A062699 -id:A062699 - cp's OEIS frontend

A171260 Numbers n such that sigma(n) = 15*phi(n) (where sigma=A000203, phi=A000010).

840, 11880, 12180, 25080, 32130, 67830, 79170, 172260, 282744, 312840, 363660, 569160, 596904, 634410, 696696, 843780, 846090, 959310, 996840, 1119690, 1201560, 1402440, 1542840, 1607340, 1929312, 2104440, 2247210, 2363790, 3309240, 3368040, 3883680

Offset: 1

Farideh Firoozbakht and M. F. Hasler, Mar 19 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.

Cf. A062699, A068391, A068390, A136547, A104900, A136540, A104901, A163667, A171256, A171257, A104902, A171258, A171259, A104903.

Select[Range[10^6], DivisorSigma[1, #] == 15 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)

for(k=1,3e6, sigma(k) - 15*eulerphi(k) || print1(k", "));

A065818 Numbers k such that 3phi(k) = 2sigma(k).

Original entry on oeis.org

5, 119, 3553, 4147, 8323, 198679, 218569, 248501, 298129, 468809, 547261, 1098349, 1398061, 1947379, 1990417, 3076549, 3453289, 3994753, 6969529, 7690249, 8790439, 11905457, 13097327, 14346577, 14732011, 14988967, 15286973, 16145269, 20851493, 21622867, 23083081, 24924599, 26218777, 26326867

Offset: 1

Labos Elemer, Nov 23 2001

nonn

			n = 3553, 3*phi(3553) = 8640 = 2*4320 = 2*sigma(3553).

Jud McCranie, Table of n, a(n) for n = 1..10000 (first 75 terms from Harry J. Smith)

Cf. A000010, A000203, A062699, A065819.

Select[Range[20000000], 3EulerPhi[#] == 2DivisorSigma[1, #] &]  (* Harvey P. Dale, Apr 18 2011 *)

isok(k) = { 3*eulerphi(k) == 2*sigma(k) } \\ Harry J. Smith, Oct 31 2009

a(22)-a(28) from Harry J. Smith, Oct 31 2009

A065822 Numbers k such that 5phi(k) = 4sigma(k).

Original entry on oeis.org

323, 377, 22591, 42619, 49751, 106711, 119647, 180947, 2782057, 2980823, 2981233, 3794737, 5112427, 5285743, 5732179, 5964229, 6073267, 6669797, 6769927, 7049407, 8025547, 8350633, 8954023, 9373213, 10039471, 10140517, 10842901

Offset: 1

Labos Elemer, Nov 23 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..335 (terms 1..60 from Harry J. Smith)

Cf. A000010, A000203, A065818, A065819, A062699.

Select[Range[1085*10^4],5EulerPhi[#]==4DivisorSigma[1,#]&] (* Harvey P. Dale, Aug 11 2019 *)

{ n=0; for (m=1, 10^9, if (5*eulerphi(m) == 4*sigma(m), write("b065822.txt", n++, " ", m); if (n==60, return)) ) } \\ Harry J. Smith, Nov 01 2009

is(n) = {my(f=factor(n)); 5*eulerphi(f) == 4*sigma(f);} \\ Amiram Eldar, Jun 26 2024

a(9)-a(27) from Harry J. Smith, Nov 01 2009

A065824 Smallest solution m to (n+1)phi(m) = nsigma(m), or -1 if no solution exists.

Original entry on oeis.org

3, 5, 7, 323, 11, 13, 899, 17, 19, 1763, 23, 5249, 3239, 29, 31, 979801, 5459, 37, 10763, 41, 43, 9179, 47, 9701, 10403, 53, 12319, 5646547, 59, 61, 24569, 19109, 67, 19043, 71, 73, 22499, 50819, 79, 41309, 83, 32639, 46979, 89, 34579, 39059, 125969

Offset: 1

Labos Elemer, Nov 23 2001

If p = a(n) is a prime solution, then (n+1)*(p-1) = n*(p+1) and p = 2n+1, so position for p if it is in fact a minimal solution is at n = (p-1)/2. E.g. 29 appears at 14th position shown by A005097. On the other hand large and (seemingly always composite) solutions arise at indices shown essentially by A047845. Also, differences between the sites of two consecutive small prime solutions appears to be d/2, half the difference between consecutive primes (A001223).

Donovan Johnson, Table of n, a(n) for n = 1..456

Cf. A000010, A000203, A062699, A065818, A065819, A065822, A065823.

