A015770 -id:A015770 - cp's OEIS frontend

A015774 Numbers k such that phi(k) | sigma_15(k).

1, 2, 3, 6, 12, 14, 15, 30, 35, 38, 42, 46, 54, 56, 70, 78, 87, 95, 105, 114, 126, 134, 135, 138, 140, 147, 161, 168, 174, 182, 184, 190, 209, 210, 215, 216, 222, 230, 248, 258, 264, 270, 285, 294, 297, 299, 315, 322, 357, 398, 402, 414, 418, 420

Offset: 1

Robert G. Wilson v

nonn

sigma_15(n) is the sum of the 15th powers of the divisors of n.

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

Cf. A000010, A013963.

Cf. A020492, A015759, A015761, A015762, A015763, A015764, A015765, A015766, A015767, A015768, A015769, A015770, A015771, A015773, A094470.

Select[Range[420], Divisible[DivisorSigma[15, #], EulerPhi[#]] &] (* Amiram Eldar, Nov 02 2019 *)

Offset corrected by Amiram Eldar, Nov 02 2019

A015771 Numbers k such that phi(k) | sigma_13(k).

Original entry on oeis.org

1, 2, 3, 6, 12, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, 270, 357, 418, 420, 570, 594, 616, 630, 714, 744, 749, 812, 840, 910, 1045, 1240, 1254, 1485, 1498, 1672, 1848, 2090, 2214, 2247, 2376, 2436, 2568, 2580, 2730

Offset: 1

Robert G. Wilson v

nonn

sigma_13(n) is the sum of the 13th powers of the divisors of n.

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

Cf. A000010, A013961.

Cf. A020492, A015759, A015761, A015762, A015763, A015764, A015765, A015766, A015767, A015768, A015769, A015770, A015773, A015774, A094470.

with(numtheory); a := []: for n from 1 to 3000 do if sigma[13](n) mod phi(n) = 0 then a := [op(a), n]; fi; od: a;

Select[Range[3000], Divisible[DivisorSigma[13, #], EulerPhi[#]] &] (* Amiram Eldar, Nov 02 2019 *)

{n: A000010(n) | A013961(n)}. - R. J. Mathar, Sep 21 2017

Offset corrected by Amiram Eldar, Nov 02 2019

A015762 Numbers n such that phi(n) | sigma_4(n).

Original entry on oeis.org

1, 2, 3, 6, 249, 498

Offset: 1

Robert G. Wilson v

nonn

sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).
sigma_{8j+4}(x)/phi(x) is an integer for j=0..500, x=1,2,3,6,249,498, and this is conjectured to hold for possible larger terms of A015762 and all j. Compare with comments to A015759, A091285, A015770. - Labos Elemer, May 27 2004
For any odd n in this sequence, 2n is also in the sequence, since phi(2n) = phi(n) and sigma_4(2n) = 17 sigma_4(n). More generally, if gcd(m,n) = 1 and m and n both are in this sequence, then mn is also in the sequence. No odd prime > 3 can be in the sequence, since if p = 2r + 1, then sigma_4(p) = 8r(2r^3 + 4r^2 + 3r + 1) + 2 is divisible by phi(p) = 2r only for r = 1. The term a(5) = 3*83 is the only odd semiprime term with a factor < 10^5. - M. F. Hasler, Aug 21 2017
a(7) > 3*10^11, if it exists. - Giovanni Resta, Aug 23 2017

Cf. A020492, A015759, A015761, A015763, A015764, A015765, A015766, A015767, A015768, A015769, A015770, A015771, A015773, A015774, A094470.

[n: n in [1..1000]| DivisorSigma(4, n) mod EulerPhi(n) eq 0]; // Vincenzo Librandi, Aug 22 2017

Select[Range[500],Divisible[DivisorSigma[4,#],EulerPhi[#]]&] (* Harvey P. Dale, Dec 16 2012 *)

select( is(n)=sigma(n,4)%eulerphi(n)==0, [1..10^4])  \\ M. F. Hasler, Aug 21 2017

A015767 Numbers k such that phi(k) | sigma_9(k).

Original entry on oeis.org

1, 2, 3, 6, 12, 14, 15, 30, 35, 38, 42, 54, 56, 70, 78, 87, 95, 105, 114, 126, 135, 140, 147, 168, 174, 182, 190, 209, 210, 215, 216, 222, 248, 258, 264, 266, 270, 285, 294, 297, 315, 342, 357, 378, 418, 420, 430, 447, 455, 456, 494, 518, 540, 546

Offset: 1

Robert G. Wilson v

nonn

sigma_9(n) = A013957(n) is the sum of the 9th powers of the divisors of n.

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

Cf. A000010, A013957.

Cf. A020492, A015759, A015761, A015762, A015763, A015764, A015765, A015766, A015768, A015769, A015770, A015771, A015773, A015774, A094470.

Select[Range[500], Divisible[DivisorSigma[9, #], EulerPhi[#]] &] (* Amiram Eldar, Nov 02 2019 *)

Offset corrected by Amiram Eldar, Nov 02 2019

A015768 Numbers k such that phi(k) | sigma_10(k).

Original entry on oeis.org

1, 2, 3, 6, 22, 33, 66, 166, 250, 375, 498, 750, 1506, 1826, 5478, 8250, 10375, 10873, 16566, 17762, 20750, 21746, 31125, 32619, 41666, 53286, 62250, 62499, 65238, 98250, 110066, 119603, 124998, 177153, 195382, 228250, 239206

Offset: 1

Robert G. Wilson v

nonn

sigma_10(k) = A013958(k) is the sum of the 10th powers of the divisors of k.

