A015765 -id:A015765 - cp's OEIS frontend

A074868 Non-balanced numbers in A015765.

295, 590, 767, 885, 1038, 1416, 1534, 1589, 1770, 2065, 2301, 2422, 3178, 3186, 3245, 3304, 3448, 3540, 4130, 4602, 4767, 5192, 5230, 5448, 5516, 5605, 6195, 6291, 6356, 6490, 6574, 6860, 7266, 7945, 7965, 8236, 8260, 8437, 8968, 9145, 9204, 9342

Offset: 1

Labos Elemer, Dec 05 2002

nonn

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

Cf. A000010, A000203, A013955, A015765, A020492, A078539.

Cf. A074866, A077801, A077803, A078544, A078549, A078550.

q[n_] := Sign[Mod[DivisorSigma[{1, 7}, n], EulerPhi[n]]] == {1, 0}; Select[Range[10000], q] (* Amiram Eldar, Apr 11 2024 *)

is(n) = {my(f = factor(n), phi = eulerphi(f)); (sigma(f) % phi) && !(sigma(f, 7) % phi);} \\ Amiram Eldar, Apr 11 2024

sigma_7(a(n)) mod phi(a(n)) = 0; sigma(a(n)) mod phi(a(n)) <> 0.

A015759 Numbers k such that phi(k) | sigma_2(k).

Original entry on oeis.org

1, 2, 3, 6, 22, 33, 66, 750, 27798250, 41697375, 76745867, 83394750, 153491734, 207656250, 230237601, 460475202, 917342250, 969062500, 2907187500, 4528006153, 5952812500, 9056012306, 13584018459, 17858437500, 27168036918, 31979062500, 57559400250

Offset: 1

Robert G. Wilson v

nonn

sigma_2(k) is the sum of the squares of the divisors of k (A001157).

All of these terms are solutions to relations for all j as follows: {sigma(j,x)/phi(x) is an integer for exponents j=4k+2}. Proof is possible by individual managements in the knowledge of divisors of x and phi(x). Compare with A015765, A015768, etc. - Labos Elemer, May 25 2004

Cf. A000010, A001157, A093643.

Cf. A015765, A015768, A094470.

Do[ If[ IntegerQ[ DivisorSigma[2, n]/EulerPhi[n]], Print[n]], {n, 1, 10^7}]
Empirical test for very high power sums of divisors [e.g., d^2802]. Table[{4*j+2, Union[Table[IntegerQ[DivisorSigma[4*j+2, Part[t, k]]/EulerPhi[Part[t, k]]], {k, 1, 13}]]}, {j, 0, 700}] Output = {True} for all 4j+2. Here t=A015759. (* Labos Elemer, May 20 2004 *)

a(9)-a(13) from Labos Elemer, May 20 2004
a(14)-a(18) from Donovan Johnson, Feb 05 2010
a(19)-a(27) from Donovan Johnson, Jun 18 2011

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

Original entry on oeis.org

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

A015770 Numbers k such that phi(k) divides sigma_12(k).

Original entry on oeis.org

1, 2, 3, 6, 249, 498, 118578, 99295058, 297885174, 4005374907

Offset: 1

Robert G. Wilson v

sigma_12(n) = A013960(n) is the sum of the 12th powers of the divisors of n.
sigma(24j+12,x)/phi(x) is an integer for j in the range 0, ..., 500 for x = 1, 2, 3, 6, 249, 498, 118578 and supposed to hold for possible larger terms of A015770 and all j. Compare with comments to A015759, A091285, A015762. - Labos Elemer, May 27 2004
a(11) > 5*10^9. - Giovanni Resta, Aug 22 2017
All known terms of A015762 (and also of this sequence) are squarefree. In that case, sigma_12(x)/sigma_4(x) = Product_{primes p|x} (p^8 - p^4 + 1) is an integer, so x is also in this sequence. - M. F. Hasler, Aug 22 2017

Cf. A020492, A015759, A015761, A015762, A015763, A015764, A015765, A015766, A015767, A015768, A015769, A015771, A015773, A015774, A094470.

Select[Range[1200000],Divisible[DivisorSigma[12,#],EulerPhi[#]]&] (* Harvey P. Dale, Dec 04 2015 *)

Corrected by Harvey P. Dale, Dec 04 2015

Offset corrected by and a(8)-a(10) from Giovanni Resta, Aug 22 2017

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 *)

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A074868 Non-balanced numbers in A015765.

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A015759 Numbers k such that phi(k) | sigma_2(k).

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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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Magma

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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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