A081383 -id:A081383 - cp's OEIS frontend

A081396 Number of common prime factors (ignoring multiplicity) of sigma(n) = A000203(n) and phi(n) = A000010(n).

0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 1, 1, 1, 2, 2

Offset: 1

Labos Elemer, Mar 28 2003

nonn

			n=209: sigma(209) = 240 = 2*2*2*2*3*5, phi(209) = 180 = 2*2*3*3*5, common factor set = {2,3,5}, so a(209)=3.

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

Cf. A000010, A000203, A001221, A009223, A081383, A082055.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] Table[Length[Intersection[ba[EulerPhi[w]], ba[DivisorSigma[1, w]]]], {w, 1, 100}]

a(n)=omega(gcd(sigma(n),eulerphi(n))) \\ Charles R Greathouse IV, Feb 19 2013

a(n) = A001221(A009223(n)). - Antti Karttunen, Jan 22 2020

Data section extended up to a(105) by Antti Karttunen, Jan 22 2020

A081397 a(n) = least k such that GCD of sigma(k) = A000203(k) and phi(k) = A000010(k) equals n-th primorial = A002110(n).

Original entry on oeis.org

3, 14, 488, 16072, 284229, 29221621, 25014312576

Offset: 1

Labos Elemer, Mar 28 2003

a(8) <= 1524139588224. - Donovan Johnson, Dec 14 2009

			k=284229: sigma(284229) = 482790 = 2*3*5*7*11*19, phi(284229) = 166320 = 2*2*2*3*3*3*3*5*7*11, common factor set = {2,3,5,7,11}, gcd(sigma,phi) = 2310.

Cf. A000010, A000203, A002110, A081383, A081396.

a(n) = Min_{k : A081396(k) = A002110(n)}.

a(6) corrected and a(7) from Donovan Johnson, Dec 14 2009

A081398 Numbers k for which the number of common prime factors of sigma(k) and phi(k) is exactly six (ignoring multiplicity).

Original entry on oeis.org

2003639, 2179316, 2180057, 2382974, 2689561, 2720567, 2761873, 2933675, 3145572, 3435381, 3925463, 4007278, 4137111, 4212692, 4360114, 4947971, 5172881, 5379122, 5441134, 5458673, 5523746, 5675816, 5748831, 5867350, 5957435, 6010917, 6537948, 6540171, 6561511

Offset: 1

Labos Elemer, Mar 28 2003

nonn

Numbers k such that A081396(k) = 6. - Amiram Eldar, Mar 25 2024

			k = 400: sigma(400) = 6846840 = 2*2*2*3*3*5*7*11*13*19, phi(400) = 1755600 = 2*2*2*2*3*5*5*7*11*19, the common prime set = {2,3,5,7,11,19} with six primes.

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

Cf. A000010, A000203, A001221, A009223, A002110, A081383, A081396, A081397.

ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[ffi[x][[2*w - 1]], {w, 1, lf[x]}] ; Do[s = Length[Intersection[ba[EulerPhi[n]], ba[DivisorSigma[1, n]]]]; If[Greater[s, 5], Print[{n, s}]], {n, 1, 10000000}]

is(n) = {my(f = factor(n)); omega(gcd(sigma(f), eulerphi(f))) == 6;} \\ Amiram Eldar, Mar 25 2024

3925463 inserted and more terms added by Amiram Eldar, Mar 25 2024

cp's OEIS Frontend

A081396 Number of common prime factors (ignoring multiplicity) of sigma(n) = A000203(n) and phi(n) = A000010(n).

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A081397 a(n) = least k such that GCD of sigma(k) = A000203(k) and phi(k) = A000010(k) equals n-th primorial = A002110(n).

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A081398 Numbers k for which the number of common prime factors of sigma(k) and phi(k) is exactly six (ignoring multiplicity).

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