A007372 -id:A007372 - cp's OEIS frontend

A007368 Smallest k such that sigma(x) = k has exactly n solutions.

2, 1, 12, 24, 96, 72, 168, 240, 336, 360, 504, 576, 1512, 1080, 1008, 720, 2304, 3600, 5376, 2520, 2160, 1440, 10416, 13392, 3360, 4032, 3024, 7056, 6720, 2880, 6480, 10800, 13104, 5040, 6048, 4320, 13440, 5760, 18720, 20736, 19152, 22680, 43680

Offset: 0

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

It's not obvious that a(n) exists for all n; I'd like to see a proof. - David Wasserman, Jun 07 2002
Note that k-1 is frequently prime. See A115374 for the least prime. For each n, it appears that there are an infinite number of k such that sigma(x)=k has exactly n solutions. - T. D. Noe, Jan 21 2006
According to Sierpiński, H. J. Kanold proved that there is a k such that sigma(x)=k has n or more solutions. Sierpiński states that Erdős proved that if, for some k, sigma(x)=k has exactly n solutions, then there are an infinite number of such k. - T. D. Noe, Oct 18 2006
Index of the first occurrence of n in A054973. - Jaroslav Krizek, Apr 25 2009

			a(10) = 504; {204, 220, 224, 246, 284, 286, 334, 415, 451, 503} is the set of x such that sigma(x) = 504.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 0..5000 (terms up to a(429) from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Kevin Ford, Sergei Konyagin, On two conjectures of Sierpiński concerning the arithmetic functions σ and ϕ, arXiv:1910.08452 [math.NT], 2019.
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964, page 166.
R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992

Cf. A000203, A054973, A002191, A007609.
Cf. A115374 (least prime p such that sigma(x)=sigma(p) has exactly n solutions).
Cf. A007369, A007370, A007371, A007372 (n such that sigma(x)=k has 0, 1, 2 and 3 solutions).
Cf. A184393, A184394, A201915 (smallest solution, largest solution, triangle of solutions for sigma(x)=a(n)).

Needs["Statistics`DataManipulation`"]; s=DivisorSigma[1, Range[10^5]]; f=Frequencies[s]; fs=Sort[f]; tfs=Transpose[fs][[1]]; utfs=Union[tfs]; firstMissing=First[Complement[Range[Last[utfs]], utfs]]; pos=1; Table[While[tfs[[pos]]T. D. Noe *)
terms = 100; cnt = DivisorSigma[1, Range[terms^3]] // Tally // Sort; a[0] = 2; a[n_] := SelectFirst[cnt, #[[2]] == n&][[1]]; Table[a[n], {n, 0, terms - 1}] (* Jean-François Alcover, Jul 18 2017 *)

More terms from David W. Wilson

A007370 Numbers k such that sigma(x) = k has a unique solution.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 13, 14, 15, 20, 28, 30, 36, 38, 39, 40, 44, 57, 62, 63, 68, 74, 78, 91, 93, 102, 110, 112, 121, 127, 133, 138, 150, 158, 160, 162, 164, 171, 174, 176, 183, 194, 195, 198, 200, 204, 212, 217, 222, 230, 242, 255, 256, 258, 260, 266, 278, 282

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
Wacław Sierpiński, Elementary Theory of Numbers, Państ. Wydaw. Nauk., Warsaw, 1964, p. 165.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Wacław Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A000203.

Number of solutions: A007369 (0), this sequence (1), A007371 (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), A060676 (12).

a = Table[ 0, {250} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 251, a[ [ s ] ]++ ], {n, 1, 250} ]; Select[ Range[ 250 ], a[ [ # ] ] == 1 & ]

list(lim)=my(v=vectorsmall(lim\1), u=List(), s); for(k=1,#v,s=sigma(k); if(s<=#v, v[s]++)); for(k=1,#v,if(v[k]==1, listput(u,k))); Vec(u) \\ Charles R Greathouse IV, Jun 15 2015

is(k) = invsigmaNum(k) == 1 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp

A007371 Numbers k such that sigma(x) = k has exactly 2 solutions.

Original entry on oeis.org

12, 18, 31, 32, 54, 56, 80, 98, 104, 108, 114, 124, 126, 128, 132, 140, 152, 156, 182, 186, 210, 264, 272, 280, 308, 320, 342, 378, 390, 392, 399, 403, 408, 416, 440, 444, 448, 492, 522, 532, 570, 572, 594, 608, 630, 632, 726, 762, 770, 774, 780, 784, 800

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A000203.

Number of solutions: A007369 (0), A007370 (1), this sequence (2), A007372 (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), A060676 (12).

a = Table[ 0, {750} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 751, a[ [ s ] ]++ ], {n, 1, 750} ]; Select[ Range[ 750 ], a[ [ # ] ] == 2 & ]

is(n)=sum(k=1,n,sigma(k)==n)==2 \\ Charles R Greathouse IV, Mar 09 2014

is(k) = invsigmaNum(k) == 2 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

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

Original entry on oeis.org

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

cp's OEIS Frontend

A007368 Smallest k such that sigma(x) = k has exactly n solutions.

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A007370 Numbers k such that sigma(x) = k has a unique solution.

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A007371 Numbers k such that sigma(x) = k has exactly 2 solutions.

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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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