A007369 -id:A007369 - cp's OEIS frontend

A060660 Numbers k such that sigma(x) = k has exactly 4 solutions.

96, 120, 180, 312, 372, 420, 434, 456, 540, 546, 560, 624, 702, 728, 798, 816, 930, 1064, 1120, 1170, 1404, 1632, 1638, 1674, 1710, 1776, 1792, 1944, 2100, 2240, 2544, 2560, 2664, 2760, 2800, 2844, 2856, 2940, 2952, 3000, 3040, 3048, 3060, 3080, 3096

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			96 = sigma(42) = sigma(62) = sigma(69) = sigma(77).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), A007372 (3), this sequence (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), A060676 (12).

a = Table[ 0, {5000} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 5001, a[ [ s ] ]++ ], {n, 1, 5000} ]; Select[ Range[ 5000 ], a[ [ # ] ] == 4 & ]

is(k) = invsigmaNum(k) == 4 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A060661 Numbers k such that sigma(x) = k has exactly 5 solutions.

Original entry on oeis.org

72, 144, 192, 216, 588, 600, 648, 792, 936, 992, 1056, 1224, 1302, 1320, 1560, 1736, 1980, 2040, 2088, 2112, 2268, 2448, 2730, 2790, 2912, 3038, 3136, 3312, 3472, 3520, 3534, 3552, 3672, 3792, 3816, 3936, 4056, 4092, 4340, 4440, 4864, 4872, 4920, 4960

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			72 = sigma(30) = sigma(46) = sigma(51) = sigma(55) = sigma(71).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), A007372 (3), A060660 (4), this sequence (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), A060676 (12).

a = Table[ 0, {5000} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 5001, a[ [ s ] ]++ ], {n, 1, 5000} ]; Select[ Range[ 5000 ], a[ [ # ] ] == 5 & ]
With[{upto=5000},Select[Union[Transpose[Select[Tally[DivisorSigma[ 1, Range[ upto]]],#[[2]]==5&]][[1]]],#<=upto&]] (* Harvey P. Dale, Jan 27 2015 *)

is(k) = invsigmaNum(k) == 5 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A060662 Numbers k such that sigma(x) = k has exactly 6 solutions.

Original entry on oeis.org

168, 252, 288, 384, 768, 1248, 1584, 1860, 2052, 2480, 2904, 3906, 3968, 4116, 4176, 4224, 4256, 4284, 4392, 4416, 4620, 5824, 5850, 5856, 5928, 6084, 6192, 6216, 6600, 6636, 6660, 6888, 6944, 7104, 7182, 7308, 7840, 7992, 8184, 8976, 9114, 9480, 9856

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			168 = sigma(60) = sigma(78) = sigma(92) = sigma(123) = sigma(143) = sigma(167).

Robert Israel, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), A007372 (3), A060660 (4), A060661 (5), this sequence (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), A060676 (12).

N:= 10^4: # for terms <= N
V:= Vector(N):
for n from 1 to N-1 do
 s:= numtheory:-sigma(n);
 if s <= N then V[s]:= V[s]+1 fi
od:
select(t -> V[t]=6, [$1..N]); # Robert Israel, Nov 21 2019

a = Table[ 0, {10000} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 10001, a[ [ s ] ]++ ], {n, 1, 10000} ]; Select[ Range[ 10000 ], a[ [ # ] ] == 6 & ]

is(k) = invsigmaNum(k) == 6 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp

A060663 Numbers k such that sigma(x) = k has exactly 7 solutions.

Original entry on oeis.org

240, 684, 744, 912, 1092, 1176, 1200, 1368, 1596, 2340, 2376, 2496, 2700, 3072, 3348, 4212, 5460, 5520, 5586, 5642, 5712, 6000, 6160, 6264, 6804, 6864, 7068, 7254, 7584, 7632, 7728, 8112, 8232, 8370, 8512, 8680, 8712, 8832, 8960, 9744, 9936

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			240 = sigma(114) = sigma(135) = sigma(158) = sigma(177) = sigma(203) = sigma(209) = sigma(239).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), this sequence (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), A060676 (12).

a = Table[ 0, {10000} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 10001, a[ [ s ] ]++ ], {n, 1, 10000} ]; Select[ Range[ 10000 ], a[ [ # ] ] == 7 & ]
With[{upto=10000},Select[Tally[DivisorSigma[1,Range[upto]]],#[[2]]==7 && #[[1]] <= upto&]][[All,1]]//Sort (* Harvey P. Dale, Jun 22 2019 *)

is(k) = invsigmaNum(k) == 7 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp

A060664 Numbers k such that sigma(x) = k has exactly 8 solutions.