See also A005097, A047845, A014076, A001223.

max = 10^7; a[n_] := For[m = 3, m <= max, m++, If[(n+1)*EulerPhi[m] == n*DivisorSigma[1, m], Print[m]; Return[m]]] /. Null -> -1; Array[a, 50] (* Jean-François Alcover, Oct 08 2016 *)

from itertools import count
from math import prod
from sympy import factorint
def A065824(n):
    for m in count(1):
        f = factorint(m)
        if (n+1)*m*prod((p-1)**2 for p in f)==n*prod(p**(e+2)-p for p,e in f.items()):
            return m # Chai Wah Wu, Aug 12 2024

(n+1)*A000010(a(n)) = n*A000203(a(n)), smallest x=a(n) solutions.

A065823 Numbers k such that 6phi(k) = 5sigma(k).

Original entry on oeis.org

11, 527, 923, 36859, 40549, 55309, 88519, 120139, 138301, 280579, 293501, 313807, 529789, 719927, 2458859, 4864117, 6191413, 6811243, 7297877, 8402663, 8624107, 9487477, 10475821, 12356441, 12940957, 13624717, 13971229, 14869033, 15293137

Offset: 1

Labos Elemer, Nov 23 2001

nonn

Not all terms are squarefree: a(74) = 137640191 = 13^2 * 89 * 9151. - Charles R Greathouse IV, Nov 13 2015

Apart from the first term, no terms are divisible by 2, 3, 5, 7, or 11. - Charles R Greathouse IV, Nov 13 2015

Harry J. Smith and Charles R Greathouse IV, Table of n, a(n) for n = 1..651 (first 70 terms from Smith)

Subsequence of A008364.

Cf. A000010, A000203, A065818, A065819, A062699, A065822, A274535(5*sigma(n)).

n=0; for (m=1, 10^9, if (6*eulerphi(m) == 5*sigma(m), write("b065823.txt", n++, " ", m); if (n==70, return))) \\ Harry J. Smith, Nov 01 2009

is(n)=my(f=factor(n)); 6*eulerphi(f)==5*sigma(f) \\ Charles R Greathouse IV, Nov 13 2015

Terms a(16)-a(29) from Harry J. Smith, Nov 01 2009

A110597 Balanced numbers (A020492) k such that k mod 12 = 1.

Original entry on oeis.org

1, 1045, 29029, 50065, 64285, 87685, 1390753, 2011009, 3189625, 7711405, 39298441, 53238625, 68393065, 75416341, 96345613, 225938245, 228404605, 231562825, 233591605, 279999445, 458406445, 462027565, 470527057, 491291125, 513574369, 663605761, 666373825

Offset: 1

Walter Kehowski, Sep 13 2005

nonn

For the first 27 terms, the quotient sigma(n)/phi(n) is 1, 2 or 3.

Amiram Eldar, Table of n, a(n) for n = 1..8224 (terms below 6.5*10^14, calculated using data from Jud McCranie)

Intersection of A017533 and A020492.

Cf. A000010, A000203, A062699.

with(numtheory); BNM1:=[]: for z from 1 to 1 do for m from 0 to 500000 do n:=12*m+1; if sigma(n) mod phi(n) = 0 then BNM1:=[op(BNM1),n] fi; od; od; BNM1;

Select[Range[10^7], Mod[#, 12] == 1 && Divisible[DivisorSigma[1, #], EulerPhi[#]] &] (* Amiram Eldar, Dec 04 2019 *)

forstep(n=1,1e5,12, if(sigma(n)%eulerphi(n)==0, print1(n", "))) \\ Charles R Greathouse IV, Nov 27 2013

a(10)-a(27) from Donovan Johnson, Aug 30 2012

A110598 Balanced numbers k such that k mod 12 = 5.

Original entry on oeis.org

137885, 145145, 3501605, 6605945, 6953765, 8409305, 10055045, 11413205, 11569805, 16540205, 18545285, 19648805, 21902705, 22806905, 25965005, 26655005, 29811665, 45680921, 71569745, 79989845, 91681289, 196492205, 214218389, 223086125, 229554941, 233601641

Offset: 1

Walter Kehowski, Sep 13 2005

nonn

For the first 26 terms, the quotient sigma(k)/phi(k) is 2 or 3.

Amiram Eldar, Table of n, a(n) for n = 1..7013 (terms below 6.5*10^14, calculated using data from Jud McCranie)

Intersection of A017581 and A020492.

