A009223 -id:A009223 - cp's OEIS frontend

A082064 Greatest common prime-divisor of phi(n) and sigma(n) = A000203(n); a(n)=1 if no common prime-divisor exists.

1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 3, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 1, 3, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 5, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 3, 1, 2, 2, 2, 3, 3

Offset: 1

Labos Elemer, Apr 07 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A000203, A006530, A009223.

Cf. also A082061, A082062, A082063, A082065, A082066, A082070.

ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; f1[x_] := EulerPhi[n]; f2[x_] := DivisorSigma[1, x]; Table[Max[Intersection[ba[f1[w]], ba[f2[w]]]], {w, 1, 128}]
(* Second program: *)
Array[If[CoprimeQ[#1, #2], 1, Max@ Apply[Intersection, Map[FactorInteger[#][[All, 1]] &, {#1, #2}]]] & @@ {EulerPhi@ #, DivisorSigma[1, #]} &, 105] (* Michael De Vlieger, Nov 03 2017 *)

A006530(n) = if(1==n, n, vecmax(factor(n)[, 1]));
A082064(n) = A006530(gcd(eulerphi(n), sigma(n))); \\ Antti Karttunen, Nov 03 2017

a(n) = A006530(A009223(n)). - Antti Karttunen, Nov 03 2017

Changed "was found" to "exists" in definition. - N. J. A. Sloane, Jan 29 2022

A009213 a(n) = gcd(d(n), phi(n)), where d is the number of divisors of n (A000005) and phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 4, 3, 4, 2, 2, 2, 2, 4, 1, 2, 6, 2, 2, 4, 2, 2, 8, 1, 4, 2, 6, 2, 8, 2, 2, 4, 4, 4, 3, 2, 2, 4, 8, 2, 4, 2, 2, 6, 2, 2, 2, 3, 2, 4, 6, 2, 2, 4, 8, 4, 4, 2, 4, 2, 2, 6, 1, 4, 4, 2, 2, 4, 8, 2, 12, 2, 4, 2, 6, 4, 8, 2, 2, 1, 4, 2, 12, 4, 2, 4, 8, 2, 12, 4, 2, 4, 2, 4, 4, 2, 6, 6, 1, 2, 8, 2, 8, 8

Offset: 1

David W. Wilson

nonn

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

Cf. A000005, A000010,

Cf. also A009191, A009195, A009223, A318459.

Table[GCD[DivisorSigma[0,n],EulerPhi[n]],{n,110}] (* Harvey P. Dale, May 12 2025 *)

A009213(n) = gcd(numdiv(n), eulerphi(n)); \\ Antti Karttunen, Sep 07 2018

A323406 Greatest common divisor of Product (p_i^e_i)-1 and Product (p_i^e_i)+1, when n = Product (p_i^e_i): a(n) = gcd(A047994(n), A034448(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 6, 8, 1, 2, 2, 2, 6, 4, 2, 2, 2, 2, 6, 2, 2, 2, 8, 2, 1, 4, 2, 24, 2, 2, 6, 8, 2, 2, 12, 2, 30, 4, 2, 2, 2, 2, 6, 8, 2, 2, 2, 8, 6, 4, 2, 2, 24, 2, 6, 16, 1, 12, 4, 2, 6, 4, 24, 2, 2, 2, 6, 8, 2, 12, 24, 2, 6, 2, 2, 2, 4, 4, 6, 8, 2, 2, 4, 8, 6, 4, 2, 24, 2, 2, 6, 40, 2, 2, 8, 2, 42, 48

Offset: 1

Antti Karttunen, Jan 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A034448, A047994.

Cf. also A009223, A066086, A126865, A322362, A323166, A323409.

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); };
A034448(n) = { my(f=factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A323406(n) = gcd(A034448(n), A047994(n));

a(n) = gcd(A034448(n), A047994(n)), where A034448 is unitary sigma, and A047994 is unitary phi.

