A006532 -id:A006532 - cp's OEIS frontend

A007678 Number of regions in regular n-gon with all diagonals drawn.

0, 0, 1, 4, 11, 24, 50, 80, 154, 220, 375, 444, 781, 952, 1456, 1696, 2500, 2466, 4029, 4500, 6175, 6820, 9086, 9024, 12926, 13988, 17875, 19180, 24129, 21480, 31900, 33856, 41416, 43792, 52921, 52956, 66675, 69996, 82954, 86800, 102050, 97734, 124271, 129404, 149941

Offset: 1

N. J. A. Sloane, Bjorn Poonen (poonen(AT)math.princeton.edu)

This sequence and A006533 are two equivalent ways of presenting the same sequence.
A quasipolynomial of order 2520. - Charles R Greathouse IV, Jan 15 2013
Also the circuit rank of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 08 2018
This sequence only counts polygons, in contrast to A006533 which also counts the n segments of the circumscribed circle delimited by the edges of the regular n-gon. Therefore a(n) = A006533(n) - n. See also A006561 which counts the intersection points, and A350000 which considers iterated "cutting along diagonals". - M. F. Hasler, Dec 13 2021
The Petrie polygon orthographic projection of a regular n-simplex is a regular (n+1)-gon with all diagonals drawn. Hence a(n+1) is the number of regions in the Petrie polygon of a regular n-simplex. - Mohammed Yaseen, Nov 05 2022

Jean Meeus, Wiskunde Post (Belgium), Vol. 10, 1972, pp. 62-63.
C. A. Pickover, The Mathematics of Oz, Problem 58 "The Beauty of Polygon Slicing", Cambridge University Press, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
Robert Dougherty-Bliss, First draft of Python program to produce colored drawings of these figures, Github, Feb 09 2020.
Martin Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5.
M. F. Hasler, Interactive illustration of A006561(n) & A006533(n); colored version for n=6 and for n=8.
Sascha Kurz, m-gons in regular n-gons (in German).
J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11, no. 1 (1998), pp. 135-156; DOI:10.1137/S0895480195281246. [Author's copy]. The latest arXiv version arXiv:math/9508209 has corrected some typos in the published version.
Bjorn Poonen and Michael Rubinstein, Mathematica programs for these sequences
J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238.
M. Rubinstein, Drawings for n=4,5,6,...
Scott R. Shannon, Colored illustration for n = 17
Scott R. Shannon, Colored illustration for n = 18
Scott R. Shannon, Colored illustration for n = 19
Scott R. Shannon, Colored illustration for n = 23
Scott R. Shannon, Colored illustration for n = 27
Scott R. Shannon, Colored illustration for n = 40
Scott R. Shannon, Colored illustration for n = 41 (1st version)
Scott R. Shannon, Colored illustration for n = 41 (2nd version)
Scott R. Shannon, Colored illustration for n = 41 (3rd version). This variation has coloring based on the number of edges of the polygon: red = 3-gon, orange = 4-gon, yellow = 5-gon, light-green = 6-gon etc.
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 18.
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
Prasad Balakrishnan Warrier, The physiognomy of the Erdős-Szekeres conjecture (happy ending problem), Math. Student (Indian Math. Soc., 2024) Vol. 93, Nos. 3-4, 28-48.
Eric Weisstein's World of Mathematics, Circuit Rank
Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph
Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals
Sequences formed by drawing all diagonals in regular polygon
Sequences related to chord diagrams

Cf. A001006, A054726, A006533, A006561, A006600, A007569 (number of vertices), A006522, A135565 (number of line segments).
A062361 gives number of triangles, A331450 and A331451 give distribution of polygons by number of sides.
A333654, A335614, A335646, A337330 give the number of internal n-gon to k-gon contacts for n>=3, k>=n.
A187781 gives number of distinct regions.

del[m_,n_]:=If[Mod[n,m]==0,1,0]; R[n_]:=If[n<3, 0, (n^4-6n^3+23n^2-42n+24)/24 + del[2,n](-5n^3+42n^2-40n-48)/48 - del[4,n](3n/4) + del[6,n](-53n^2+310n)/12 + del[12,n](49n/2) + del[18,n]*32n + del[24,n]*19n - del[30,n]*36n - del[42,n]*50n - del[60,n]*190n - del[84,n]*78n - del[90,n]*48n - del[120,n]*78n - del[210,n]*48n]; Table[R[n], {n,1,1000}] (* T. D. Noe, Dec 21 2006 *)

