A207800 -id:A207800

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A207790

Permutation of positive numbers. See comments.

+10
4

1, 2, 4, 3, 8, 5, 6, 7, 16, 11, 9, 13, 12, 17, 10, 19, 32, 23, 14, 29, 18, 31, 15, 37, 24, 41, 21, 43, 20, 47, 22, 53, 64, 59, 25, 61, 27, 67, 26, 71, 36, 73, 33, 79, 28, 83, 34, 89, 48, 97, 35, 101, 30, 103, 38, 107, 40, 109, 39, 113, 42, 127, 46, 131, 128, 137, 49, 139, 44, 149, 51, 151

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(1)=1; on places 2,4,6,8,... we put primes (A000040); on places 3,7,11,15,... we put products of two primes (A001358); on places 5,13,21,29,... we put products of three primes (A014612); on places 9,25,41,57,... we put products of four primes (A014613); on places 17,49,81,... we put products of five primes (A014614); etc.

Primes with the index not exceeding n have density 1/2, semiprimes have density 1/4, etc.

By our system, here and in A207800, A207801, A207802 we used the order: a(1)=1; the first appearance of a new kind of numbers in places of the form 2^k+1, k=0,1,2,..., with period of the appearance 2^{k+1}.

LINKS

Ivan Neretin, Table of n, a(n) for n = 1..8192

FORMULA

For n>1, a(n) = A078840(A249725(n-1)). - Ivan Neretin, Apr 30 2016

MATHEMATICA

mx = 72; a = Array[1 &, mx]; cnt = mx - 1; offs = Table[2^(i - 1) + 1, {i, 1, mx}]; n = 1; While[cnt > 0, n++; idx = PrimeOmega[n]; pos = offs[[idx]]; If[pos > mx, Continue[]]; offs[[idx]] += 2^idx; a[[pos]] = n; cnt--]; a (* Ivan Neretin, May 06 2015 *)

CROSSREFS

Cf. A057114.

KEYWORD

nonn

AUTHOR

Vladimir Shevelev and Peter J. C. Moses, Feb 20 2012

STATUS

approved

A207801

Permutation of positive numbers. See comments.

+10
3

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 14, 21, 16, 11, 18, 27, 20, 35, 22, 33, 24, 49, 26, 39, 28, 55, 30, 45, 32, 13, 34, 51, 36, 65, 38, 57, 40, 77, 42, 63, 44, 85, 46, 69, 48, 121, 50, 75, 52, 95, 54, 81, 56, 91, 58, 87, 60, 115, 62, 93, 64, 17, 66, 99, 68, 125, 70, 105, 72, 119, 74

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(1)=1; on places 2,4,6,8,... we put the same numbers, i.e., numbers with the least prime divisor p_1=2; on places 3,7,11,15,... we put numbers with the least prime divisor p_2=3; on places 5,13,21,29,... we put numbers with the least prime divisor p_3=5; etc.

For general description of the order, see comment in A207790.

LINKS

Ivan Neretin, Table of n, a(n) for n = 1..8192

FORMULA

For n>1, a(n) = A083140(1+A209268(n-1)). Equivalently, a(n) = A083221(1+A249725(n-1)). - Ivan Neretin, Apr 30 2016

MATHEMATICA

a = Array[1 &, mx = 74]; cnt = mx - 1; offs = Table[2^(i - 1) + 1, {i, 1, mx}]; n = 1; While[cnt > 0, n++; idx = PrimePi[FactorInteger[n][[1, 1]]]; pos = offs[[idx]]; If[pos > mx, Continue[]]; offs[[idx]] += 2^idx; a[[pos]] = n; cnt--]; a (* Ivan Neretin, May 06 2015 *)

CROSSREFS

Cf. A207790, A207800.

KEYWORD

nonn

AUTHOR

Vladimir Shevelev and Peter J. C. Moses, Feb 20 2012

STATUS

approved

A207802

Permutation of positive numbers. See comments.

+10
1

1, 2, 3, 4, 5, 8, 6, 16, 7, 32, 9, 64, 10, 128, 12, 256, 11, 512, 18, 1024, 15, 2048, 24, 4096, 14, 8192, 27, 16384, 20, 32768, 36, 65536, 13, 131072, 48, 262144, 25, 524288, 54, 1048576, 21, 2097152, 72, 4194304, 30, 8388608, 81, 16777216, 22, 33554432, 96, 67108864, 40, 134217728, 108

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(1)=1; on places 2,4,6,8,... we put 2^m, m=1,2,3,..., i.e., numbers with the greatest prime divisor p_1=2; on places 3,7,11,15,... we put numbers with the greatest prime divisor p_2=3; on places 5,13,21,29,... we put numbers with the greatest prime divisor p_3=5; etc.

For general description of the order, see the comment in A207790.

LINKS

Table of n, a(n) for n=1..55.

FORMULA

For n>1, a(n) = A125624(A209268(n-1)). - Ivan Neretin, Apr 30 2016

CROSSREFS

Cf. A207790, A207800, A207801.

KEYWORD

nonn

AUTHOR

Vladimir Shevelev and Peter J. C. Moses, Feb 20 2012

EXTENSIONS

a(30) fixed by Ivan Neretin, Apr 30 2016

STATUS

approved

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