268335.
Similar sequences related to psi: A000082, A033196, A323332, A371413, A371415.
Similar sequences related to exponentially odd numbers: A366438, A366439, A366534, A366535, A367417, A368711, A374456.

f[p_, e_] := If[OddQ[e], (p+1) * p^(e-1), 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100], # > 0 &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, (f[i, 1]+1) * f[i, 1]^(f[i, 2] - 1), 0));}
lista(kmax) = {my(s1); for(k = 1, kmax, s1 = s(k); if(s1 > 0, print1(s1, ", ")));}

a(n) = A001615(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 / A065463^2 = 2.01515877170903249510... .

cp's OEIS Frontend

A371413 Dedekind psi function applied to the cubefull numbers (A036966).

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A371414 Euler phi function applied to the cubefull exponentially odd numbers (A335988).

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A374457 The Dedekind psi function values of the exponentially odd numbers (A268335).

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