/* Only for odd n > 3, not suitable for other values of n! */ { a(n)=local(nr,x,fn,cn,fn2); nr=0; fn=floor(n/2); cn=ceil(n/2); fn2=(fn-1)^2-1; nr=fn2*n+fn+(n-2)*fn+cn; x=(n-5)/2; if (x>0,nr+=x*(x+1)*(2*x+1)/6*n); nr; } \\ Jon Perry, Jul 08 2003

apply( {A007678(n)=if(n%2, (((n-6)*n+23)*n-42)*n/24+1, ((n^3/2 -17*n^2/4 +22*n -if(n%4, 31, 40) +!(n%6)*(310 -53*n))/12 +!(n%12)*49/2 +!(n%18)*32 +!(n%24)*19 -!(n%30)*36 -!(n%42)*50 -!(n%60)*190 -!(n%84)*78 -!(n%90)*48 -!(n%120)*78 -!(n%210)*48)*n)}, [1..44]) \\ M. F. Hasler, Aug 06 2021

def d(n,m): return not n % m
def A007678(n): return (1176*d(n,12)*n - 3744*d(n,120)*n + 1536*d(n,18)*n - d(n,2)*(5*n**3 - 42*n**2 + 40*n + 48) - 2304*d(n,210)*n + 912*d(n,24)*n - 1728*d(n,30)*n - 36*d(n,4)*n - 2400*d(n,42)*n - 4*d(n,6)*n*(53*n - 310) - 9120*d(n,60)*n - 3744*d(n,84)*n - 2304*d(n,90)*n + 2*n**4 - 12*n**3 + 46*n**2 - 84*n)//48 + 1 # Chai Wah Wu, Mar 08 2021

For odd n > 3, a(n) = sumstep {i=5, n, 2, (i-2)*floor(n/2)+(i-4)*ceiling(n/2)+1} + x*(x+1)*(2*x+1)/6*n), where x = (n-5)/2. Simplifying the floor/ceiling components gives the PARI code below. - Jon Perry, Jul 08 2003
For odd n, a(n) = (24 - 42*n + 23*n^2 - 6*n^3 + n^4)/24. - Graeme McRae, Dec 24 2004
a(n) = A006533(n) - n. - T. D. Noe, Dec 23 2006
For odd n, binomial transform of [1, 10, 29, 36, 16, 0, 0, 0, ...] = [1, 11, 50, 154, ...]. - Gary W. Adamson, Aug 02 2011
a(n) = A135565(n) - A007569(n) + 1. - Max Alekseyev
See the Mma code in A006533 for the explicit Poonen-Rubenstein formula that holds for all n. - N. J. A. Sloane, Jan 23 2020

More terms from Graeme McRae, Dec 26 2004

a(1) = a(2) = 0 prepended by Max Alekseyev, Dec 01 2011

A020477 Numbers whose sum of divisors is a cube.

Original entry on oeis.org

1, 7, 102, 110, 142, 159, 187, 381, 690, 714, 770, 994, 1034, 1054, 1065, 1113, 1164, 1173, 1265, 1293, 1309, 1633, 1643, 2667, 3638, 3937, 4505, 4830, 4855, 5373, 5671, 5730, 5997, 6486, 6517, 6906, 7130, 7238, 7378, 7455, 7755, 7905, 8148, 8211, 8426

Offset: 1

David W. Wilson

nonn

			Factor 381; divisors are 1, 3, 127, 381. Sum is 512. Integral cube root of n is 8. So 381 is in sequence.

David Wells, Curious and Interesting Numbers (Revised), Penguin Books, page 118.

K. D. Bajpai, Table of n, a(n) for n = 1..10800 (first 1000 terms from T. D. Noe)
Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.
Carol Nelson, David E. Penney, Carl Pomerance, 714 and 715, J. Recreational Mathematics 7(2), Spring 1974, 87-89 [copy from Wayback machine]
C. Nelson, D. E. Penney, and C. Pomerance (1974) 714 and 715, J. Recreational Mathematics 7(2), 87-89 (see top of page 89); alternative copy.

Cf. A000203, A006532.

Do[If[IntegerQ[DivisorSigma[1, n]^(1/3)], Print[n]], {n, 1, 10^4}]
Select[Range[10000],IntegerQ[Surd[DivisorSigma[1,#],3]]&] (* Harvey P. Dale, Nov 16 2014 *)

isok(n) = ispower(sigma(n), 3); \\ Michel Marcus, Jul 03 2014

A006533 Place n equally-spaced points around a circle and join every pair of points by a chord; this divides the circle into a(n) regions.

