A015774 -id:A015774 - cp's OEIS frontend

A078550 Non-balanced numbers in A015774.

38, 46, 54, 87, 95, 114, 126, 134, 135, 138, 147, 161, 174, 182, 184, 209, 215, 216, 222, 230, 258, 285, 294, 297, 299, 315, 322, 398, 402, 414, 430, 437, 455, 456, 483, 540, 546, 551, 552, 598, 609, 623, 627, 632, 635, 644, 645, 670, 690, 762, 783, 805

Offset: 1

Labos Elemer, Dec 05 2002

nonn

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

Cf. A000010, A000203, A013963, A015774, A020492, A078539.

Cf. A074866, A074868, A077801, A077803, A078544, A078549.

q[n_] := Sign[Mod[DivisorSigma[{1, 15}, n], EulerPhi[n]]] == {1, 0}; Select[Range[1000], q] (* Amiram Eldar, Apr 11 2024 *)

is(n) = {my(f = factor(n), phi = eulerphi(f)); (sigma(f) % phi) && !(sigma(f, 15) % phi);} \\ Amiram Eldar, Apr 11 2024

sigma_15(a(n)) mod phi(a(n)) = 0; sigma(a(n)) mod phi(a(n)) <> 0.

Name corrected by Amiram Eldar, Apr 11 2024

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

A078539 Least non-balanced x (i.e., not in A020492) such that sigma(2n-1,x)/phi(x) is an integer.

Original entry on oeis.org

38, 46, 295, 38, 235, 749, 38, 3687, 6128, 38, 1415, 46, 38, 4254, 10451, 38, 46, 8351, 38, 334, 4511, 38, 3398, 295, 38, 1286, 46, 38, 148870, 11015, 38, 46, 35519, 38, 10239, 14072, 38, 235, 76088, 38, 5991, 46, 38, 718, 295, 38, 46, 11654, 38, 30761

Offset: 2

Labos Elemer, Dec 02 2002

nonn

			n=7: 2n-1 = 13, cases of sigma(13,x)/phi(x) is an integer listed in A015771: 1, 2, 3,6, 12, etc,; the first term which is non-balanced, i.e., not in A020492 is a(7) = 749 = A020492(31); increasing n, the trend of a(n) is roughly the same. If 2n-1 = 3s, i.e., is divisible by 3, then a(3s) = 38. Similar relationships hold for 2n - 1 = 5s, 7s, 11s, etc.

Amiram Eldar, Table of n, a(n) for n = 2..1200

Cf. A000005, A000010, A015759-A015774, A020492, A078538.

Table[fl=1; Do[s1=DivisorSigma[1, n]/EulerPhi[n]; sk=DivisorSigma[2*k-1, n]/EulerPhi[n]; If[ !IntegerQ[s1]&&IntegerQ[sk]&&Equal[fl, 1], Print[{n, 2*k-1}]; fl=0], {n, 1, 1000000}], {k, 2, 100}]

a(n) = min{x; sigma(1,x) mod phi(x) = 0 but sigma(2n-1, x) mod phi(x) is not 0}.

a(31) corrected by Amiram Eldar, Jul 21 2019

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 *)

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

cp's OEIS Frontend

A078550 Non-balanced numbers in A015774.

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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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A078539 Least non-balanced x (i.e., not in A020492) such that sigma(2n-1,x)/phi(x) is an integer.

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

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