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.
%I A239050 #86 Feb 25 2025 21:01:32 %S A239050 4,12,16,28,24,48,32,60,52,72,48,112,56,96,96,124,72,156,80,168,128, %T A239050 144,96,240,124,168,160,224,120,288,128,252,192,216,192,364,152,240, %U A239050 224,360,168,384,176,336,312,288,192,496,228,372,288,392,216,480,288,480,320,360,240,672,248,384,416,508 %N A239050 a(n) = 4*sigma(n). %C A239050 4 times the sum of divisors of n. %C A239050 a(n) is also the total number of horizontal cells in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) where the structure of every three-dimensional quadrant arises after the 90-degree zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a square formed by four cells (see links and examples). - _Omar E. Pol_, Jul 04 2016 %H A239050 Antti Karttunen, <a href="/A239050/b239050.txt">Table of n, a(n) for n = 1..10000</a> %H A239050 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr02.jpg">Diagram of the triangle before the 90-degree-zig-zag folding (rows: 1..28)</a> %H A239050 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr01.jpg">Folding the first eight rows of triangle</a> %H A239050 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a> %F A239050 a(n) = 4*A000203(n) = 2*A074400(n). %F A239050 a(n) = A000203(n) + A272027(n). - _Omar E. Pol_, Jul 04 2016 %F A239050 Dirichlet g.f.: 4*zeta(s-1)*zeta(s). - _Ilya Gutkovskiy_, Jul 04 2016 %F A239050 Conjecture: a(n) = sigma(3*n) = A144613(n) iff n is not a multiple of 3. - _Omar E. Pol_, Oct 02 2018 %F A239050 The conjecture above is correct. Write n = 3^e*m, gcd(3, m) = 1, then sigma(3*n) = sigma(3^(e+1))*sigma(m) = ((3^(e+2) - 1)/2)*sigma(m) = ((3^(e+2) - 1)/(3^(e+1) - 1))*sigma(3^e*m), and (3^(e+2) - 1)/(3^(e+1) - 1) = 4 if and only if e = 0. - _Jianing Song_, Feb 03 2019 %e A239050 For n = 4 the sum of divisors of 4 is 1 + 2 + 4 = 7, so a(4) = 4*7 = 28. %e A239050 For n = 5 the sum of divisors of 5 is 1 + 5 = 6, so a(5) = 4*6 = 24. %e A239050 . %e A239050 Illustration of initial terms: _ _ _ _ _ _ %e A239050 . _ _ _ _ _ _ |_|_|_|_|_|_| %e A239050 . _ _ _ _ _|_|_|_|_|_|_|_ _ _| |_ _ %e A239050 . _ _ _ _ _|_|_|_|_|_ |_|_| |_|_| |_| |_| %e A239050 . _ _ |_|_|_|_| |_| |_| |_| |_| |_| |_| %e A239050 . |_|_| |_| |_| |_| |_| |_| |_| |_| |_| %e A239050 . |_|_| |_|_ _|_| |_| |_| |_| |_| |_| |_| %e A239050 . |_|_|_|_| |_|_ _ _ _|_| |_|_ _|_| |_| |_| %e A239050 . |_|_|_|_| |_|_|_ _ _ _|_|_| |_|_ _|_| %e A239050 . |_|_|_|_|_|_| |_ _ _ _ _ _| %e A239050 . |_|_|_|_|_|_| %e A239050 . %e A239050 n: 1 2 3 4 5 %e A239050 S(n): 1 3 4 7 6 %e A239050 a(n): 4 12 16 28 24 %e A239050 . %e A239050 For n = 1..5, the figure n represents the reflection in the four quadrants of the symmetric representation of S(n) = sigma(n) = A000203(n). For more information see A237270 and A237593. %e A239050 The diagram also represents the top view of the first four terraces of the stepped pyramid described in Comments section. - _Omar E. Pol_, Jul 04 2016 %p A239050 with(numtheory): seq(4*sigma(n), n=1..64); # _Omar E. Pol_, Jul 04 2016 %t A239050 Array[4 DivisorSigma[1, #] &, 64] (* _Michael De Vlieger_, Nov 16 2017 *) %o A239050 (PARI) a(n) = 4 * sigma(n); \\ _Omar E. Pol_, Jul 04 2016 %o A239050 (Magma) [4*SumOfDivisors(n): n in [1..70]]; // _Vincenzo Librandi_, Jul 30 2019 %Y A239050 Alternating row sums of A239662. %Y A239050 Partial sums give A243980. %Y A239050 k times sigma(n), k=1..6: A000203, A074400, A272027, this sequence, A274535, A274536. %Y A239050 k times sigma(n), k = 1..10: A000203, A074400, A272027, this sequence, A274535, A274536, A319527, A319528, A325299, A326122. %Y A239050 Cf. A008438, A017113, A062731, A112610, A144613, A193553, A196020, A235791, A236104, A237270, A237593, A239052, A239053, A239660, A239662, A244050, A262626. %K A239050 nonn,easy %O A239050 1,1 %A A239050 _Omar E. Pol_, Mar 09 2014