cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

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

Views

Author

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

Keywords

References

  • 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).

Links

Crossrefs

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).

Programs

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

Views

Author

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

Keywords

References

  • 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).

Links

Crossrefs

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).

Programs

  • Mathematica
    a = Table[ 0, {750} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 751, a[ [ s ] ]++ ], {n, 1, 750} ]; Select[ Range[ 750 ], a[ [ # ] ] == 2 & ]
  • PARI
    is(n)=sum(k=1,n,sigma(k)==n)==2 \\ Charles R Greathouse IV, Mar 09 2014
  • PARI
    is(k) = invsigmaNum(k) == 2 \\ Amiram Eldar, Nov 17 2024, using Max Alekseyev's invphi.gp

A007372 Numbers k such that sigma(x) = k has exactly 3 solutions.

Original entry on oeis.org

24, 42, 48, 60, 84, 90, 224, 228, 234, 248, 270, 294, 324, 450, 468, 528, 558, 620, 640, 660, 810, 882, 888, 896, 968, 972, 1020, 1050, 1104, 1116, 1140, 1216, 1232, 1240, 1274, 1332, 1392, 1400, 1452, 1456, 1464, 1482, 1524, 1530, 1600, 1694, 1716, 1760
Offset: 1

Views

Author

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

Keywords

References

  • 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).

Links

Crossrefs

Cf. A000203.
Number of solutions: A007369 (0), A007370 (1), A007371 (2), this sequence (3), A060660 (4), A060661 (5), A060662 (6), A060663 (7), A060664 (8), A060665 (9), A060666 (10), A060678 (11), A060676 (12).

Programs

  • Mathematica
    a = Table[ 0, {2500} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 2501, a[ [ s ] ]++ ], {n, 1, 2500} ]; Select[ Range[ 2500 ], a[ [ # ] ] == 3 & ]
  • PARI
    is(n)=sum(k=1,n,sigma(k)==n)==3 \\ Charles R Greathouse IV, Mar 09 2014
  • PARI
    is(k) = invsigmaNum(k) == 3 \\ 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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

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

Links

Crossrefs

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).

Programs

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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

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

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • PARI
    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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

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

Links

Crossrefs

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).

Programs

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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

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

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • PARI
    is(k) = invsigmaNum(k) == 7 \\ 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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Comments

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

Examples

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

Links

Crossrefs

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).

Programs

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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

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

Links

Crossrefs

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).

Programs

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

Views

Author

Robert G. Wilson v, Apr 18 2001

Keywords

Examples

			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).

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • PARI
    is(k) = invsigmaNum(k) == 12 \\ Amiram Eldar, Nov 18 2024, using Max Alekseyev's invphi.gp
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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