Cf. A000010, A000203, A062699, A068391.

with(numtheory); BNM5:=[]: for z from 1 to 1 do for m from 1 to 1000000 do n:=12*m+5; if sigma(n) mod phi(n) = 0 then BNM5:=[op(BNM5),n] fi; od; od; BNM5;

Select[Range[5,12000000,12],MemberQ[{2,3},DivisorSigma[1,#]/EulerPhi[#]]&]  (* Harvey P. Dale, May 06 2012 *)

a(10)-a(26) from Donovan Johnson, Aug 30 2012

A110599 Balanced numbers n such that n mod 12 = 7.

Original entry on oeis.org

24871, 58435, 140335, 1529983, 2086903, 3722875, 3830827, 8697535, 13932919, 16408315, 21578755, 27882595, 76319155, 126245119, 183531439, 192871987, 198394675, 207619555, 229523371, 337800463, 361504507, 416690995, 440127655, 535044055, 693298315, 729802255

Offset: 1

Walter Kehowski, Sep 13 2005

nonn

For the first 26 terms, the quotient (sigma(n)/phi(n)) is 2 or 3.

Amiram Eldar, Table of n, a(n) for n = 1..7901 (terms below 6.5*10^14, calculated using data from Jud McCranie)

Intersection of A017605 and A020492.

Cf. A000010, A000203, A062699.

with(numtheory); BNM7:=[]: for z from 1 to 1 do for m from 1 to 1000000 do n:=12*m+7; if sigma(n) mod phi(n) = 0 then BNM7:=[op(BNM7),n] fi; od; od; BNM7;

Select[Range[10^7], Mod[#, 12] == 7 && Divisible[DivisorSigma[1, #], EulerPhi[#]] &] (* Amiram Eldar, Dec 04 2019 *)

Duplicate terms removed and a(8)-a(26) from Donovan Johnson, Aug 30 2012

A292390 Numbers n such that psi(n) = 2*phi(n).

Original entry on oeis.org

3, 9, 27, 35, 81, 175, 243, 245, 729, 875, 1045, 1225, 1715, 2187, 4375, 5225, 6125, 6561, 8575, 11495, 12005, 19683, 19855, 21875, 24871, 26125, 29029, 30625, 42875, 50065, 57475, 58435, 59049, 60025, 64285, 84035, 87685, 99275, 109375, 126445, 130625, 137885, 140335, 153125

Offset: 1

Robert G. Wilson v, Amiram Eldar and Altug Alkan, Sep 15 2017

Squarefree terms are 3, 35, 1045, 24871, 29029, 50065, 58435, 64285, ... Squarefree terms of this sequence are in A062699. Note that A062699 also has terms that are not squarefree: 2011009, 3189625, 3722875, ...
If n is in the sequence, then so are all numbers that have the same set of prime factors as n. - Robert Israel, Sep 15 2017
All terms are odd. Terms divisible by 3 are powers of 3. - Robert Israel, Sep 18 2017

			3^k is a term for all k > 0 since psi(3^k) = 4*3^(k-1) = 2*phi(3^k).

Robert Israel, Table of n, a(n) for n = 1..400

Cf. A000010, A001615, A062699, A291051, A291932.

pp:= n -> mul((p+1)/(p-1), p = numtheory:-factorset(n)):
select(pp=2, [seq(i,i=1..200000,2)]); # Robert Israel, Sep 15 2017

psi[n_] := n*Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]; Select[ Range@ 200000, 2EulerPhi[#] == psi[#] &]

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
isok(n) = a001615(n)==2*eulerphi(n); \\ after Charles R Greathouse IV at A001615

cp's OEIS Frontend

A171260 Numbers n such that sigma(n) = 15*phi(n) (where sigma=A000203, phi=A000010).

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A065818 Numbers k such that 3*phi(k) = 2*sigma(k).

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A065822 Numbers k such that 5*phi(k) = 4*sigma(k).

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A065824 Smallest solution m to (n+1)*phi(m) = n*sigma(m), or -1 if no solution exists.

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A065823 Numbers k such that 6*phi(k) = 5*sigma(k).

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A110597 Balanced numbers (A020492) k such that k mod 12 = 1.

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A110598 Balanced numbers k such that k mod 12 = 5.

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A110599 Balanced numbers n such that n mod 12 = 7.

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A065818 Numbers k such that 3phi(k) = 2sigma(k).

A065822 Numbers k such that 5phi(k) = 4sigma(k).

A065824 Smallest solution m to (n+1)phi(m) = nsigma(m), or -1 if no solution exists.

A065823 Numbers k such that 6phi(k) = 5sigma(k).