A055008 -id:A055008 - cp's OEIS frontend

A073815 Least number x such that gcd(phi(x), sigma(x)) = n.

1, 3, 18, 12, 200, 14, 3364, 15, 722, 328, 9801, 42, 25281, 116, 1800, 165, 36992, 810, 4414201, 88, 196, 29161, 541696, 35, 2928200, 1413, 103968, 172, 98942809, 488, 1547536, 336, 19602, 17536, 814088, 370, 49042009, 55297, 1521, 319, 3150464641

Offset: 1

Labos Elemer, Nov 12 2002

nonn

Values are frequently identical to terms of A077102. Since gcd(a,b) and gcd(a+b,a-b) may differ, so may the smallest solutions. A077102(m) and a(m) differ at m = 1, 2, 4, 8, 16, 28, 32, 40, etc.

Giovanni Resta, Table of n, a(n) for n = 1..120

Cf. A000203, A000010, A009223, A055008, A077099-A077102, A051612, A065387, A023897, A020492.

f[x_] := Apply[GCD, {DivisorSigma[1, x], EulerPhi[x]}] t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10^13}];

a(n)=my(x=n);while(gcd(eulerphi(x),sigma(x))!=n, x++); x \\ Charles R Greathouse IV, Dec 09 2013

a(n) = Min{x; A055008(x)=n}. a(n)=Min{x; gcd(A000203(x), A000010(x))=n}

a(n) = Min{x: A023897(x)= n}, smallest balanced number (A020492) for which the quotient equals n.

A065299 Numbers k such that sigma(k)*phi(k) is squarefree.

Original entry on oeis.org

1, 2, 4, 9, 121, 242, 529, 1058, 2209, 3481, 4418, 5041, 6889, 6962, 10082, 11449, 13778, 17161, 22898, 27889, 32041, 34322, 51529, 55778, 57121, 64082, 96721, 103058, 114242, 120409, 128881, 146689, 175561, 185761, 193442, 196249, 218089

Offset: 1

Labos Elemer, Oct 29 2001

nonn

			All solutions are either squares or twice squares. Proper subset of A055008 or A028982. Several squares (of primes) and 2*squares are not here. E.g., 242 is here because phi(242) = 110, sigma(242) = 399, 2*5*11*3*7*19 is squarefree; 18 is not here, since 2*3*3*13 is not squarefree.

Harry J. Smith and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (terms 1..500 from Smith)

A062354(m) is squarefree.

Cf. A000010, A000203, A008683, A055008, A028982, A049195-A049199, A049149.

a[x_] := Abs[MoebiusMu[DivisorSigma[1, x]*EulerPhi[x]]] Do[s=as[n]; If[Equal[s, 1], Print[{n, Sqrt[n]}]], {n, 1, 1000000}]
Select[Range[250000],SquareFreeQ[DivisorSigma[1,#]*EulerPhi[#]]&] (* Harvey P. Dale, Jul 15 2015 *)

n=0; for (m = 1, 10^9, s=abs(moebius(sigma(m)*eulerphi(m))); if (s==1, write("b065299.txt", n++, " ", m); if (n==500, return))) \\ Harry J. Smith, Oct 15 2009

is(f)=my(n=#f~, v=List()); for(i=1,n, if(f[i,1]>2, listput(v,f[i,1]-1)); if(f[i,2]>2, return(0), f[i,2]>1, listput(v,f[i,1])); listput(v, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))); for(i=2,#v, for(j=1,i-1, if(gcd(v[i],v[j])>1, return(0)))); for(i=1,#v, if(!issquarefree(v[i]), return(0))); 1
sq(f)=f[,2]*=2; f
double(f)=if(#f~ && f[1,1]==2, f[1,2]++, f=concat([2,1],f)); f
list(lim)=my(v=List()); forsquarefree(n=1,sqrtint(lim\1), if(is(sq(n[2])), listput(v,n[1]^2))); forsquarefree(n=1,sqrtint(lim\2), if(is(double(sq(n[2]))), listput(v,2*n[1]^2))); Set(v) \\ Charles R Greathouse IV, Feb 05 2018

Solutions to abs(A008683(A000203(x)*A000010(x))) = 1.

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 *)

A248861 Numbers k such that phi(k)^phi(k) == 1 (mod sigma(k)).