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

A297170 a(n) = gcd(phi(n), sigma(n)-n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 4, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 4, 2, 4, 1, 4, 1, 2, 1, 1, 5, 4, 1, 1, 1, 2, 1, 2, 1, 6, 1, 20, 3, 2, 1, 4, 2, 1, 1, 2, 1, 6, 1, 8, 1, 4, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 1, 4, 1, 6, 1, 2, 2, 4, 1, 4, 1, 2, 1, 4, 1, 24, 3, 4, 5, 2, 1, 4, 1, 1, 3, 1, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, Mar 03 2018

nonn

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

Cf. A000010, A000203, A001065, A009223.

A297170(n) = gcd(eulerphi(n),sigma(n)-n);

a(n) = gcd(A000010(n), A001065(n)).

A307640 Least number k such that n divides gcd(sigma(k), phi(k), tau(k)).

Original entry on oeis.org

1, 3, 18, 15, 3344, 45, 24128, 30, 882, 3344, 1012736, 126, 1953792, 24128, 16200, 168, 452263936, 2016, 1852571648, 3344, 40768, 1012736, 27007123456, 420, 1490000, 1953792, 103968, 24128, 2739920699392, 30096, 8348342681600, 840, 9114624, 452263936, 6163776, 2016

Offset: 1

Marius A. Burtea, Apr 19 2019

nonn

For each n >= 1 there are infinitely many numbers s such that n divides sigma(s), phi(s) and tau(s).

From Dirichlet's theorem there are infinitely many numbers m for which the numbers p = n*m + 1 are prime. Then sigma(p^(n-1)), phi(p^(n-1)) and tau(p^(n-1)) numbers are divisible by n.

			For n = 2, sigma(3) = 4, phi(3) = 2, tau(3) = 4 are divisible by 2.
For n = 5, sigma(3344) = 7440, phi (3344) = 1440, tau (3344) = 20 are divisible by 5 and by 10.
For n = 11, sigma(1012736) = 2161632 = 11 * 196512, phi(1012736) = 11 * 43008, tau(1012736) = 11 * 4 are divisible by 11.

Laurențiu Panaitopol, Alexandru Gica, Arithmetic problems and number theory. Ideas and methods of solving, Ed. Gil, Zalău, 2006, ch. 13, p. 79, pr. 18. (in Romanian).

Cf. A000005, A000010, A000203, A009223, A222713.

for m in [1..16] do
      for n in [1..2000000] do
              if IsIntegral(SumOfDivisors(n)/m) and IsIntegral(EulerPhi(n)/m) and IsIntegral(NumberOfDivisors(n)/m) then
             m,n;
             break;
            end if;
      end for;
end for;

Array[Block[{i = 1}, While[Mod[GCD[DivisorSigma[1, i], EulerPhi@ i,DivisorSigma[0, i]], #] != 0, i++]; i] &, 16] (*Adaptation after A222713*)

isok(n,k) = ! frac(gcd(sigma(k), gcd(eulerphi(k), numdiv(k)))/n);
a(n) = my(k=1); while(!isok(n,k), k++); k; \\ Michel Marcus, Apr 20 2019

a(n) = {if(n==1,return(1)); my(res = oo, f = factor(n), hpf = f[#f~, 1]); forprime(p = 2, oo, if(p ^ (hpf - 1) > res, return(res)); forstep(i = p ^ (hpf - 1), res, p ^ (hpf - 1), if(isok(n, i), res = min(res, i);  next(2) ) ) ) } \\ uses isok from above \\ David A. Corneth, Apr 22 2019

More terms from David A. Corneth, Apr 21 2019

cp's OEIS Frontend

A082064 Greatest common prime-divisor of phi(n) and sigma(n) = A000203(n); a(n)=1 if no common prime-divisor exists.

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A009213 a(n) = gcd(d(n), phi(n)), where d is the number of divisors of n (A000005) and phi is Euler's totient function (A000010).

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A323406 Greatest common divisor of Product (p_i^e_i)-1 and Product (p_i^e_i)+1, when n = Product (p_i^e_i): a(n) = gcd(A047994(n), A034448(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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A297170 a(n) = gcd(phi(n), sigma(n)-n).

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A307640 Least number k such that n divides gcd(sigma(k), phi(k), tau(k)).

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