Original entry on oeis.org

336, 432, 672, 756, 840, 1536, 1620, 1764, 1848, 2280, 2394, 2604, 2808, 3264, 4080, 4480, 4860, 5328, 6528, 6624, 7128, 8316, 8568, 8880, 10608, 11040, 11448, 12288, 12420, 12636, 13176, 13200, 13248, 13536, 13860, 14196, 14208, 14448, 14700

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			336 = sigma(132) = sigma(140) = sigma(182) = sigma(188) = sigma(195) = sigma(249) = sigma(287) = sigma(299).

Robert Israel, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), this sequence (8), A060665 (9), A060666 (10), A060678 (11), A060676 (12).

N:= 30000: # to get terms <= N
V:= Vector(N):
for k from 1 to N-1 do
  t:= numtheory:-sigma(k);
  if t <= N then V[t]:= V[t]+1 fi
od:
select(t -> V[t]=8, [$1..N]); # Robert Israel, Sep 22 2019

a = Table[ 0, {20000} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 20001, a[ [ s ] ]++ ], {n, 1, 20000} ]; Select[ Range[ 20000 ], a[ [ # ] ] == 8 & ]

is(k) = invsigmaNum(k) == 8 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp

A060665 Numbers k such that sigma(x) = k has exactly 9 solutions.

Original entry on oeis.org

360, 480, 1488, 1800, 1824, 2184, 2232, 2640, 3120, 3420, 3696, 3744, 3960, 4200, 5292, 5580, 5808, 6144, 7344, 7980, 8100, 8352, 8448, 8784, 9144, 10164, 10296, 11592, 11664, 11970, 12432, 13968, 14520, 14560, 15504, 15600, 15912, 16224

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

Do we have a(n) ~ c*n where c ~= 700? - David A. Corneth, Sep 23 2019

			360 = sigma(120) = sigma(174) = sigma(184) = sigma(190) = sigma(267) = sigma(295) = sigma(319) = sigma(323) = sigma(359).

David A. Corneth, Table of n, a(n) for n = 1..10046 (first 8577 terms from Robert Israel, terms <= 7*10^6)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), this sequence (9), A060666 (10), A060678 (11), A060676 (12).

N:= 60000: # to get terms <= N
V:= Vector(N):
for k from 1 to N-1 do
  t:= numtheory:-sigma(k);
  if t <= N then V[t]:= V[t]+1 fi
od:
select(t -> V[t]=9, [$1..N]); # Robert Israel, Sep 22 2019

a = Table[ 0, {20000} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 20001, a[ [ s ] ]++ ], {n, 1, 20000} ]; Select[ Range[ 20000 ], a[ [ # ] ] == 9 & ]

upto(n) = {my(v = vecsort(vector(n, i, sigma(i))), res = List()); for(i = 2, #v - 9, if(v[i-1] <= n && v[i-1] != v[i] && v[i] == v[i + 8] && v[i] != v[i+9], listput(res, v[i]))); res} \\ David A. Corneth, Sep 23 2019

is(k) = invsigmaNum(k) == 9 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp

A060666 Numbers k such that sigma(x) = k has exactly 10 solutions.

Original entry on oeis.org

504, 864, 960, 1152, 1260, 2400, 3276, 3888, 4992, 6696, 7020, 7644, 8892, 9672, 9984, 11172, 11200, 11376, 11616, 11856, 12936, 13728, 13888, 14136, 14280, 15480, 15876, 15984, 17808, 19488, 21336, 22608, 23688, 24738, 24840, 25080

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			504 = sigma(204) = sigma(220) = sigma(224) = sigma(246) = sigma(284) = sigma(286) = sigma(334) = sigma(415) = sigma(451) = sigma(504).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), this sequence (10), A060678 (11), A060676 (12).

a = Table[ 0, {30000} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 30001, a[ [ s ] ]++ ], {n, 1, 30000} ]; Select[ Range[ 30000 ], a[ [ # ] ] == 10 & ]

is(k) = invsigmaNum(k) == 10 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp

A060676 Numbers k such that sigma (x) = k has exactly 12 solutions.

