A083221 -id:A083221 - cp's OEIS frontend

A249817 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),A246277(n)).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 42, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 54, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 66, 67, 68, 111, 70, 71, 72, 73, 74, 51, 76, 77, 78, 79, 80, 45, 82, 83, 84, 155, 86, 135

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

a(n) tells which number in square array A083221 (the sieve of Eratosthenes) is at the same position where n is in array A246278. As both arrays have even numbers as their topmost row and primes as their leftmost column, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A083140 is at the same position where n is in the array A246279, as they are the transposes of above two arrays.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A249818.
There are three different "deep" versions of this permutation, recursing on values of A055396(n) and/or A246277(n), namely: A250245, A250247 and A250249.
Other similar or related permutations: A249815.
Cf. A000040, A005843, A020639, A055396, A078898, A083140, A083221, A246277, A246278, A246279, A249821.
Differs from its inverse A249818 for the first time at n=33, where a(33) = 39, while A249818(33) = 45.

lim = 87; a083221 = Table[Take[Prime[n] Select[Range[Ceiling[lim/2]^2], GCD[# Prime@ n, Product[Prime@ i, {i, 1, n - 1}]] == 1 &], Ceiling[lim/2]], {n, Ceiling[lim/2]}]; a055396[n_] PrimePi[FactorInteger[n][[1, 1]]]; a246277[n_] := Which[n == 1, 0, EvenQ@ n, n/2, True, a246277[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ Transpose@ FactorInteger@ n, Last@ Transpose@ FactorInteger@ n]]]; Table[a083221[[a055396@ n, a246277@ n]], {n, 2, lim}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A055396 and Yasutoshi Kohmoto at A083140 *)

(define (A249817 n) (if (= 1 n) n (A083221bi (A055396 n) (A246277 n)))) ;; Code for A083221bi given in A083221
;; Alternative version:
(define (A249817 n) (if (= 1 n) n (A083221bi (A055396 n) (A249821bi (A055396 n) (A078898 n))))) ;; Code for A249821bi given in A249821.

a(1) = 1, a(n) = A083221(A055396(n), A246277(n)).
a(1) = 1, a(n) = A083221(A055396(n), A249821(A055396(n), A078898(n))).
As a composition of other permutations:
a(1) = 1, and for n > 1, a(n) = 1 + A249815(n-1).
Other identities. For all n >= 1:
a(A005843(n)) = A005843(n) and a(A000040(n)) = A000040(n). [Fixes even numbers and primes, among other numbers. Cf. comments above].
A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].

A250245 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),a(A246277(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 54, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 42, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 78, 67, 68, 111, 70, 71, 72, 73, 74, 51, 76, 77, 126, 79, 80, 45, 82

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

The first 7-cycle occurs at: (33 39 63 57 99 81 45) which is mirrored by the cycle (66 78 126 114 198 162 90) with double-size terms.
The cycle which contains 55 as its smallest term, goes as: 55, 65, 95, 185, 425, 325, 205, 455, 395, 1055, 2945, 6035, 30845, ...
while to the other direction (A250246) it goes as: 55, 125, 245, 115, 625, 8575, 40375, ...
The cycle which contains 69 as its smallest term, goes as: 69, 111, 183, 351, 261, 273, 387, 489, 939, 1863, 909, 1161, 981, 1281, 4167, ...
while to the other direction (A250246) it goes as: 69, 135, 87, 105, 225, 207, 231, 195, 525, 1053, 3159, 24909, ...

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A250246.
Other similar permutations: A250244, A250247, A250249, A243071, A252755.
Cf. A005843, A020639, A055396, A064989, A083221, A246277, A250469.
Differs from the "vanilla version" A249817 for the first time at n=42, where a(42) = 54, while A249817(42) = 42.
Differs from A250246 for the first time at n = 33, where a(33) = 39, while A250246(33) = 45.
Differs from A250249 for the first time at n=73, where a(73) = 73, while A250249(73) = 103.

a(1) = 1, a(n) = A083221(A055396(n), a(A246277(n))).
a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(a(A064989(2n+1))). - Antti Karttunen, Jan 18 2015
As a composition of related permutations:
a(n) = A252755(A243071(n)).
Other identities. For all n >= 1:
a(n) = a(2n)/2. [The even bisection halved gives the sequence back.]
A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].