Original entry on oeis.org

1, 2, 4, 8, 16, 30, 57, 88, 163, 230, 386, 456, 794, 966, 1471, 1712, 2517, 2484, 4048, 4520, 6196, 6842, 9109, 9048, 12951, 14014, 17902, 19208, 24158, 21510, 31931, 33888, 41449, 43826, 52956, 52992, 66712, 70034, 82993, 86840, 102091, 97776, 124314, 129448, 149986, 155894, 179447, 179280

Offset: 1

N. J. A. Sloane, Bjorn Poonen (poonen(AT)math.princeton.edu)

This sequence and A007678 are two equivalent ways of presenting the same sequence. - N. J. A. Sloane, Jan 23 2020

In contrast to A007678, which only counts the polygons, this sequence also counts the n segments of the circle bounded by the arc of the circle and the straight line, both joining two neighboring points on the circle. Therefore a(n) = A007678(n) + n. - M. F. Hasler, Dec 12 2021

Jean Meeus, Wiskunde Post (Belgium), Vol. 10, 1972, pp. 62-63.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
M. F. Hasler, Interactive illustration of A006561(n) & A006533(n); colored version for n=6 and for n=8.
Sascha Kurz, m-gons in regular n-gons
J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
Burkard Polster (Mathologer), The hardest "What comes next?" (Euler's pentagonal formula), Youtube video, Oct 17 2020.
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, No. 1 (1998), pp. 135-156. (Author's copy.).
Bjorn Poonen and Michael Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.
Bjorn Poonen and Michael Rubinstein, Mathematica programs for these sequences
Michael Rubinstein, Drawings for n=4,5,6,...
Prasad Balakrishnan Warrier, The physiognomy of the Erdős-Szekeres conjecture (happy ending problem), Math. Student (Indian Math. Soc., 2024) Vol. 93, Nos. 3-4, 28-48.
Sequences formed by drawing all diagonals in regular polygon

Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

del[m_,n_]:=If[Mod[n,m]==0,1,0];
R[n_]:=(n^4-6n^3+23n^2-18n+24)/24 + del[2,n](-5n^3+42n^2-40n-48)/48 - del[4,n](3n/4) + del[6,n](-53n^2+310n)/12 + del[12,n](49n/2) + del[18,n]*32n + del[24,n]*19n - del[30,n]*36n - del[42,n]*50n - del[60,n]*190n - del[84,n]*78n - del[90,n]*48n - del[120,n]*78n - del[210,n]*48n;
Table[R[n], {n,1,1000}] (* T. D. Noe, Dec 21 2006 *)

apply( {A006533(n)=if(n%2, (((n-6)*n+23)*n-18)*n/24+1, ((n^3/2 -17*n^2/4 +22*n -if(n%4, 19, 28) +!(n%6)*(310 -53*n))/12 +!(n%12)*49/2 +!(n%18)*32 +!(n%24)*19 -!(n%30)*36 -!(n%42)*50 -!(n%60)*190 -!(n%84)*78 -!(n%90)*48 -!(n%120)*78 -!(n%210)*48)*n)}, [1..44]) \\ M. F. Hasler, Aug 06 2021

def d(n,m): return not n % m
def A006533(n): return (1176*d(n,12)*n - 3744*d(n,120)*n + 1536*d(n,18)*n - d(n,2)*(5*n**3 - 42*n**2 + 40*n + 48) - 2304*d(n,210)*n + 912*d(n,24)*n - 1728*d(n,30)*n - 36*d(n,4)*n - 2400*d(n,42)*n - 4*d(n,6)*n*(53*n - 310) - 9120*d(n,60)*n - 3744*d(n,84)*n - 2304*d(n,90)*n + 2*n**4 - 12*n**3 + 46*n**2 - 36*n)//48 + 1 # Chai Wah Wu, Mar 08 2021

Poonen and Rubinstein give an explicit formula for a(n) (see Mma code).

a(n) = A007678(n) + n. - T. D. Noe, Dec 23 2006

Added more terms from b-file. - N. J. A. Sloane, Jan 23 2020

Edited definition. - N. J. A. Sloane, Mar 17 2024

A008848 Squares whose sum of divisors is a square.

