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A264102 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one. 3
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..58.

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

...

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

Cf. A001748, A001750, A030078, A172190.

For symmetric representation of sigma: A235791, A236104, A237270, A237271, A237591, A237593, A241008, A246955.

Subsequence of A280107.

Sequence in context: A064507 A249729 A280107 * A103246 A206347 A247316

Adjacent sequences:  A264099 A264100 A264101 * A264103 A264104 A264105

KEYWORD

nonn,tabf

AUTHOR

Hartmut F. W. Hoft, Nov 03 2015

STATUS

approved

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Last modified April 7 10:55 EDT 2020. Contains 333301 sequences. (Running on oeis4.)