A068390 -id:A068390 - cp's OEIS frontend

A171258 Numbers n such that sigma(n) = 13*phi(n) (where sigma=A000203, phi=A000010).

630, 5544, 11160, 18810, 27000, 57000, 80388, 161820, 178020, 182880, 242820, 265608, 388620, 391500, 447678, 465192, 522522, 671760, 690120, 711000, 775170, 826500, 901170, 1051830, 1102290, 1157130, 1418160, 1578330, 1679400, 1812384, 1874520, 1993824

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, A171259, A171260, A104903.

Select[Range[2*10^6],DivisorSigma[1,#]==13EulerPhi[#]&] (* Harvey P. Dale, Mar 29 2018 *)

for(k=1,2e6, sigma(k) - 13*eulerphi(k) || print1(k", "));

A171259 Numbers n such that sigma(n) = 14*phi(n) (where sigma=A000203, phi=A000010).

Original entry on oeis.org

420, 2730, 5940, 12540, 24024, 38610, 48360, 66528, 77490, 81510, 133920, 140448, 141372, 156420, 163590, 282720, 284580, 298452, 348348, 498420, 600780, 681912, 701220, 771420, 792480, 901530, 918918, 1016730, 1052220, 1150968, 1372680, 1439592, 1654620

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, A171260, A104903.

Select[Range[10^6], DivisorSigma[1, #] == 14 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)

for(k=1,2e6, sigma(k) - 14*eulerphi(k) || print1(k", "));

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

Original entry on oeis.org

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", "));

A292422 Numbers of the form x = 2^kpq, p,q odd primes, such that sigma(x)/phi(x) = 4.

Original entry on oeis.org

418, 3596, 3956, 5396, 8636, 41656, 56536, 393104, 2072608, 2316448, 6543008, 17434528, 135394432, 217023616, 1264918784, 1490909824, 2710540544, 11444858368, 17669583104, 34797058304, 37698861568, 70572901376, 132968907776, 226965472256, 233356030976, 552070776832, 596357220352, 601188468736

Offset: 1

Jud McCranie, Sep 16 2017

nonn

A subset of A068390 and A020492 (balanced numbers). Conjectured to be infinite by Broughan and Zhou.
From David A. Corneth, Sep 21 2019: (Start)
Exactly 130 terms are of the form 2^35 * p * q.
We have phi and sigma are multiplicative and sigma(2^k) / phi(2^k) = 4 - 2/2^k, and sigma(p)/phi(p) = 1 + 2 / (p-1).
So we need (4 - 2/2^k) * (1 + 2 / (p-1)) <= 4 which gives a lower bound on p depending on k; p > nextprime(4*2^k).
We can then, given k and p, solve for q. Without loss of generality, p < q. Then search over the primes and stop for that value of k when p > q.
This method may be refined using insights from the article and/or given some k, solve the system (1 + 2 / (p-1)) * (1 + 2 / (q - 1)) = (a*m) / (b*m) for p and q where a/b is in lowest terms, m > 0. (End)
Furthermore, p < 8*2^k - 2. - David A. Corneth, Sep 26 2019

			418 = 2*11*19; sigma(418) = 720 = 4*phi(418).

David A. Corneth, Table of n, a(n) for n = 1..221
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.
David A. Corneth, Some more terms, including terms b-file
David A. Corneth, Some more terms ordered by their exponent of 2, specifying whether it is known that they are all such terms.

Cf. A068390, A020492.

Subsequence of A293391 (sigma(x)/phi(x) square).

is(n) = my(f = factor(n)); #f~ == 3 && f[2, 2] == 1 && f[3, 2] == 1 && f[1,1] == 2 && sigma(f) / eulerphi(f) == 4 \\ David A. Corneth, Sep 21 2019

A164647 Numbers n such that sigma(n)/phi(n) = 16/9.

Original entry on oeis.org

1463, 2945, 8255, 70091, 81809, 89999, 122759, 187625, 193039, 196469, 388585, 494665, 671365, 2311673, 2442583, 2687113, 4209985, 4705285, 4902247, 5393017, 5667389, 5866003, 9248323, 10795967, 11345411, 11670275, 11773027, 13290485, 13741273, 15978487

Offset: 1

M. F. Hasler, Aug 22 2009

nonn

A subsequence of A011257.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000010 (=phi), A000203 (=sigma), A068390, A163667, A164646.

for( n=1,10^7, sigma(n)==16/9*eulerphi(n) && print1(n","))

More terms from Farideh Firoozbakht, Sep 22 2009

A079546 a(n) = sigma(n) - 4*phi(n).