Original entry on oeis.org

1, 81, 400, 32400, 1705636, 3648100, 138156516, 295496100, 1055340196, 1476326929, 2263475776, 2323432804, 2592846400, 2661528100, 7036525456, 10994571025, 17604513124, 39415749156, 61436066769, 85482555876, 90526367376, 97577515876, 98551417041

Offset: 1

N. J. A. Sloane

nonn

Solutions to sigma(x^2) = (2k+1)^2. - Labos Elemer, Aug 22 2002
Intersection of A006532 and A000290. The product of any two coprime terms is also in this sequence. - Charles R Greathouse IV, May 10 2011
Also intersection of A069070 and A000290. - Michel Marcus, Oct 06 2013
Conjectures: (1) a(2) = 81 is the only prime power (A246655) in this sequence. (2) 81 and 400 are only terms x for which sigma(x) is in A246655. (3) x = 1 is the only such term that sigma(x) is also a term. See also comments in A074386, A336547 and A350072. - Antti Karttunen, Jul 03 2023, (2) corrected in May 11 2024

			n=81: sigma(81) = 1+3+9+27+81 = 121 = 11^2.
n=400: sigma(400) = sigma(16)*sigma(25) = 31*31 = 961.
n=32400 (= 81*400): sigma(32400) = 116281 = 341^2 = 121*961.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 10.
I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.

Donovan Johnson, Table of n, a(n) for n = 1..400

Terms of A008847 squared.
Cf. A028982, A001248, A000203, A074386, A231484 (odd terms), A246655, A336547, A350072.
Subsequence of A000290, of A006532, and of A069070.

Do[s=DivisorSigma[1, n^2]; If[IntegerQ[Sqrt[s]]&&Mod[s, 2]==1, Print[n^2]], {n, 1, 10000000}] (* Labos Elemer *)
Select[Range[320000]^2,IntegerQ[Sqrt[DivisorSigma[1,#]]]&] (* Harvey P. Dale, Feb 22 2015 *)

for(n=1,1e6,if(issquare(sigma(n^2)), print1(n^2", "))) \\ Charles R Greathouse IV, May 10 2011

a(n) = A008847(n)^2.

A048251 a(n) is the smallest number whose sum of divisors is 6^n.

Original entry on oeis.org

1, 5, 22, 102, 510, 3210, 17490, 112890, 600270, 3466470, 20205570, 118879530, 697118730, 3949737330, 24217298490, 143487592710, 841422307110, 4973562896610, 29520886859310, 180254162529210, 1052751138726210, 6301225298627490, 37854941354933010, 224270177470178070

Offset: 0

Labos Elemer

nonn

Terms of this sequence are products of distinct terms in A005105. - Ray Chandler, Sep 01 2010

			sigma(k) = 1296 = 6^4 for each k in {510, 642, 710, 742, 782, 795, 862, 935, 1177, 1207, 1219}; the smallest of these is a(4)=510.

Ray Chandler, Table of n, a(n) for n = 0..1000

Cf. A006532, A020477, A019422, A019423, A018427, A048252-A048256.

a(n) = {my(k=1); while (sigma(k) != 6^n, k++); k;} \\ Michel Marcus, May 14 2018

a(n) = Min{k : A000203(k) = 6^n}.

a(9)-a(14) from Donovan Johnson, Sep 02 2008
a(15)-a(24) from Walter Kehowski, Aug 22 2010
Edited and extended by Ray Chandler, Sep 01 2010
Error in sequence corrected by N. J. A. Sloane, Oct 04 2010

A006535 Number of one-sided hexagonal polyominoes with n cells.

Original entry on oeis.org

1, 1, 3, 10, 33, 147, 620, 2821, 12942, 60639, 286190, 1364621, 6545430, 31586358, 153143956, 745700845, 3644379397, 17869651166, 87877879487, 433301253231, 2141584454057, 10607707971062, 52646117638427, 261756764824964, 1303625908234997, 6502430891223011, 32480041218465452, 162454295068924189, 813541940383789255, 4078750395194965720

Offset: 1

N. J. A. Sloane

J. Meeus, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Mason, Table of n, a(n) for n = 1..36
J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyhex

Cf. A000228, A030225, A001207.

a(n) = 2*A000228(n) - A030225(n).

a(7)-a(12) from David W. Wilson
a(13) from Achim Flammenkamp, Feb 15 1999
a(14)-a(20) from Joseph Myers, Sep 21 2002
a(21)-a(30) from John Mason, Jul 18 2023

A019423 Numbers whose sum of divisors is a fifth power.

Original entry on oeis.org

1, 21, 31, 651, 889, 3210, 3498, 3710, 3882, 3910, 4310, 4922, 4982, 5182, 5457, 5885, 6035, 6095, 6307, 6797, 7117, 7327, 7597, 24573, 27559, 71193, 82110, 90510, 94981, 97410, 98671, 99301, 99510, 100110, 103362, 104622, 107778, 108438, 108822

Offset: 1

David W. Wilson

nonn

Marius A. Burtea, Table of n, a(n) for n = 1..3648 (terms 1..1000 from Donovan Johnson; a(1211) re-indexed and duplicate a(2311) removed by _Georg Fischer_, Mar 21 2022).
Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.

