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A083140 Sieve of Eratosthenes arranged as an array and read by antidiagonals in the up direction; n-th row has property that smallest prime factor is prime(n). 47
2, 3, 4, 5, 9, 6, 7, 25, 15, 8, 11, 49, 35, 21, 10, 13, 121, 77, 55, 27, 12, 17, 169, 143, 91, 65, 33, 14, 19, 289, 221, 187, 119, 85, 39, 16, 23, 361, 323, 247, 209, 133, 95, 45, 18, 29, 529, 437, 391, 299, 253, 161, 115, 51, 20, 31, 841, 667, 551, 493, 377, 319, 203 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

A permutation of natural numbers >= 2.

Let Pn(n) = A002110 denote the primorial function. The subset in the n-th row as a fraction of the natural numbers is EulerPhi(Pn(n-1))/Pn(n).  For example, for n=5, EulerPhi(Pn(5-1)) = 48, Pn(5)=2310, so row 5 contains 48/2310 of the set of all natural numbers > 11.  The sum of EulerPhi(Pn(n-1))/Pn(n) for all n gives the fraction of natural numbers that aren't prime.  The infinite series is equivalent to EulerPhi(Pn(n))/Pn(n) for n terms. - Jamie Morken, Jan 15 2018

For row n > 4, for values x in the row starting at x=n+1, A079047(n) consecutive values are equal to prime(n)*x. For example, for n=5, row 5: 1,121,143,187,... for x starting at n+1 dividing each value by prime(n) = 121/11, 143/11, 187/11,... = 11,13,17,... This is equal to A000040(n) in the range A000040(n) to A000040(n+A079047(n)-1). For row n < 4, appending the first A005867(n) values to match the length of A079047(n) gives the primes gap sequence for length A079047(n) shifted by n. - Jamie Morken, Feb 02 2018

LINKS

Ivan Neretin, Table of n, a(n) for n = 2..5051

Index entries for sequences that are permutations of the natural numbers

Index entries for sequences generated by sieves

FORMULA

From Jamie Morken, Jan 15 2018: (Start)

Formulas for the count of values in the n-th row as a fraction x of the natural numbers:

x = EulerPhi(Pn(n-1))/Pn(n)

for n=1 x = 1/2 = (2-1)/(2);

for n=2 x = 1/6 = (2-1)/(2*3);

for n=3 x = 1/15 = (3-1)/(2*3*5);

for n=4 x = 4/105 = ((5-1)*(3-1))/(2*3*5*7);

for n=5 x = 8/385 = ((7-1)*(5-1)*(3-1))/(2*3*5*7*11);

for n=6 x = 16/1001 = ((11-1)*(7-1)*(5-1)*(3-1))/(2*3*5*7*11*13).

A related series for n=5:

1 - (((2-1)/(2)) + ((2-1)/(2*3)) + ((3-1)/(2*3*5)) + (((5-1)*(3-1))/(2*3*5*7)) + (((7-1)*(5-1)*(3-1))/(2*3*5*7*11))

1 - 0.792207792207... = 0.207792207792... = EulerPhi(Pn(5))/Pn(5) = 480/2310. (End)

For n > 2, T(n,k) = k*A038111(n)/A038110(n) - A306701(n, k mod A005867(n-1)) * prime(n)/A038110(n). - Jamie Morken, Mar 27 2019

EXAMPLE

Array begins:

   2   4   6   8  10  12  14  16  18  20  22  24 .... (A005843 \ {0})

   3   9  15  21  27  33  39  45  51  57  63  69 .... (A016945)

   5  25  35  55  65  85  95 115 125 145 155 175 .... (A084967)

   7  49  77  91 119 133 161 203 217 259 287 301 .... (A084968)

  11 121 143 187 209 253 319 341 407 451 473 517 .... (A084969)

  13 169 221 247 299 377 403 481 533 559 611 689 .... (A084970)

MATHEMATICA

a = Join[ {Table[2n, {n, 1, 12}]}, Table[ Take[ Prime[n]*Select[ Range[100], GCD[ Prime[n] #, Product[ Prime[i], {i, 1, n - 1}]] == 1 &], 12], {n, 2, 12}]]; Flatten[ Table[ a[[i, n - i]], {n, 2, 12}, {i, n - 1, 1, -1}]]

(* second program: *)

rows = 12; Clear[T]; Do[For[m = p = Prime[n]; k = 1, k <= rows, m += p, If[ FactorInteger[m][[1, 1]] == p, T[n, k++] = m]], {n, rows}]; Table[T[n - k + 1, k], {n, rows}, {k, n}] // Flatten (* Jean-Fran├žois Alcover, Mar 08 2016 *)

CROSSREFS

Cf. A083141 (main diagonal), A083221 (transpose), A004280, A038179, A084967, A084968, A084969, A084970, A084971.

Arrays of integers grouped into rows by various criteria:

by greatest prime factor: A125624,

by lowest prime factor: this sequence (upward antidiagonals), A083221 (downward antidiagonals),

by number of distinct prime factors: A125666,

by number of prime factors counted with multiplicity: A078840,

by prime signature: A095904,

by ordered prime signature: A096153,

by number of divisors: A119586,

by number of 1's in binary expansion: A066884 (upward), A067576 (downward),

by distance to next prime: A192179.

Cf. also A002110.

Sequence in context: A119586 A095904 A096153 * A246279 A285112 A253565

Adjacent sequences:  A083137 A083138 A083139 * A083141 A083142 A083143

KEYWORD

nonn,tabl,nice

AUTHOR

Yasutoshi Kohmoto, Jun 05 2003

EXTENSIONS

More terms from Hugo Pfoertner and Robert G. Wilson v, Jun 13 2003

STATUS

approved

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Last modified April 19 22:22 EDT 2019. Contains 322291 sequences. (Running on oeis4.)