Amiram Eldar, Table of n, a(n) for n = 1..623

Cf. A000010, A013958.

Cf. A020492, A015759, A015761, A015762, A015763, A015764, A015765, A015766, A015767, A015769, A015770, A015771, A015773, A015774, A094470.

[k:k in [1..240000]| IsIntegral(DivisorSigma(10,k)/EulerPhi(k))]; // Marius A. Burtea, Nov 06 2019

Select[Range[240000], Divisible[DivisorSigma[10, #], EulerPhi[#]] &] (* Amiram Eldar, Nov 06 2019 *)

A015769 Numbers k such that phi(k) | sigma_11(k).

Original entry on oeis.org

1, 2, 3, 6, 12, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190, 210, 235, 248, 264, 270, 329, 357, 418, 420, 470, 570, 594, 616, 630, 658, 695, 705, 714, 744, 799, 807, 812, 840, 910, 940, 987, 1045, 1240, 1254, 1316, 1390, 1410, 1485, 1529

Offset: 1

Robert G. Wilson v

nonn

sigma_11(n) = A013959(n) is the sum of the 11th powers of the divisors of n.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000010, A013959.

Cf. A020492, A015759, A015761, A015762, A015763, A015764, A015765, A015766, A015767, A015768, A015770, A015771, A015773, A015774, A094470.

Select[Range[2000],Divisible[DivisorSigma[11,#],EulerPhi[#]]&] (* Harvey P. Dale, Feb 07 2015 *)

Offset corrected by Amiram Eldar, Nov 02 2019

A015773 Numbers k such that phi(k) | sigma_14(k).

Original entry on oeis.org

1, 2, 3, 6, 22, 33, 66, 118, 177, 354, 454, 750, 1298, 1362, 1372, 1947, 3894, 4116, 4994, 8706, 14982, 15092, 26786, 33906, 44250, 45276, 56750, 65542, 77858, 80358, 98961, 116787, 170250, 171500, 196626, 197922, 233574, 242844

Offset: 1

Robert G. Wilson v

nonn

sigma_14(k) is the sum of the 14th powers of the divisors of k.

Amiram Eldar, Table of n, a(n) for n = 1..360

Cf. A000010, A013962.

Cf. A020492, A015759, A015761, A015762, A015763, A015764, A015765, A015766, A015767, A015768, A015769, A015770, A015771, A015774, A094470.

Select[Range[250000],Divisible[DivisorSigma[14,#],EulerPhi[#]]&] (* Harvey P. Dale, Feb 02 2019 *)

A015764 Numbers n such that phi(n) | sigma_6(n).

Original entry on oeis.org

1, 2, 3, 6, 22, 33, 66, 262, 750, 786, 8646, 56946, 222386, 626406, 667158, 737286, 1223123, 2446246, 2939046, 3669369, 6804006, 7338738, 27798250, 31684246, 41697375, 44970486, 53817126, 62128086, 76745867, 83394750, 95052738, 139991987, 153491734, 174684203

Offset: 1

Robert G. Wilson v

nonn

sigma_6(n) is the sum of the 6th powers of the divisors of n.

Cf. A000010 (phi(n)), A013954 (sigma_6(n)).

Cf. A020492, A015759, A015761, A015762, A015763, A015765, A015766, A015767, A015768, A015769, A015770, A015771, A015773, A015774, A094470.

with(numtheory): A015764:=n->`if`(sigma[6](n) mod phi(n) = 0,n,NULL): seq(A015764(n), n=1..10^5); # Wesley Ivan Hurt, Mar 10 2015

Select[Range[10^5], Divisible[DivisorSigma[6, #], EulerPhi[#]] &] (* Amiram Eldar, Jan 20 2019 *)

More terms from Labos Elemer, May 03 2002

a(23)-a(34) from Amiram Eldar, Jan 20 2019

A015766 Numbers k such that phi(k) | sigma_8(k).

Original entry on oeis.org

1, 2, 3, 6, 19689, 39378

Offset: 1

Robert G. Wilson v

sigma_8(n) is the sum of the 8th powers of the divisors of n.
sigma(16j+8,x)/phi(x) is an integer for j = 0, ..., 500 and 6 actual terms of this sequence. Compare to A015759, A015762, A015770 and A091285. - Labos Elemer, May 27 2004
No additional terms up to 5 million. - Harvey P. Dale, Jan 31 2016

Cf. A015759, A015762, A015770, A091285.

Select[Range[40000],Divisible[DivisorSigma[8,#],EulerPhi[#]]&] (* Harvey P. Dale, Jan 31 2016 *)

cp's OEIS Frontend

A015774 Numbers k such that phi(k) | sigma_15(k).

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A015771 Numbers k such that phi(k) | sigma_13(k).

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A015762 Numbers n such that phi(n) | sigma_4(n).

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A015767 Numbers k such that phi(k) | sigma_9(k).

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A015768 Numbers k such that phi(k) | sigma_10(k).

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Magma

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A015769 Numbers k such that phi(k) | sigma_11(k).

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A015773 Numbers k such that phi(k) | sigma_14(k).

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Mathematica

A015764 Numbers n such that phi(n) | sigma_6(n).

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Maple