A264102 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one.
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
Examples
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.
Links
- Hartmut F. W. Hoft, Diagram of symmetric representations of sigma
- Hartmut F. W. Hoft, Proof of formula for 4 regions of width 1
Crossrefs
Programs
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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 *)
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.
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