OFFSET
1,1
COMMENTS
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).
For a proof of the formula for this sequence see the link.
LINKS
Hartmut F. W. Hoft, Diagram of symmetric representations of sigma
Hartmut F. W. Hoft, Proof of formula for 4 regions of width 1
FORMULA
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.
EXAMPLE
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.
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.
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.
m=0| 3 5 7 11 13
-----------------------------
7 | 21
11 | 33 55
13 | 39 65
17 | 51 85 119
19 | 57 95 133
23 | 69 115 161 253
29 | 87 145 203 319 377
31 | 93 155 217 341 403
37 | 111 185 259 407 481
41 | 123 205 287 451 533
...
89 | 267 445 623 979 1157
...
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.
m=1| 3 5 7 11 13
-------------------------------
23 | 230
29 | 290 406
31 | 310 434
37 | 370 518
41 | 410 574
43 | 430 602
47 | 470 658 1034
53 | 530 742 1166 1378
...
89 | 890 1246 1958 2314
...
m=2| 3 5 7 11 13
-------------------------------
89 | 3916
97 | 4268
101| 4444
103| 4532
107| 4708 5564
109| 4796 5668
...
The fourth layer for m = 3 starts with number 37672 in column p = 17 and row q = 277.
The subsequence of the 2-parameter case n = 2^m * p^3 with 2^(m+1) < p gives rise to the following irregular triangle:
p\m| 0 1 2 3
----------------------------------
3 | 27
5 | 125 250
7 | 343 686
11 | 1331 2662 5324
13 | 2197 4394 8788
17 | 4913 9826 19652 39304
19 | 6859 13718 27436 54872
23 | 12167 24334 48668 97336
29 | 24389 48778 97556 195112
...
MATHEMATICA
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]
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]
(* 2^(4m+3)<=bound is a simpler test, but computes some empty stalks *)
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]
a264102[335] (* data *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Hartmut F. W. Hoft, Nov 03 2015
STATUS
approved