Original entry on oeis.org

1, 2, 8, 36, 128, 225, 289, 578, 900, 2025, 2601, 3600, 10404, 32768, 41616, 45369, 57600, 242064, 665856, 725904, 783225, 1134225, 1140624, 1782225, 1988100, 2903616, 3132900, 4862025, 6155361, 6275025, 7128900, 7868025, 8625969, 10208025, 13505625

Offset: 1

Farideh Firoozbakht, Dec 12 2014

nonn

2^m is a term of the sequence if and only if m=2^j-1 where j is a nonnegative integer. Hence the sequence is infinite.
289 is a term of the sequence which is of the form p^2 where p is prime. What is the next such term?
578 is a term of the sequence which is not of the form 2^m or m^2. What is the next such term?
A248862 gives primes p such that 900*p^2 is a term of the sequence.
Subsequence of A055008. - Jason Yuen, Jul 01 2024

Jason Yuen, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A177006, A248862, A055008.

Prepend[Select[Range[30000], Mod[EulerPhi[#]^EulerPhi[#], DivisorSigma[1, #]] == 1 &], 1] (* Michael De Vlieger, Dec 13 2014 *)

isok(n) = my(in = eulerphi(n)); lift(Mod(in, sigma(n))^in - 1) == 0; \\ Michel Marcus, Dec 13 2014

A260963 Numbers n such that gcd(sigma(n), n*(n+1)/2 - sigma(n)) = 1, where sigma(n) is sum of positive divisors of n.

Original entry on oeis.org

1, 4, 9, 10, 16, 21, 22, 25, 34, 36, 46, 49, 57, 58, 64, 70, 81, 82, 85, 93, 94, 100, 106, 118, 121, 129, 130, 133, 142, 144, 154, 166, 169, 178, 201, 202, 205, 214, 217, 225, 226, 237, 238, 250, 253, 256, 262, 265, 274, 289, 298, 301, 309, 310, 322, 324, 325

Offset: 1

Paolo P. Lava, Aug 27 2015

			sigma(10) = 18, 10*11/2 - sigma(10) = 55 - 18 = 37 and gcd(18,37) = 1 because 18 = 2*9 and 37 is prime.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A000203, A024816, A055008, A058075.

with(numtheory): P:=proc(q) local n; for n from 1 to q do
if gcd(sigma(n),n*(n+1)/2-sigma(n))=1 then print(n); fi; od; end: P(10^9);

Select[Range@ 360, GCD[DivisorSigma[1, #], # (# + 1)/2 - DivisorSigma[1, #]] == 1 &] (* Michael De Vlieger, Aug 27 2015 *)

A077103 Numbers n such that gcd(a,b) is not equal to gcd(a+b,a-b), where a=sigma(n)=A000203(n) and b=phi(n)=A000010(n).

Original entry on oeis.org

1, 2, 12, 15, 30, 39, 44, 55, 56, 76, 78, 87, 95, 99, 110, 111, 125, 140, 143, 147, 159, 171, 172, 174, 175, 183, 184, 190, 198, 215, 216, 222, 236, 247, 250, 252, 264, 268, 286, 287, 294, 295, 303, 315, 318, 319, 327, 332, 335, 336, 342, 350, 357, 363, 364

Offset: 1

Labos Elemer, Nov 12 2002

nonn

			n=76: a=sigma(76)=140, b=phi(76)=36, a+b=176, a-b=104, gcd(a,b) = gcd(140,36) = 4 < gcd(a+b,a-b) = gcd(176,104) = 8.

Cf. A000010, A000203, A051612, A065387, A055008, A077099.

Do[s=GCD[a=DivisorSigma[1, n], b=EulerPhi[n]]; s1=GCD[a+b, a-b]; If[ !Equal[s, s1], Print[{n, a, b, a+b, a-b, s, s1, s1/s}]], {n, 1, 1000}]

gcd(A000010(n), A000203(n)) is not equal to gcd(A065387(n), A051612(n)); or A055008(n) is not equal to A077099(n).

cp's OEIS Frontend

A073815 Least number x such that gcd(phi(x), sigma(x)) = n.

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A065299 Numbers k such that sigma(k)*phi(k) is squarefree.

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

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A248861 Numbers k such that phi(k)^phi(k) == 1 (mod sigma(k)).

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A260963 Numbers n such that gcd(sigma(n), n*(n+1)/2 - sigma(n)) = 1, where sigma(n) is sum of positive divisors of n.

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A077103 Numbers n such that gcd(a,b) is not equal to gcd(a+b,a-b), where a=sigma(n)=A000203(n) and b=phi(n)=A000010(n).

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