A046678 -id:A046678 - cp's OEIS frontend

A046679 Numbers k such that the number of divisors of k and sum of squares of divisors of k are relatively prime.

1, 2, 8, 9, 16, 18, 64, 72, 81, 128, 144, 625, 729, 1024, 1152, 1296, 1458, 2401, 4096, 5184, 5625, 5832, 6561, 8192, 9216, 10000, 11664, 13122, 15625, 21609, 28561, 31250, 32768, 38416, 40000, 46656, 50625, 52488, 59049, 65536, 83521, 90000

Offset: 1

Labos Elemer

nonn

It can be shown that this is a subsequence of A028982.

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

Cf. A028982, A046678, A046680, A046681, A046683, A046685.

Select[Range[91000],CoprimeQ[DivisorSigma[0,#], DivisorSigma[2,#]]&] (* Harvey P. Dale, May 11 2011 *)

isok(n) = gcd(sigma(n, 2), numdiv(n)) == 1; \\ Michel Marcus, Sep 24 2019

a(1)=1 added by Amiram Eldar, Sep 24 2019

A046680 Numbers k such that the number of divisors of k and sum of cubes of divisors of k are relatively prime.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 25, 36, 81, 100, 121, 128, 144, 162, 225, 256, 289, 324, 400, 484, 512, 529, 625, 729, 841, 900, 1024, 1089, 1156, 1250, 1296, 1458, 1681, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2809, 2916, 3025, 3364, 3481, 3600, 4096, 4356, 4624

Offset: 1

Labos Elemer

nonn

It can be shown that this is a subsequence of A028982.

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

Cf. A028982, A046678, A046679, A046681, A046683, A046685.

Select[Range[5000], CoprimeQ[DivisorSigma[0, #], DivisorSigma[3, #]] &] (* Amiram Eldar, Aug 08 2019 *)

isok(n) = gcd(numdiv(n), sigma(n, 3)) == 1; \\ Michel Marcus, Sep 24 2019

A046681 Numbers k such that the number of divisors of k and sum of 4th powers of divisors of k are relatively prime.

Original entry on oeis.org

1, 2, 8, 9, 18, 64, 72, 128, 625, 729, 1024, 1152, 1250, 1458, 4096, 5000, 5625, 5832, 6561, 8192, 9216, 11250, 13122, 15625, 31250, 32768, 40000, 45000, 46656, 52488, 59049, 65536, 80000, 93312, 117649, 118098, 125000, 235298, 262144, 294912

Offset: 1

Labos Elemer

nonn

It can be shown that this is a subsequence of A028982.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A028982, A046678, A046679, A046680, A046683, A046685.

Select[Range[300000],CoprimeQ[DivisorSigma[0,#],DivisorSigma[4,#]]&] (* Harvey P. Dale, May 30 2012 *)

isok(n) = gcd(sigma(n, 4), numdiv(n)) == 1; \\ Michel Marcus, Sep 24 2019

A046683 Numbers k such that the sum of squares of divisors of k and sum of cubes of divisors of k are relatively prime.

Original entry on oeis.org

1, 2, 4, 9, 18, 25, 36, 100, 121, 225, 289, 484, 529, 841, 900, 1089, 1156, 1681, 2116, 2209, 2601, 2809, 3364, 3481, 4356, 4761, 5041, 6724, 6889, 7225, 7569, 7921, 8836, 10201, 10404, 11236, 11449, 12769, 13225, 13924, 15129, 17161, 18769, 19044

Offset: 1

Labos Elemer

nonn

It can be shown that this is a subsequence of A028982.

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

Cf. A028982, A046678, A046679, A046680, A046681, A046685.

sdcdQ[n_]:=Module[{d=Divisors[n]},CoprimeQ[Total[d^2],Total[d^3]]]; Select[ Range[ 20000],sdcdQ] (* Harvey P. Dale, Apr 09 2018 *)

isok(n) = gcd(sigma(n, 2), sigma(n, 3)) == 1; \\ Michel Marcus, Sep 24 2019

A046685 Numbers k such that the sum of cubes of divisors of k and the sum of 4th powers of divisors of k are relatively prime.

Original entry on oeis.org

1, 2, 4, 8, 9, 18, 25, 100, 121, 225, 289, 484, 529, 841, 1089, 1156, 1681, 2116, 2209, 2601, 2809, 3364, 3481, 4761, 5041, 6724, 6889, 7225, 7569, 7921, 8836, 10201, 11236, 11449, 12769, 13225, 13924, 15129, 17161, 18769, 19881, 20164, 21025

Offset: 1

Labos Elemer

nonn

It can be shown that this is a subsequence of A028982.
From Robert Israel, Jul 09 2018: (Start)
The only terms that are not in A062503 are 2, 8 and 18.
No term is divisible by a term of A002476.
p^2 is a term for every p in A003627. (End)

Robert Israel, Table of n, a(n) for n = 1..10000
Mathematics StackExchange, Are 1+p^3+p^6 and 1+p^4+p^8 coprime?

Cf. A002476, A003627, A028982, A046678, A046679, A046680, A046681, A046683, A062503.

N:= 10^6: # to get all terms <= N
sort(select(filter, [seq(t^2,t=1..isqrt(N)),seq(2*t^2,t=1..isqrt(N/2))])); # Robert Israel, Jul 09 2018

Select[Range[25000], CoprimeQ[DivisorSigma[3, #], DivisorSigma[4, #]] &] (* Michael De Vlieger, Aug 10 2023 *)

isok(n) = gcd(sigma(n, 3), sigma(n, 4)) == 1; \\ Michel Marcus, Sep 24 2019

A225983 Numbers k such that gcd(phi(k), tau(k)) = 1.