A302042 A032742 analog for a nonstandard factorization process based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 9, 11, 1, 12, 5, 13, 7, 14, 1, 15, 1, 16, 15, 17, 7, 18, 1, 19, 11, 20, 1, 21, 1, 22, 21, 23, 1, 24, 7, 25, 25, 26, 1, 27, 25, 28, 27, 29, 1, 30, 1, 31, 13, 32, 11, 33, 1, 34, 33, 35, 1, 36, 1, 37, 17, 38, 11, 39, 1, 40, 39, 41, 1, 42, 35, 43, 35, 44, 1, 45, 49, 46, 45, 47, 13, 48

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

Like [A020639(n), A032742(n)] also ordered pair [A020639(n), a(n)] is unique for each n. Iterating n, a(n), a(a(n)), a(a(a(n))), ..., until 1 is reached, and taking the smallest prime factor (A020639) of each term gives a multiset of primes in ascending order, unique for each natural number n >= 1. Permutation pair A250245/A250246 maps between this non-standard prime factorization of n and the ordinary prime factorization of n.

			For n = 66, A020639(66) [its smallest prime factor] is 2. Because A055396(66) = A000720(2) = 1, a(66) is just A078898(66) = 66/2 = 33.
For n = 33, A020639(33) = 3 and A055396(33) = 2, so a(33) = A250469(A078898(33)) = A250469(6) = 15. [15 is under 6 in array A083221].
For n = 15, A020639(15) = 3 and A055396(15) = 2, so a(15) = A250469(A078898(15)) = A250469(3) = 5. [5 is under 3 is array A083221].
For n = 5, A020639(5) = 5 and A055396(5) = 3, so a(5) = A250469(A250469(A078898(5))) = A250469(A250469(1)) = 1.
Collecting the primes given by A020639 we get a multiset of factors: [2, 3, 3, 5]. Note that 2*3*3*5 = 90 = A250246(66).
If we start from n = 66, iterating the map n -> A302044(n) [instead of n -> A302042(n)] and apply A020639 to each term obtained we get just a single instance of each prime: [2, 3, 5]. Then by applying A302045 to the same terms we get the corresponding exponents (multiplicities) of those primes: [1, 2, 1].

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A020639, A032742, A055396, A078898, A083221, A250245, A250246, A250469, A268674, A276151, A280496, A302032.

Cf. also following analogs: A302041 (omega), A253557 (bigomega), A302043, A302044, A302045 (exponent of the least prime present), A302046 (prime signature filter), A302050 (Moebius mu), A302051 (tau), A302052 (char.fun of squares), A302039, A302055 (Arith. derivative).

\\ Assuming A250469 and its inverse A268674 have been precomputed, then the following is fast enough:
A302042(n) = if(1==n,n,my(k=0); while((n%2), n = A268674(n); k++); n = n/2; while(k>0, n = A250469(n); k--); (n));

A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A078898(n) = if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k));
\\ Faster if we precompute A078898 as an ordinal transform of A020639:
up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));

For n > 1, a(n) = A250469^(r)(A078898(n)), where r = A055396(n)-1 and A250469^(r)(n) stands for applying r times the map x -> A250469(x), starting from x = n.
a(n) = A250245(A032742(A250246(n))) = A250245(A280496(n)) = A250245(A250246(n) / A020639(n)).
a(n) = n - A302043(n).

A249821 Square array of permutations: A(row,col) = A246277(A083221(row,col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ... .

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 5, 3, 2, 1, 6, 4, 5, 3, 2, 1, 7, 7, 7, 5, 3, 2, 1, 8, 11, 11, 7, 5, 3, 2, 1, 9, 6, 13, 11, 7, 5, 3, 2, 1, 10, 13, 17, 13, 11, 7, 5, 3, 2, 1, 11, 17, 4, 17, 13, 11, 7, 5, 3, 2, 1, 12, 10, 19, 19, 17, 13, 11, 7, 5, 3, 2, 1, 13, 19, 23, 23, 19, 17, 13, 11, 7, 5, 3, 2, 1, 14, 9, 6, 29, 23, 19, 17, 13, 11, 7, 5, 3, 2, 1, 15, 8, 29, 31, 29, 23, 19, 17, 13, 11, 7, 5, 3, 2, 1

