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A264102 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one.

%I A264102 #16 Oct 04 2018 06:52:11
%S A264102 21,27,33,39,51,55,57,65,69,85,87,93,95,111,115,119,123,125,129,133,
%T A264102 141,145,155,159,161,177,183,185,201,203,205,213,215,217,219,230,235,
%U A264102 237,249,250,253,259,265,267,287,290,291,295,301,303,305,309,310,319,321,327,329,335
%N A264102 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one.
%C A264102 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 A264102 For a proof of the formula for this sequence see the link.
%H A264102 Hartmut F. W. Hoft, <a href="/A264102/a264102.pdf">Diagram of symmetric representations of sigma</a>
%H A264102 Hartmut F. W. Hoft, <a href="/A264102/a264102_1.pdf">Proof of formula for 4 regions of width 1</a>
%F A264102 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 A264102 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 A264102 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 A264102 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 A264102 m=0| 3    5    7    11   13
%e A264102 -----------------------------
%e A264102 7  | 21
%e A264102 11 | 33   55
%e A264102 13 | 39   65
%e A264102 17 | 51   85   119
%e A264102 19 | 57   95   133
%e A264102 23 | 69   115  161  253
%e A264102 29 | 87   145  203  319  377
%e A264102 31 | 93   155  217  341  403
%e A264102 37 | 111  185  259  407  481
%e A264102 41 | 123  205  287  451  533
%e A264102 ...
%e A264102 89 | 267  445  623  979  1157
%e A264102 ...
%e A264102 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 A264102 m=1| 3    5    7    11   13
%e A264102 -------------------------------
%e A264102 23 |     230
%e A264102 29 |     290  406
%e A264102 31 |     310  434
%e A264102 37 |     370  518
%e A264102 41 |     410  574
%e A264102 43 |     430  602
%e A264102 47 |     470  658  1034
%e A264102 53 |     530  742  1166  1378
%e A264102 ...
%e A264102 89 |     890  1246 1958  2314
%e A264102 ...
%e A264102 m=2| 3    5    7    11   13
%e A264102 -------------------------------
%e A264102 89 |               3916
%e A264102 97 |               4268
%e A264102 101|               4444
%e A264102 103|               4532
%e A264102 107|               4708  5564
%e A264102 109|               4796  5668
%e A264102 ...
%e A264102 The fourth layer for m = 3 starts with number 37672 in column p = 17 and row q = 277.
%e A264102 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 A264102 p\m| 0      1       2       3
%e A264102 ----------------------------------
%e A264102 3  | 27
%e A264102 5  | 125    250
%e A264102 7  | 343    686
%e A264102 11 | 1331   2662    5324
%e A264102 13 | 2197   4394    8788
%e A264102 17 | 4913   9826    19652   39304
%e A264102 19 | 6859   13718   27436   54872
%e A264102 23 | 12167  24334   48668   97336
%e A264102 29 | 24389  48778   97556   195112
%e A264102 ...
%e A264102 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 A264102 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 A264102 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 A264102 (* 2^(4m+3)<=bound is a simpler test, but computes some empty stalks *)
%t A264102 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 a264102[335] (* data *)
%Y A264102 Cf. A001748, A001750, A030078, A172190.
%Y A264102 For symmetric representation of sigma: A235791, A236104, A237270, A237271, A237591, A237593, A241008, A246955.
%Y A264102 Subsequence of A280107.
%K A264102 nonn,tabf
%O A264102 1,1
%A A264102 _Hartmut F. W. Hoft_, Nov 03 2015