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%I #16 Oct 04 2018 06:52:11
%S 21,27,33,39,51,55,57,65,69,85,87,93,95,111,115,119,123,125,129,133,
%T 141,145,155,159,161,177,183,185,201,203,205,213,215,217,219,230,235,
%U 237,249,250,253,259,265,267,287,290,291,295,301,303,305,309,310,319,321,327,329,335
%N Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one.
%C The areas of the first two regions are (2^(m+1) - 1) * (p * q + 1) / 2 and (2^(m+1) - 1) * (p + q) / 2, respectively. Twice their sum equals sigma(n) = (2^(m+1) - 1) * (p + 1) * (q + 1).
%C For a proof of the formula for this sequence see the link.
%H Hartmut F. W. Hoft, <a href="/A264102/a264102.pdf">Diagram of symmetric representations of sigma</a>
%H Hartmut F. W. Hoft, <a href="/A264102/a264102_1.pdf">Proof of formula for 4 regions of width 1</a>
%F n = 2^m * p * q where m >= 0, p > 2 is prime, 2^(m+1) < p < 2^(m+1) * p < q, and either q is prime or q = p^2.
%e 65 = 5*13 is in the sequence since m = 0 and 2 < 5 < 10 < 13. The first two regions in the symmetric representation of sigma(65) = 84 start with legs 1 and 5 of the Dyck path and have areas 33 and 9, respectively.
%e 406 = 2*7*29 is in the sequence since m=1 and 4 < 7 < 28 < 29. The first two regions in the symmetric representation of sigma(406) = 720 start with legs 1 and 7 and have areas 306 and 54, respectively. Note also that 406 is a triangular number and the middle two regions meet at the center of the Dyck path.
%e One case in the formula for the sequence is the 3-parameter expression n = 2^m * p * q with p and q distinct primes satisfying the stated conditions. That subsequence can be visualized as a skew tetrahedron since the start of each "line" on an irregular "triangular" side of the "tetrahedron" is determined by a different prime number and each layer is determined by a different power of two. Below are the first three layers with primes p designating columns and primes q rows.
%e m=0| 3 5 7 11 13
%e -----------------------------
%e 7 | 21
%e 11 | 33 55
%e 13 | 39 65
%e 17 | 51 85 119
%e 19 | 57 95 133
%e 23 | 69 115 161 253
%e 29 | 87 145 203 319 377
%e 31 | 93 155 217 341 403
%e 37 | 111 185 259 407 481
%e 41 | 123 205 287 451 533
%e ...
%e 89 | 267 445 623 979 1157
%e ...
%e Column 1 is A001748 except for the first three terms and column 2 is A001750 except for the first four terms in the two resepctive sequences.
%e m=1| 3 5 7 11 13
%e -------------------------------
%e 23 | 230
%e 29 | 290 406
%e 31 | 310 434
%e 37 | 370 518
%e 41 | 410 574
%e 43 | 430 602
%e 47 | 470 658 1034
%e 53 | 530 742 1166 1378
%e ...
%e 89 | 890 1246 1958 2314
%e ...
%e m=2| 3 5 7 11 13
%e -------------------------------
%e 89 | 3916
%e 97 | 4268
%e 101| 4444
%e 103| 4532
%e 107| 4708 5564
%e 109| 4796 5668
%e ...
%e The fourth layer for m = 3 starts with number 37672 in column p = 17 and row q = 277.
%e The subsequence of the 2-parameter case n = 2^m * p^3 with 2^(m+1) < p gives rise to the following irregular triangle:
%e p\m| 0 1 2 3
%e ----------------------------------
%e 3 | 27
%e 5 | 125 250
%e 7 | 343 686
%e 11 | 1331 2662 5324
%e 13 | 2197 4394 8788
%e 17 | 4913 9826 19652 39304
%e 19 | 6859 13718 27436 54872
%e 23 | 12167 24334 48668 97336
%e 29 | 24389 48778 97556 195112
%e ...
%e The first column in this triangle is A030078 except for the first term and the second column is A172190 except for the first two terms respectively in the two sequences.
%t mpStalk[m_, p_, bound_] := Module[{q=NextPrime[2^(m+1)*p], list={}}, While[2^m*p*q<=bound, AppendTo[list, 2^m*p*q]; q=NextPrime[q]]; If[2^m*p^3<=bound, AppendTo[list, 2^m*p^3]]; list]
%t mTriangle[m_, bound_] := Module[{p=NextPrime[2^(m+1)], list={}}, While[2^m*p*NextPrime[2^(m+1)*p]<=bound, list=Union[list, mpStalk[m, p, bound]]; p=NextPrime[p]]; list]
%t (* 2^(4m+3)<=bound is a simpler test, but computes some empty stalks *)
%t a264102[bound_] := Module[{m=0, list={}}, While[2^m*NextPrime[2^(m+1)]*NextPrime[2^(m+1)*NextPrime[2^(m+1)]]<=bound, list=Union[list, mTriangle[m, bound]]; m++]; list]
%t a264102[335] (* data *)
%Y Cf. A001748, A001750, A030078, A172190.
%Y For symmetric representation of sigma: A235791, A236104, A237270, A237271, A237591, A237593, A241008, A246955.
%Y Subsequence of A280107.
%K nonn,tabf
%O 1,1
%A _Hartmut F. W. Hoft_, Nov 03 2015
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