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 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. 5
 1, 2, 6, 3, 30, 10, 15, 5, 210, 42, 70, 14, 105, 21, 35, 7, 2310, 330, 462, 66, 770, 110, 154, 1155, 22, 165, 231, 33, 385, 55, 77, 11, 30030, 2730, 4290, 6006, 390, 546, 858, 10010, 78, 910, 1430, 2002, 130, 15015, 182, 286, 1365, 2145, 26, 3003, 195, 273, 429 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Let gpf(m) = A006530(m) and let phi(m) = A000010(m) for m in A005117. Row n contains m in A005117 such that A000720(A006530(m)) = n, sorted such that phi(m)/m increases as k increases. 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. 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. 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. 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. LINKS Michael De Vlieger, Table of n, a(n) for n = 0..16384 Michael De Vlieger, Small plot of m in A307540 at x = pi(gpf(m)), y = phi(m)/m. Michael De Vlieger, Enlarged plot of m in A307540 at x = pi(gpf(m)), y = phi(m)/m. FORMULA For n > 0, row lengths = A000079(n - 1). T(n, 1) = A002110(n) = p_n#. T(n, 2) = A306237(n) = p_n#/prime(n - 1). T(n, 2^(n - 1) - 1) = A006094(n). T(n, 2^(n - 1)) = A000040(n) = prime(n) for n >= 1. Last even term in row n = A077017(n). First odd term in row n = A070826(n). EXAMPLE Triangle begins: 1; 2; 6, 3; 30, 10, 15, 5; 210, 42, 70, 14, 105, 21, 35, 7; ... First terms of this sequence appear bottom to top in the chart below. The values of n appear in the header, values m = T(n,k) followed parenthetically by phi(m)/m appear in column n. The x axis plots according to primepi(gpf(m)), while the y axis plots k according to phi(m)/m: 0 1 2 3 4 . . . . . --- 1 ------------------------------------------------ (1/1) . . . . . . . . . . . . . . . . . . 7 . . . 5 (6/7) . . . (4/5) . . . . . . . . . . 35 . . 3 . (24/35) . . (2/3) . . . . . . . . . . . . . . . . 21 . . . . (4/7) . . . 15 . . . . (8/15) . . 2 . . . . (1/2) . . . . . . . . . . . . 105 . . . . (16/35) . . . . 14 . . . 10 (3/7) . . . (2/5) . . . . . . . . . . 70 . . 6 . (12/35) . . (1/3) . . . . . . 42 . . . 30 (2/7) . . . (4/15) . . . . . 210 . . . . (8/35) ... MATHEMATICA 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 CROSSREFS Cf. A000010, A000040, A000079, A000720, A002110, A005117, A006094, A006530, A007947, A059894, A070826, A077017, A306237. Sequence in context: A283478 A125666 A371799 * A304087 A284003 A172031 Adjacent sequences: A307537 A307538 A307539 * A307541 A307542 A307543 KEYWORD nonn,tabf,easy AUTHOR Michael De Vlieger, Apr 13 2019 STATUS approved

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Last modified August 9 02:38 EDT 2024. Contains 375024 sequences. (Running on oeis4.)