Offset: 1

Antti Karttunen, Nov 06 2014

Permutation A249817 preserves the smallest prime factor of n, i.e., A055396(A249817(n)) = A055396(n), in other words, keeps all the terms that appear on any row of A246278 on the same row of A083221. Permutations in this table are induced by changes that A249817 does onto each row of the latter table, thus permutation on row r of this table can be used to sort row r of A246278 into ascending order. I.e., A246278(r, A(r,c)) = A083221(r,c) [the corresponding row in the Sieve of Eratosthenes, where each row appears in monotone order].
The multi-set of cycle-sizes of permutation A249817 is a disjoint union of cycle-sizes of all permutations in this array. For example, A249817 has a 7-cycle (33 39 63 57 99 81 45) which originates from the 7-cycle (6 7 11 10 17 14 8) of A064216, which occurs as the second row in this table.
On each row, 4 is the first composite number (and the first term less than previous, apart from row 1), and on row n it occurs in position A250474(n). This follows because A001222(A246277(n)) = A001222(n)-1 and because on each row of A083221 (see A083140) all terms between the square of prime (second term on each row) and the first cube (of the same prime, this cube mapping in this array to 4) are nonsquare semiprimes (A006881), this implies that the corresponding terms in this array must be primes.
Also, as the smaller prime factor of the terms on row n of A083221 is constant, A020639(n), and for all i < j: A246277(p_{i} * p_{j}) < A246277(p_i * p_{j+1}), the primes on any row appear in monotone order.

			The top left corner of the array:
1, 2, 3, 4, 5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ...
1, 2, 3, 5, 4,  7, 11,  6, 13, 17, 10, 19,  9,  8, 23, 29, 14, 15, 31, ...
1, 2, 3, 5, 7, 11, 13, 17,  4, 19, 23,  6, 29, 31, 37, 41,  9, 43, 10, ...
1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,  4, 41, 43, 47, 53, 59, ...
...

Inverse permutations can be found from table A249822.
Row k+1 is a left-to-right composition of the first k rows of A251721.
Row 1: A000027 (an identity permutation), Row 2: A064216, Row 3: A249823, Row 4: A249825.
The initial growing part of each row converges towards A008578.
Cf. A001222, A002260, A004736, A006881, A008578, A020639, A083221, A246277, A249817, A250474.

(define (A249821 n) (A249821bi (A002260 n) (A004736 n)))
(define (A249821bi row col) (A246277 (A083221bi row col))) ;; Code for A083221bi given in A083221.

A(row,col) = A246277(A083221(row,col)).

A001222(A(row,col)) = A001222(A083221(row,col)) - 1. [This follows directly from the properties of A246277.]

A302041 An omega analog for a nonstandard factorization based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A001221, A069010, A250246, A252754, A302044.
Cf. A302040 (positions of terms < 2).
Cf. A253557 (a similar analog for bigomega), A302050, A302051, A302052, A302039, A302055 (other analogs).
Differs from A302031 for the first time at n=59, where a(59) = 1, while A302031(59) = 2.

\\ Assuming A250469 and its inverse A268674 have been precomputed, then the following is reasonably fast:
A302044(n) = if(1==n,n,my(k=0); while((n%2), n = A268674(n); k++); n = (n/2^valuation(n, 2)); while(k>0, n = A250469(n); k--); (n));
A302041(n) = if(1==n, 0,1+A302041(A302044(n)));

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A000265(n) = (n/2^valuation(n, 2));
A302044(n) = { my(c = A000265(A078898(n))); if(1==c,1,my(p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p)); };
A302041(n) = if(1==n, 0,1+A302041(A302044(n)));

\\ Or, using also some of the code from above:
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A055396(n) = if(1==n,0,primepi(A020639(n)));
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
A302041(n) = omega(A250246(n));

a(1) = 0; for n > 1, a(n) = 1 + a(A302044(n)).
a(n) = A001221(A250246(n)).
a(n) = A069010(A252754(n)).

A302044 A028234 analog for factorization process based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 5, 1, 1, 9, 1, 5, 1, 11, 1, 3, 1, 13, 7, 7, 1, 15, 1, 1, 5, 17, 7, 9, 1, 19, 11, 5, 1, 21, 1, 11, 1, 23, 1, 3, 1, 25, 25, 13, 1, 27, 1, 7, 7, 29, 1, 15, 1, 31, 13, 1, 11, 33, 1, 17, 5, 35, 1, 9, 1, 37, 17, 19, 11, 39, 1, 5, 11, 41, 1, 21, 7, 43, 35, 11, 1, 45, 1, 23, 1, 47, 13, 3, 1, 49, 19, 25, 1, 51, 1, 13, 25

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

Iterating n, a(n), a(a(n)), a(a(a(n))), ..., until 1 is reached, and taking the smallest prime factor (A020639) of each term gives a sequence of distinct primes in ascending order, while applying A302045 to the same terms gives the corresponding exponents (multiplicities) of those primes. Permutation pair A250245/A250246 maps between this non-standard prime factorization and the ordinary factorization of n. See also comments and examples in A302042.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A000265, A020639, A055396, A078898, A028234, A083221, A250245, A250246, A250469, A302034, A302042, A302045.

