A009195 -id:A009195 - cp's OEIS frontend

A295313 a(n) = gcd(sigma(n), phi(sigma(n))).

1, 1, 2, 1, 2, 4, 4, 1, 1, 6, 4, 4, 2, 8, 8, 1, 6, 3, 4, 6, 16, 12, 8, 4, 1, 6, 8, 8, 2, 24, 16, 9, 16, 18, 16, 1, 2, 4, 8, 6, 6, 32, 4, 12, 6, 24, 16, 4, 3, 3, 24, 14, 18, 8, 24, 8, 16, 6, 4, 24, 2, 32, 8, 1, 12, 48, 4, 18, 32, 48, 24, 3, 2, 6, 4, 4, 32, 24, 16, 6, 11, 18, 12, 32, 36, 4, 8, 12, 6, 18, 16, 24, 64, 48, 8

Offset: 1

Antti Karttunen, Nov 22 2017

nonn

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

Cf. A000010, A000203, A009195, A295311, A295314, A295315.

a(n) = my(sn = sigma(n)); gcd(sn, eulerphi(sn)); \\ Michel Marcus, Nov 23 2017

a(n) = A009195(A000203(n)).

A295314 a(n) = sigma(n) / gcd(sigma(n), phi(sigma(n))).

Original entry on oeis.org

1, 3, 2, 7, 3, 3, 2, 15, 13, 3, 3, 7, 7, 3, 3, 31, 3, 13, 5, 7, 2, 3, 3, 15, 31, 7, 5, 7, 15, 3, 2, 7, 3, 3, 3, 91, 19, 15, 7, 15, 7, 3, 11, 7, 13, 3, 3, 31, 19, 31, 3, 7, 3, 15, 3, 15, 5, 15, 15, 7, 31, 3, 13, 127, 7, 3, 17, 7, 3, 3, 3, 65, 37, 19, 31, 35, 3, 7, 5, 31, 11, 7, 7, 7, 3, 33, 15, 15, 15, 13, 7, 7, 2, 3

Offset: 1

Antti Karttunen, Nov 22 2017

nonn

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

Cf. A000010, A000203, A009195, A109395, A295313, A295315.

a(n) = my(sn = sigma(n)); sn/gcd(sn, eulerphi(sn)); \\ Michel Marcus, Nov 23 2017

a(n) = A000203(n) / A295313(n) = A109395(A000203(n)).

A295315 a(n) = phi(sigma(n)) / gcd(sigma(n), phi(sigma(n))).

Original entry on oeis.org

1, 2, 1, 6, 1, 1, 1, 8, 12, 1, 1, 3, 3, 1, 1, 30, 1, 8, 2, 2, 1, 1, 1, 4, 30, 2, 2, 3, 4, 1, 1, 4, 1, 1, 1, 72, 9, 4, 3, 4, 2, 1, 5, 2, 4, 1, 1, 15, 12, 20, 1, 3, 1, 4, 1, 4, 2, 4, 4, 2, 15, 1, 6, 126, 2, 1, 8, 2, 1, 1, 1, 32, 18, 6, 15, 12, 1, 2, 2, 10, 10, 2, 2, 3, 1, 10, 4, 4, 4, 4, 3, 2, 1, 1, 4, 2, 3, 12, 4

Offset: 1

Antti Karttunen, Nov 22 2017

nonn

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

Cf. A000010, A000203, A009195, A062401, A076512, A295312, A295313, A295314.

a(n) = my(sn = sigma(n)); eulerphi(sn) / gcd(sn, eulerphi(sn)); \\ Michel Marcus, Nov 23 2017

a(n) = A062401(n) / A295313(n) = A062401(n) / A009195(A000203(n)).

a(n) = A076512(A000203(n)).

A318459 a(n) = gcd(n, tau(n), phi(n)), where tau = A000005 and phi = A000010.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 07 2018, after Labos Elemer's A074389

nonn

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

Cf. A000005, A000010, A009191, A009195, A009213.

Cf. also A074389.

Table[GCD[n,DivisorSigma[0,n],EulerPhi[n]],{n,110}] (* Harvey P. Dale, Jul 30 2019 *)

A318459(n) = gcd([n, numdiv(n), eulerphi(n)]);

a(n) = gcd(n, A000005(n), A000010(n)).

a(n) = gcd(n,A009213(n)) = gcd(A000005(n),A009195(n)) = gcd(A000010(n),A009191(n)).

A326198 Number of positive integers that are reachable from n with some combination of transitions x -> x-phi(x) and x -> gcd(x,phi(x)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 24 2019

nonn

			From n = 12 we can reach any of the following numbers > 0: 12 (with an empty sequence of transitions), 8 (as A051953(12) = 8), 4 (as A009195(12) = A009195(8) = A051953(8) = 4), 2 (as A009195(4) = A051953(4) = 2) and 1 (as A009195(2) = A051953(2) = 1), thus a(12) = 5.
The directed acyclic graph formed from those two transitions with 12 as its unique root looks like this:
    12
    / \
   |   8
    \ /
     4
     |
     2
     |
     1

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000010, A009195, A051953, A071575, A300243, A326195.

Cf. also A326189, A326196, A327160, A327161.

A326198aux(n,xs) = if(vecsearch(xs,n),xs, xs = setunion([n],xs); if(1==n,xs, my(a=gcd(n,eulerphi(n)), b=n-eulerphi(n)); xs = A326198aux(a,xs); if((a==b),xs, A326198aux(b,xs))));
A326198(n) = length(A326198aux(n,Set([])));

a(n) > max(A071575(n), A326195(n)).

