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%I #37 Apr 07 2020 23:23:45
%S 0,1,3,2,7,5,6,4,15,11,13,9,14,10,12,8,31,23,27,19,29,21,25,30,17,22,
%T 26,18,28,20,24,16,63,47,55,59,39,43,51,61,35,45,53,57,37,62,41,49,46,
%U 54,33,58,38,42,50,60,34,44,52,56,36,40,48,32,127,95,111,119
%N Irregular triangle read by rows: T(n,k) = A087207(A307540(n,k)).
%C Let gpf(m) = A006530(m) and let phi(m) = A000010(m) for m in A005117.
%C Row n contains m in A005117 such that A006530(m) = n, sorted such that phi(m)/m increases as k increases.
%C 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 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 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} (as A059894) to achieve codes M -> m for each row n.
%C 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.
%C This sequence interprets the code M as a binary value. The sequence is a permutation of the natural numbers since the ratio phi(m)/m is unique for squarefree m.
%C This sequence and A059894 are identical for 1 <= n <= 23.
%C Numbers of terms in rows n of this sequence and A059894 (partitioned by powers of 2) that are coincident: 1, 2, 4, 8, 14, 14, 10, 26, 14, 20, 10, 16, 22, 12, 18, 18, 16, 14, 18, 18, 18, 14, 16, ...}.
%C The graphs of this sequence and A059894 are similar.
%C The graph of this sequence feature squares of size 2^(j-1) at (x,y) = (h,h) where h = 2^j, integers, that have pi-radian rotational symmetry.
%H Antti Karttunen, <a href="/A307544/b307544.txt">Table of n, a(n) for n = 0..16383</a>
%H Michael De Vlieger, <a href="/A307544/a307544.png">Plot comparing A059894 and A307544</a>.
%H Antti Karttunen, <a href="/A307544/a307544.txt">Data supplement: n, a(n) computed for n = 0..65535</a> (including terms 0..16384 previously computed by Michael De Vlieger)
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F For n > 0, row lengths = 2^(n - 1).
%F T(n,1) = 2^n - 1 = A000225(n).
%F T(n,2^(n - 1)) = 2^(n - 1).
%e 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. In square brackets, we write the multiplicities of primes in positional order with the smallest prime at right (big-endian). The x axis plots k according to primepi(gpf(m)), while the y axis plots k according to phi(m)/m:
%e 0 1 2 3 4
%e . . . . .
%e --- 1 ------------------------------------------------
%e (1/1) . . . .
%e [0] . . . .
%e . . . . .
%e . . . . 7
%e . . . 5 (6/7)
%e . . . (4/5) [1000]
%e . . . [100] .
%e . . . . 35
%e . . 3 . (24/35)
%e . . (2/3) . [1100]
%e . . [10] . .
%e . . . . .
%e . . . . 21
%e . . . . (4/7)
%e . . . 15 [1010]
%e . . . (8/15) .
%e . 2 . [110] .
%e . (1/2) . . .
%e . [1] . . 105
%e . . . . (16/35)
%e . . . . [1110]
%e . . . . 14
%e . . . 10 (3/7)
%e . . . (2/5) [1001]
%e . . . [101] .
%e . . . . 70
%e . . 6 . (12/35)
%e . . (1/3) . [1101]
%e . . [11] . 42
%e . . . 30 (2/7)
%e . . . (4/15) [1011]
%e . . . [111] 210
%e . . . . (8/35)
%e . . . . [1111]
%e ...
%e a(1) = 0 since T(0,1) = 1 = empty product.
%e a(2) = 1 since T(1,1) = 2 = 2^1 -> binary "1" = decimal 1.
%e a(3) = 3 since T(2,1) = 6 = 2^1 * 3^1 -> binary "11" = decimal 3.
%e a(4) = 2 since T(2,2) = 3 = 2^0 * 3^1 -> binary "10" = decimal 2.
%e a(5) = 7 since T(3,1) = 30 = 2^1 * 3^1 * 5^1 -> binary "111" = decimal 7, etc.
%e Graph of first 32 terms: (Begin)
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e x
%e (End)
%e From _Antti Karttunen_, Jan 10 2020: (Start)
%e Arranged as a binary tree:
%e 0
%e |
%e ...................1...................
%e 3 2
%e 7......../ \........5 6......../ \........4
%e / \ / \ / \ / \
%e / \ / \ / \ / \
%e / \ / \ / \ / \
%e 15 11 13 9 14 10 12 8
%e 31 23 27 19 29 21 25 30 17 22 26 18 28 20 24 16
%e etc.
%e (End)
%t Prepend[Array[SortBy[#, Last] &@ Map[{#2, EulerPhi[#1]/#1} & @@ {Times @@ MapIndexed[Prime[First@ #2]^#1 &, Reverse@ #], FromDigits[#, 2]} &, Map[Prepend[Reverse@ #, 1] &, Tuples[{1, 0}, # - 1]]] &, 7], {{0, 0, 1}}][[All, All, 1]] // Flatten
%o (PARI)
%o up_to = 1023;
%o rat(n) = { my(m=1, p=2); while(n, if(n%2, m *= (p-1)/p); n >>= 1; p = nextprime(1+p)); (m); };
%o cmpA307544(a,b) = if(!a,sign(-b),if(!b,sign(a), my(as=logint(a,2), bs=logint(b,2)); if(as!=bs, sign(as-bs), sign(rat(a)-rat(b)))));
%o A307544list(up_to) = vecsort(vector(1+up_to,n,n-1), cmpA307544);
%o v307544 = A307544list(up_to);
%o A307544(n) = v307544[1+n]; \\ _Antti Karttunen_, Jan 10 2020
%Y Cf. A000010, A000040, A000079, A000225, A002110, A005117, A006094, A006530, A007947, A048672, A059894, A067255, A087207, A225679, A225680, A306237, A307540.
%K nonn,easy,look,tabf
%O 0,3
%A _Michael De Vlieger_, Apr 19 2019
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