Original entry on oeis.org

1512, 1872, 2352, 3192, 3780, 4104, 4560, 4752, 5880, 6120, 8160, 8424, 8820, 11424, 13056, 15264, 16464, 16704, 17160, 17360, 17760, 18648, 19680, 19800, 20880, 22752, 23616, 24552, 24864, 27432, 30336, 30492, 31200, 32448, 35328

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			1512 = sigma(480) = sigma(636) = sigma(736) = sigma(748) = sigma(830) = sigma(902) = sigma(1006) = sigma(1105) = sigma(1255) = sigma(1391) = sigma(1411) = sigma(1511).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), this sequence (12).

a = Table[ 0, {50000} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 50001, a[ [ s ] ]++ ], {n, 1, 50000} ]; Select[ Range[ 50000 ], a[ [ # ] ] == 12 & ]
Take[Sort[Transpose[Select[Tally[DivisorSigma[1,Range[100000]]],#[[2]] == 12&]][[1]]],50] (* Harvey P. Dale, Jan 18 2013 *)

is(k) = invsigmaNum(k) == 12 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp

A060678 Numbers k such that sigma (x) = k has exactly 11 solutions.

Original entry on oeis.org

576, 1296, 2976, 3168, 3648, 3720, 4788, 4896, 5544, 6300, 9000, 9840, 10656, 11808, 12528, 13020, 13320, 14760, 15456, 16740, 17920, 18288, 18576, 19344, 19840, 20400, 21280, 22800, 23296, 24300, 26712, 26928, 27552, 27936, 28392

Offset: 1

Robert G. Wilson v, Apr 18 2001

nonn

			576 = sigma(210) = sigma(282) = sigma(310) = sigma(322) = sigma(345) = sigma(357) = sigma(382) = sigma(385) = sigma(497) = sigma(517) = sigma(527).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), A007371 (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), this sequence (11), A060676 (12).

a = Table[ 0, {30000} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 30001, a[ [ s ] ]++ ], {n, 1, 30000} ]; Select[ Range[ 30000 ], a[ [ # ] ] == 11 & ]

is(k) = invsigmaNum(k) == 11 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp

A078426 Numbers k such that there is no solution to the equation sigma(x) = 2^k, where sigma(x) denotes the sum of the divisors of x.

Original entry on oeis.org

1, 4, 6, 11, 470, 475, 477, 480, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 522, 525, 527, 532, 1077, 1082

Offset: 1

Shyam Sunder Gupta, Dec 29 2002

nonn

Numbers that are not a sum of distinct Mersenne exponents (A000043). - Vladeta Jovovic, Jan 01 2003
Because there is a large gap between the 31st and 32nd Mersenne exponents, all k between 704338 and 756839 are in this sequence. - T. D. Noe, Oct 12 2006
A000203(A180162(a(n))) = 6^a(n), for n > 1. - Walter Kehowski, Aug 16 2010
Using all known Mersenne exponents, there are exactly 52935 terms in this sequence. When a new Mersenne prime (with exponent q) is found, there will be no new terms if the sum of the previous Mersenne exponents (A109472) is greater than q - 22.

			a(2)=4 because no positive integer value of x can satisfy sigma(x) = 2^4 = 16.

S. Kravitz, "Beware of the Fifth", Solution to Problem 2309, Journal of Recreational Mathematics, 29(1):76 Baywood NY 1998.

T. D. Noe, Table of n, a(n) for n = 1..300

Cf. A000203, A007369, A046528, A063883, A180221 (complement).

e={2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,4253, 4423,9689,9941,11213,19937,21701,23209,44497,86243,110503,132049,216091, 756839,859433,1257787,1398269}; u={0}; Do[u=Union[u, u+e[[k]]], {k,Length[e]}]; Complement[Range[e[[-1]]], u]

More terms from Vladeta Jovovic, Jan 01 2003
Edited by N. J. A. Sloane, Aug 23 2010
Edited by Max Alekseyev, Jan 24 2014

cp's OEIS Frontend

A060660 Numbers k such that sigma(x) = k has exactly 4 solutions.

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A060661 Numbers k such that sigma(x) = k has exactly 5 solutions.

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A060662 Numbers k such that sigma(x) = k has exactly 6 solutions.

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A060663 Numbers k such that sigma(x) = k has exactly 7 solutions.

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A060664 Numbers k such that sigma(x) = k has exactly 8 solutions.

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A060665 Numbers k such that sigma(x) = k has exactly 9 solutions.

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A060666 Numbers k such that sigma(x) = k has exactly 10 solutions.

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A060676 Numbers k such that sigma (x) = k has exactly 12 solutions.

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