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A264102
Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one.
7
21, 27, 33, 39, 51, 55, 57, 65, 69, 85, 87, 93, 95, 111, 115, 119, 123, 125, 129, 133, 141, 145, 155, 159, 161, 177, 183, 185, 201, 203, 205, 213, 215, 217, 219, 230, 235, 237, 249, 250, 253, 259, 265, 267, 287, 290, 291, 295, 301, 303, 305, 309, 310, 319, 321, 327, 329, 335
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.
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
...
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.
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
For symmetric representation of sigma: A235791, A236104, A237270, A237271, A237591, A237593, A241008, A246955.
Subsequence of A280107.
Sequence in context: A064507 A249729 A280107 * A364414 A365081 A352096
KEYWORD
nonn,tabf
AUTHOR
Hartmut F. W. Hoft, Nov 03 2015
STATUS
approved