This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A307540 #12 Apr 06 2020 19:57:41
%S A307540 1,2,6,3,30,10,15,5,210,42,70,14,105,21,35,7,2310,330,462,66,770,110,
%T A307540 154,1155,22,165,231,33,385,55,77,11,30030,2730,4290,6006,390,546,858,
%U A307540 10010,78,910,1430,2002,130,15015,182,286,1365,2145,26,3003,195,273,429
%N A307540 Irregular triangle T(n,k) such that squarefree m with gpf(m) = prime(n) in each row are arranged according to increasing values of phi(m)/m.
%C A307540 Let gpf(m) = A006530(m) and let phi(m) = A000010(m) for m in A005117.
%C A307540 Row n contains m in A005117 such that A000720(A006530(m)) = n, sorted such that phi(m)/m increases as k increases.
%C A307540 Let m be the squarefree kernel A007947(m') of m'. We only consider squarefree m since phi(m)/m = phi(m')/m'. Let prime p | n and prime q be a nondivisor of n.
%C A307540 Since m is squarefree, we might encode the multiplicities of its prime divisors in a positional notation M that is finite at n significant digits. For example, m = 42 can be encoded reverse(A067255(42)) = 1,0,1,1 = 7^1 * 5^0 * 3^1 * 2^1. It is necessary to reverse row m of A067255 (hereinafter simply A067255(m)) so as to preserve zeros in M = A067255(m) pertaining to small nondivisor primes q < p. The code M is a series of 0's and 1's since m is squarefree. Then it is clear that row n contains all m such that A067255(m) has n terms, and there are 2^(n - 1) possible terms for n >= 1.
%C A307540 We may use an approach that generates the binary expansion of the range 2^(n - 1) < M < 2^n - 1, or we may append 1 to the reversed (n - 1)-tuples of {1, 0} to achieve codes M -> m for each row n, which is tantamount to ordering according to A059894.
%C A307540 Originally it was thought that the codes M were in order of the latter algorithm, and we could avoid sorting. Observation shows that the m still require sorting by the function phi(m)/m indeed to be in increasing order in row n. Still, the latter approach is slightly more efficient than the former in generating the sequence.
%H A307540 Michael De Vlieger, <a href="/A307540/b307540.txt">Table of n, a(n) for n = 0..16384</a>
%H A307540 Michael De Vlieger, <a href="/A307540/a307540.png">Small plot of m in A307540 at x = pi(gpf(m)), y = phi(m)/m</a>.
%H A307540 Michael De Vlieger, <a href="/A307540/a307540_1.png">Enlarged plot of m in A307540 at x = pi(gpf(m)), y = phi(m)/m</a>.
%F A307540 For n > 0, row lengths = A000079(n - 1).
%F A307540 T(n, 1) = A002110(n) = p_n#.
%F A307540 T(n, 2) = A306237(n) = p_n#/prime(n - 1).
%F A307540 T(n, 2^(n - 1) - 1) = A006094(n).
%F A307540 T(n, 2^(n - 1)) = A000040(n) = prime(n) for n >= 1.
%F A307540 Last even term in row n = A077017(n).
%F A307540 First odd term in row n = A070826(n).
%e A307540 Triangle begins:
%e A307540 1;
%e A307540 2;
%e A307540 6, 3;
%e A307540 30, 10, 15, 5;
%e A307540 210, 42, 70, 14, 105, 21, 35, 7;
%e A307540 ...
%e A307540 First terms of this sequence appear bottom to top in the chart below. The
%e A307540 values of n appear in the header, values m = T(n,k) followed
%e A307540 parenthetically by phi(m)/m appear in column n. The x axis plots
%e A307540 according to primepi(gpf(m)), while the y axis plots k according to
%e A307540 phi(m)/m:
%e A307540 0 1 2 3 4
%e A307540 . . . . .
%e A307540 --- 1 ------------------------------------------------
%e A307540 (1/1) . . . .
%e A307540 . . . . .
%e A307540 . . . . .
%e A307540 . . . . 7
%e A307540 . . . 5 (6/7)
%e A307540 . . . (4/5) .
%e A307540 . . . . .
%e A307540 . . . . 35
%e A307540 . . 3 . (24/35)
%e A307540 . . (2/3) . .
%e A307540 . . . . .
%e A307540 . . . . .
%e A307540 . . . . 21
%e A307540 . . . . (4/7)
%e A307540 . . . 15 .
%e A307540 . . . (8/15) .
%e A307540 . 2 . . .
%e A307540 . (1/2) . . .
%e A307540 . . . . .
%e A307540 . . . . 105
%e A307540 . . . . (16/35)
%e A307540 . . . . 14
%e A307540 . . . 10 (3/7)
%e A307540 . . . (2/5) .
%e A307540 . . . . .
%e A307540 . . . . 70
%e A307540 . . 6 . (12/35)
%e A307540 . . (1/3) . .
%e A307540 . . . . 42
%e A307540 . . . 30 (2/7)
%e A307540 . . . (4/15) .
%e A307540 . . . . 210
%e A307540 . . . . (8/35)
%e A307540 ...
%t A307540 Prepend[Array[SortBy[#, Last] &@ Map[{#1, #2, EulerPhi[#1]/#1} & @@ {Times @@ MapIndexed[Prime[First@ #2]^#1 &, Reverse@ #], FromDigits@ #} &, Map[Prepend[Reverse@ #, 1] &, Tuples[{1, 0}, # - 1]]] &, 6], {{1, 0, 1}}][[All, All, 1]] // Flatten
%Y A307540 Cf. A000010, A000040, A000079, A000720, A002110, A005117, A006094, A006530, A007947, A059894, A070826, A077017, A306237.
%K A307540 nonn,tabf,easy
%O A307540 0,2
%A A307540 _Michael De Vlieger_, Apr 13 2019