A326203 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n > 2 and gcd(n,phi(n)) = 1, with f(n) = n for all other numbers.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 14, 15, 3, 16, 17, 18, 19, 20, 3, 21, 3, 22, 3, 23, 3, 24, 3, 25, 26, 27, 3, 28, 3, 29, 30, 31, 3, 32, 33, 34, 3, 35, 3, 36, 37, 38, 39, 40, 3, 41, 3, 42, 43, 44, 3, 45, 3, 46, 3, 47, 3, 48, 3, 49, 50, 51, 3, 52, 3, 53, 54, 55, 3, 56, 3, 57, 3, 58, 3, 59, 3, 60, 61, 62, 3, 63, 3, 64, 65, 66, 3, 67, 3

Offset: 1

Antti Karttunen, Jul 11 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A009195(i) = A009195(j) => A297086(i) = A297086(j),
  a(i) = a(j) => A000001(i) = A000001(j) => A297086(i) = A297086(j).

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

Cf. A000001, A000010, A003277, A009195, A297086, A305801.

Cf. also A326201, A326202.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux326203(n) = if((n>2) && (1==gcd(n,eulerphi(n))),0,n);
v326203 = rgs_transform(vector(up_to, n, Aux326203(n)));
A326203(n) = v326203[n];

A327979 a(n) = gcd(n, A002322(n)), where A002322 is Carmichael lambda, also known as psi.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 4, 1, 6, 1, 4, 3, 2, 1, 2, 5, 2, 9, 2, 1, 2, 1, 8, 1, 2, 1, 6, 1, 2, 3, 4, 1, 6, 1, 2, 3, 2, 1, 4, 7, 10, 1, 4, 1, 18, 5, 2, 3, 2, 1, 4, 1, 2, 3, 16, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 5, 2, 1, 6, 1, 4, 27, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 3, 2, 1, 8, 1, 14, 3, 20, 1, 2, 1, 4, 3

Offset: 1

Antti Karttunen, Oct 04 2019

nonn

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

Cf. A002322.

Cf. also A009191, A009194, A009195, A306695.

Array[GCD[#, CarmichaelLambda[#]] &, 100] (* Amiram Eldar, Oct 04 2019 *)

PARI
```
A327979(n) = gcd(n, lcm(znstar(n)[2]));
```

a(n) = gcd(n, A002322(n)).

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

A074391 a(n) is the smallest number such that gcd(a(n), sigma(a(n))) = n.

Original entry on oeis.org

1, 10, 15, 12, 95, 6, 91, 56, 153, 40, 473, 24, 117, 182, 135, 336, 1139, 90, 703, 380, 861, 946, 3151, 168, 3725, 468, 1431, 28, 5017, 570, 775, 992, 891, 2176, 4865, 792, 2701, 1406, 585, 280, 6683, 546, 11051, 1892, 1305, 6302, 13207, 528, 4753, 5800

Offset: 1

Labos Elemer, Aug 23 2002

nonn

a(n) is the smallest number k such that A017666(k), the denominator of sigma(k)/k, is equal to k/n. - Jaroslav Krizek, Sep 23 2014

Each term a(n) is divisible by its index n. - Michel Marcus, Jan 13 2015

			n=6: a(6)=6 because gcd(6, sigma(6))=6 and a(6)=6 is the smallest.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A000203, A017666, A073815, A050399, A009195, A009194.

A074391:=func; [A074391(n): n in[1..100]] // Jaroslav Krizek, Sep 23 2014

f:= proc(n) local k;
  for k from n by n do
    if igcd(k, numtheory:-sigma(k))=n then return k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 11 2020

f[x_] := GCD[DivisorSigma[1, x], x] t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 1000000}];

a(n) = my(k=1); while (gcd(sigma(k), k) != n, k++); k; \\ Michel Marcus, Jan 13 2015

a(n) = Min{x; gcd(x, sigma(x))} = Min{x; gcd(x, A000203(x))} = n. - corrected by Michel Marcus, Jan 13 2015

A098204 Greatest common divisor of multiperfect numbers and their totient.

Original entry on oeis.org

1, 2, 4, 8, 16, 96, 64, 864, 72, 1536, 252, 69120, 4096, 86400, 4608, 256, 768, 24576, 65536, 570240, 217728, 691200, 81920, 1105920, 262144, 245760, 516096, 9676800, 26544672, 50181120, 1741824, 2145024, 226437120, 103028889600, 12597120

Offset: 1

Labos Elemer, Sep 23 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..730

Cf. A000010, A007691, A009195, A098203.

a(n) = gcd( A007691(n), A000010(A007691(n)) ).

a(n) = A009195(A007691(n)). - Amiram Eldar, May 10 2024

More terms from Michel Marcus, Sep 19 2013

cp's OEIS Frontend

A295313 a(n) = gcd(sigma(n), phi(sigma(n))).

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A295314 a(n) = sigma(n) / gcd(sigma(n), phi(sigma(n))).

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A295315 a(n) = phi(sigma(n)) / gcd(sigma(n), phi(sigma(n))).

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A318459 a(n) = gcd(n, tau(n), phi(n)), where tau = A000005 and phi = A000010.

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A326198 Number of positive integers that are reachable from n with some combination of transitions x -> x-phi(x) and x -> gcd(x,phi(x)).

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A326203 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n > 2 and gcd(n,phi(n)) = 1, with f(n) = n for all other numbers.

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A327979 a(n) = gcd(n, A002322(n)), where A002322 is Carmichael lambda, also known as psi.

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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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A074391 a(n) is the smallest number such that gcd(a(n), sigma(a(n))) = n.

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