Original entry on oeis.org

-3, -1, -4, -1, -10, 4, -16, -1, -11, 2, -28, 12, -34, 0, -8, -1, -46, 15, -52, 10, -16, -4, -64, 28, -49, -6, -32, 8, -82, 40, -88, -1, -32, -10, -48, 43, -106, -12, -40, 26, -118, 48, -124, 4, -18, -16, -136, 60, -111, 13, -56, 2, -154, 48, -88, 24, -64, -22, -172, 104, -178, -24, -40, -1, -108, 64, -196, -2, -80, 48, -208, 99, -214

Offset: 1

N. J. A. Sloane, Jan 23 2003

If k is even and a(k) = 0 then sigma(2*k) >= 4*k, i.e., 2*k is nondeficient (A023196) (Makowski, 1987). - Amiram Eldar, Dec 05 2023

Andrzej Makowski, Remarks on some problems in the elementary theory of numbers, Acta Math. Univ. Comenian 50/51 (1987), 277-281.
József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Handbook of Number Theory I, Springer, 2005, Chapter III, p. 88.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A023196, A068390.

[DivisorSigma(1, n) - 4*EulerPhi(n): n in [1..80]]; // G. C. Greubel, Jun 19 2019

Table[DivisorSigma[1,n]-4*EulerPhi[n],{n,80}] (* Harvey P. Dale, Dec 08 2014 *)

vector(80, n, sigma(n) - 4*eulerphi(n)) \\ G. C. Greubel, Jun 19 2019

[sigma(n,1) - 4*euler_phi(n) for n in (1..80)] # G. C. Greubel, Jun 19 2019

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/12 - 12/Pi^2 = -0.393387... . - Amiram Eldar, Dec 05 2023

A164649 Numbers n such that sigma(n)/phi(n) = 36/25.

Original entry on oeis.org

5797, 10153, 20377, 50953, 383719, 405449, 446039, 486421, 608399, 973709, 1321529, 1521311, 3086369, 3228511, 3451877, 3529813, 3859513, 4552373, 4767721, 5827679, 6194321, 6479599, 6724039, 6927893, 7038241, 7919197, 11696111, 15893773, 16894141, 16924873

Offset: 1

M. F. Hasler, Aug 22 2009

nonn

A subsequence of A011257. See A164646-A164650 for related sequences.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000010 (=phi), A000203 (=sigma), A068390 (sigma/phi=4), A163667 (sigma/phi=9), A164646-A164650.

for( n=1,1e7, sigma(n)==36/25*eulerphi(n) && print1(n","))

More terms from Sean A. Irvine, May 17 2010

A164650 Numbers n such that sigma(n)/phi(n) = 49/36.

Original entry on oeis.org

679, 10127, 20273, 672203, 971261, 1133639, 1247129, 1336231, 1646743, 1701089, 2369471, 2674969, 2722499, 2989909, 3160079, 3597659, 4545749, 6333503, 7127861, 9357101, 10574629, 20070061, 52928293, 67931137, 74731807, 79940069, 80704813, 93444911, 128155333

Offset: 1

M. F. Hasler, Aug 22 2009

nonn

A subsequence of A011257.
If 7^{k+1}-1 = d*D such that p = 2*7^{k+1}*(d+1)-1 and q = 2*(7^{k+1}+D)-1 are distinct primes, then n = 7^k*p*q is a term of this sequence.
The same theorem holds for sequences of numbers such that sigma/phi=b^2/(b-1)^2 with other primes b (here b=7), cf. A068390, A164646, A164648.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000010 (=phi), A000203 (=sigma), A068390 (sigma/phi=4), A163667 (sigma/phi=9), A164646-A164649.

for( n=1,1e7, sigma(n)==49/36*eulerphi(n) && print1(n","))

A165629 Numbers n such that sigma(n)/phi(n) = 25/4, where sigma = A000203, phi = A000010.

Original entry on oeis.org

760, 11020, 18088, 21112, 58206, 65262, 71630, 100280, 123424, 142688, 262276, 303212, 332710, 630344, 679070, 761390, 1265096, 1369120, 1454060, 1454260, 1462552, 1704794, 2185750, 2386664, 2627548, 2783872, 2786056, 2909380, 2927848, 5207680, 5289220

Offset: 1

Farideh Firoozbakht and M. F. Hasler, Sep 23 2009

nonn

A subsequence of A011257. Contains the product m*n of relatively prime (gcd(m,n)=1) terms (m,n) in A068390 x A164648 and in A164646 x A165630.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Select[Range[5300000],4*DivisorSigma[1,#]==25*EulerPhi[#]&] (* Harvey P. Dale, May 09 2012 *)

for( i=1,1e9, sigma(i)*4-25*eulerphi(i) || print1(i", "))

cp's OEIS Frontend

A171258 Numbers n such that sigma(n) = 13*phi(n) (where sigma=A000203, phi=A000010).

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A171259 Numbers n such that sigma(n) = 14*phi(n) (where sigma=A000203, phi=A000010).

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

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A292422 Numbers of the form x = 2^k*p*q, p,q odd primes, such that sigma(x)/phi(x) = 4.

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A164647 Numbers n such that sigma(n)/phi(n) = 16/9.

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A079546 a(n) = sigma(n) - 4*phi(n).

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A164649 Numbers n such that sigma(n)/phi(n) = 36/25.

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A164650 Numbers n such that sigma(n)/phi(n) = 49/36.

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A292422 Numbers of the form x = 2^kpq, p,q odd primes, such that sigma(x)/phi(x) = 4.