A015759 -id:A015759 - cp's OEIS frontend

A093643 Integer quotients arising in A015759.

1, 5, 5, 25, 61, 61, 305, 4069, 91058273, 91058273, 86169701, 455291365, 430848505, 1129161898, 430848505, 2154242525, 5554554653, 3952066643, 19760333215, 5173153429, 20889495113, 25865767145, 25865767145, 104447475565, 129328835725, 241076065223, 350624513369

Offset: 1

Labos Elemer, May 20 2004

nonn

Cf. A015759.

a(10)-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

A015765 Numbers n such that phi(n) | sigma_7(n).

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, 295, 357, 418, 420, 570, 590, 594, 616, 630, 714, 744, 767, 812, 840, 885, 910, 1038, 1045, 1240, 1254, 1416, 1485, 1534, 1589, 1672, 1770, 1848, 2065, 2090, 2214, 2301, 2376, 2422, 2436, 2580, 2730, 2970

Offset: 1

Robert G. Wilson v

sigma_7(n) is the sum of the 7th powers of the divisors of n.

In contrast to other values of k for which sigma_k(n) is much less frequently divisible by phi(n) (cf. A015762 (k=4: a(7) > 10^11 if it exists), A015759 (k=2: a(23) > 10^11)), it is quite easy to compute hundreds or even thousands of terms of the present sequence. - M. F. Hasler, Aug 26 2017

M. F. Hasler, Table of n, a(n) for n = 1..10000 (first 1001 terms from Vincenzo Librandi)

Select[Range[2100],Divisible[DivisorSigma[7,#],EulerPhi[#]]&] (* Harvey P. Dale, Aug 17 2013 *)

select( is(n)=sigma(n,7)%eulerphi(n)==0, [1..3000]) \\ M. F. Hasler, Aug 26 2017

Third line of data completed by M. F. Hasler, Aug 26 2017

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

A094470 Numbers k such that phi(k) | sigma_18(k).

Original entry on oeis.org

1, 2, 3, 6, 22, 33, 66, 262, 750, 786, 2182, 6546, 8646, 56946, 72006, 162066, 222386, 626406, 667158, 737286, 857526, 1223123, 1489686, 1782726, 2446246, 2939046, 3669369, 4388406, 4780947, 6804006, 7338738, 9561894, 10761126, 12157926

Offset: 1

Labos Elemer, May 25 2004

nonn

A015759 seems to be a true subsequence here.

sigma_18(n) is the sum of the 18th powers of the divisors of n (A013966). Conjecture: analogous sequences with any 4j+2 exponent instead of 18, includes all terms of A015759 [with exponent=2].

Amiram Eldar, Table of n, a(n) for n = 1..130

Cf. A000010, A013966, A015759.

Select[Range[10^6], Divisible[DivisorSigma[18, #], EulerPhi[#]] &] (* Amiram Eldar, Mar 07 2020 *)

isok(k) = (sigma(k, 18) % eulerphi(k)) == 0; \\ Michel Marcus, Mar 07 2020

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

cp's OEIS Frontend

A093643 Integer quotients arising in A015759.

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

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A015765 Numbers n such that phi(n) | sigma_7(n).

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

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Maple

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

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Magma

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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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A094470 Numbers k such that phi(k) | sigma_18(k).

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

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