Cf. A000203, A006532, A019422, A019424, A020477, A048257, A048258.

[n:n in [1..10000]| IsPower(SumOfDivisors(n),5)]; // Marius A. Burtea, Apr 17 2019

lista(nn) = {for (i=1, nn, s = sigma(i); if (s == 1 || ispower(s, 5), print1(i, ", ")););} \\ Michel Marcus, Jun 12 2013

A008847 Numbers k such that sum of divisors of k^2 is a square.

Original entry on oeis.org

1, 9, 20, 180, 1306, 1910, 11754, 17190, 32486, 38423, 47576, 48202, 50920, 51590, 83884, 104855, 132682, 198534, 247863, 292374, 300876, 312374, 313929, 334330, 345807, 376095, 428184, 433818, 458280, 464310, 469623, 498892, 623615, 754956, 768460, 787127, 943695, 985369

Offset: 1

N. J. A. Sloane

These are the square roots of squares in A006532. - M. F. Hasler, Oct 23 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 10.
I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.

Donovan Johnson, Table of n, a(n) for n = 1..400 (first 161 terms from Zak Seidov)

Cf. A008848, A008849, A008850, A163763.

Cf. A000203, A010052, A000290.

a008847 n = a008847_list !! (n-1)
a008847_list = filter ((== 1) . a010052 . a000203 . a000290) [1..]
-- Reinhard Zumkeller, Mar 27 2013

with(numtheory): readlib(issqr): for i from 1 to 10^5 do if issqr(sigma(i^2)) then print(i); fi; od;

s = {}; Do[ If[IntegerQ[ Sqrt[ DivisorSigma[1, n^2]]], Print[n]; AppendTo[s, n]], {n, 10^6}]; s (* Jean-François Alcover, May 05 2011 *)
Select[Range[1000000],IntegerQ[Sqrt[DivisorSigma[1,#^2]]]&] (* Harvey P. Dale, Aug 22 2011 *)

is_A008847(n)=issquare(sigma(n^2)) \\ M. F. Hasler, Oct 23 2010

A163763(n) = sqrt(sigma(A008847(n)^2)). - M. F. Hasler, Oct 16 2010

a(n) = sqrt(A008848(n)). - Zak Seidov, May 01 2016

A048699 Nonprime numbers whose sum of aliquot divisors (A001065) is a perfect square.

Original entry on oeis.org

1, 9, 12, 15, 24, 26, 56, 75, 76, 90, 95, 119, 122, 124, 140, 143, 147, 153, 176, 194, 215, 243, 287, 332, 363, 386, 407, 477, 495, 507, 511, 524, 527, 536, 551, 575, 688, 738, 791, 794, 815, 867, 871, 892, 924, 935, 963, 992, 1075, 1083, 1159, 1196, 1199, 1295, 1304

Offset: 1

Enoch Haga

The sum of aliquot divisors of prime numbers is 1.

If a^2 is an odd square for which a^2-1 = p + q with p,q primes, then p*q is a term. If m = 2^k-1 is a Mersenne prime then m*(2^k) (twice an even perfect number) is a term. If b = 2^j is a square and b-7 = 3s is a semiprime then 4s is a term. - Metin Sariyar, Apr 02 2020

			a(3)=15; aliquot divisors are 1,3,5; sum of aliquot divisors = 9 and 3^2=9.

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

Cf. A001065, A006532, A020477, A048698, A073040 (includes primes).

a := []; for n from 1 to 2000 do if sigma(n) <> n+1 and issqr(sigma(n)-n) then a := [op(a), n]; fi; od: a;

nn=1400;Select[Complement[Range[nn],Prime[Range[PrimePi[nn]]]],IntegerQ[ Sqrt[DivisorSigma[1,#]-#]]&] (* Harvey P. Dale, Apr 25 2011 *)

isok(k) = !ispseudoprime(k) && issquare(sigma(k) - k); \\ Michel Marcus, May 13 2025

cp's OEIS Frontend

A178492 Primitive elements of A006532, i.e., which are not product of earlier terms (> 1) in that sequence.

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A007678 Number of regions in regular n-gon with all diagonals drawn.

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A020477 Numbers whose sum of divisors is a cube.

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A006533 Place n equally-spaced points around a circle and join every pair of points by a chord; this divides the circle into a(n) regions.

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A008848 Squares whose sum of divisors is a square.

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A048251 a(n) is the smallest number whose sum of divisors is 6^n.

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PARI

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A006535 Number of one-sided hexagonal polyominoes with n cells.

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