Original entry on oeis.org

1, 2, 4, 16, 25, 64, 81, 100, 121, 256, 289, 484, 529, 729, 841, 1024, 1156, 1296, 1600, 1681, 2116, 2209, 2401, 2809, 3025, 3364, 3481, 4096, 4624, 5041, 5184, 6400, 6724, 6889, 7225, 7744, 7921, 8464, 8836, 10201, 11236, 11449, 11664, 12100, 12769, 13225

Offset: 1

Paolo P. Lava, May 22 2013

nonn

			If n = 13924 then phi(n) = 6844 = 2^2*29*59 and tau(n) = 9 = 3^2. There is no common prime factor.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Paolo P. Lava, terms 101..1000 from T. D. Noe)

Cf. A000005, A000010, A046678, A055008.

with(numtheory); A225983:=proc(q) local n;
for n from 1 to q do if gcd(tau(n),phi(n))=1 then print(n);
fi; od; end: A225983(10^6);

t = {}; n = 0; While[Length[t] < 100, n++; If[GCD[EulerPhi[n], DivisorSigma[0, n]] == 1, AppendTo[t, n]]]; t (* T. D. Noe, May 22 2013 *)

A330606 Numbers k such that k*d(k) and sigma(k) are relatively prime, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 25, 36, 64, 81, 100, 121, 128, 144, 225, 256, 289, 324, 400, 484, 512, 529, 576, 625, 729, 841, 900, 1024, 1089, 1156, 1250, 1296, 1600, 1681, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2809, 3025, 3364, 3481, 3600, 4096, 4356, 4624, 4761

Offset: 1

Amiram Eldar, Dec 20 2019

nonn

If p is prime and p == 2 (mod 3) then p^2 is in the sequence.
Let E(x) = #{n | a(n) <= x} be the number of terms of this sequence up to x. Kanold proved that there are two constants 0 < c1 < c2 and a positive number x_0 such that c1 < E(x)/sqrt(x/log(x)) < c2 for x > x_0. De Koninck and Kátai proved that there is a positive constant c such that E(x) = c * (1 + o(1)) * sqrt(x/log(x)).
Apparently most of the terms are squares or powers of 2. Terms that are not included 1250, 4802, 31250, 57122, ...
Numbers k such that A099377(k) = A038040(k) and A099378(k) = A000203(k). - Amiram Eldar, Nov 02 2021

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 75.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Imre Kátai, On an estimate of Kanold, Int. J. Math. Anal., Vol. 5, No. 8 (2007), pp. 1-12.
Hans-Joachim Kanold, Über das harmonische Mittel der Teiler einer natürlichen Zahl II, Mathematische Annalen, Vol. 134, No. 3 (1958), pp. 225-231.

Cf. A000005, A000203, A038040, A099377, A099378, A324121.

Subsequence of A014567 and A046678.

[k:k in [1..5000]| Gcd(k*NumberOfDivisors(k),DivisorSigma(1,k)) eq 1]; // Marius A. Burtea, Dec 20 2019

Select[Range[10^4], CoprimeQ[# * DivisorSigma[0, #], DivisorSigma[1, #]] &]

A298080 Integers m such that both phi(m) and sigma(m) are coprime to tau(m).

Original entry on oeis.org

1, 2, 4, 16, 25, 64, 81, 100, 121, 256, 289, 484, 529, 729, 841, 1024, 1156, 1296, 1600, 1681, 2116, 2209, 2401, 2809, 3025, 3364, 3481, 4096, 4624, 5041, 5184, 6400, 6724, 6889, 7225, 7921, 8464, 8836, 10201, 11236, 11449, 11664, 12100, 12769, 13225, 13456, 13924

Offset: 1

Michel Marcus, Jan 12 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Imre Kátai, On the coprimality of some arithmetic functions, Publications de l'Institut Mathématique, 2010 87(101):121-128.

Intersection of A046678 and A225983.

Select[Range[14000], CoprimeQ[EulerPhi[#], (d = DivisorSigma[0, #])] && CoprimeQ[DivisorSigma[1, #], d] &] (* Amiram Eldar, Aug 08 2020 *)

isok(n) = (gcd(eulerphi(n), numdiv(n))==1) && (gcd(sigma(n), numdiv(n)) == 1);

cp's OEIS Frontend

A046679 Numbers k such that the number of divisors of k and sum of squares of divisors of k are relatively prime.

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A046680 Numbers k such that the number of divisors of k and sum of cubes of divisors of k are relatively prime.

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A046681 Numbers k such that the number of divisors of k and sum of 4th powers of divisors of k are relatively prime.

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A046683 Numbers k such that the sum of squares of divisors of k and sum of cubes of divisors of k are relatively prime.

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A046685 Numbers k such that the sum of cubes of divisors of k and the sum of 4th powers of divisors of k are relatively prime.

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Maple

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A225983 Numbers k such that gcd(phi(k), tau(k)) = 1.

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Maple

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A330606 Numbers k such that k*d(k) and sigma(k) are relatively prime, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

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Magma

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A298080 Integers m such that both phi(m) and sigma(m) are coprime to tau(m).

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