Cf. A302040 (positions of 1's).

\\ Assuming A250469 and its inverse A268674 have been precomputed, then the following is fast enough:
A302044(n) = if(1==n,n,my(k=0); while((n%2), n = A268674(n); k++); n = (n/2^valuation(n, 2)); while(k>0, n = A250469(n); k--); (n));

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A000265(n) = (n/2^valuation(n, 2));
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302044(n) = { my(c = A000265(A078898(n))); if(1==c,1,my(p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p)); };

For n > 1, a(n) = A250469^(r)(A000265(A078898(n))), where r = A055396(n)-1 and A250469^(r)(n) stands for applying r times the map x -> A250469(x), starting from x = n.

a(n) = A250245(A028234(A250246(n))).

A302051 An analog of A000005 for nonstandard factorization based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 6, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 5, 4, 2, 10, 3, 6, 6, 6, 2, 8, 4, 8, 6, 4, 2, 12, 2, 4, 4, 7, 4, 12, 2, 6, 8, 8, 2, 12, 2, 4, 4, 6, 4, 8, 2, 10, 6, 4, 2, 12, 6, 4, 8, 8, 2, 10, 4, 6, 6, 4, 4, 12, 2, 6, 4, 9, 2, 12, 2, 8, 9

Offset: 1

Antti Karttunen, Apr 01 2018

nonn

See A302042, A302044 and A302045 for a description of the factorization process.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A000005, A083221, A302042, A302044, A302045, A302052 (reduced modulo 2), A302053 (gives the positions of odd numbers).

Cf. also A253557, A302041, A302050, A302052, A302039, A302055 for other similar analogs.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A000265(n) = (n/2^valuation(n, 2));
A001511(n) = 1+valuation(n,2);
A302045(n) = A001511(A078898(n));
A302044(n) = { my(c = A000265(A078898(n))); if(1==c,1,my(p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p)); };
A302051(n) = if(1==n,n,(A302045(n)+1)*A302051(A302044(n)));

\\ Or, using also some of the code from above:
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A055396(n) = if(1==n,0,primepi(A020639(n)));
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
A302051(n) = numdiv(A250246(n));

a(1) = 1, for n > 1, a(n) = (A302045(n)+1) * a(A302044(n)).
a(n) = A000005(A250246(n)).
a(n) = A106737(A252754(n)).

A249741 Sieve of Eratosthenes minus one: a(n) = A083221(n+1) - 1.

Original entry on oeis.org

1, 3, 2, 5, 8, 4, 7, 14, 24, 6, 9, 20, 34, 48, 10, 11, 26, 54, 76, 120, 12, 13, 32, 64, 90, 142, 168, 16, 15, 38, 84, 118, 186, 220, 288, 18, 17, 44, 94, 132, 208, 246, 322, 360, 22, 19, 50, 114, 160, 252, 298, 390, 436, 528, 28, 21, 56, 124, 202, 318, 376, 492, 550, 666, 840, 30, 23, 62, 144, 216, 340, 402, 526, 588, 712, 898, 960, 36, 25

Offset: 1

Antti Karttunen, Nov 15 2014

			The top left corner of the array:
   1,   3,   5,    7,    9,   11,   13,   15,   17,   19,   21,   23,   25,
   2,   8,  14,   20,   26,   32,   38,   44,   50,   56,   62,   68,   74,
   4,  24,  34,   54,   64,   84,   94,  114,  124,  144,  154,  174,  184,
   6,  48,  76,   90,  118,  132,  160,  202,  216,  258,  286,  300,  328,
  10, 120, 142,  186,  208,  252,  318,  340,  406,  450,  472,  516,  582,
  12, 168, 220,  246,  298,  376,  402,  480,  532,  558,  610,  688,  766,
  16, 288, 322,  390,  492,  526,  628,  696,  730,  798,  900, 1002, 1036,
  18, 360, 436,  550,  588,  702,  778,  816,  892, 1006, 1120, 1158, 1272,
  22, 528, 666,  712,  850,  942,  988, 1080, 1218, 1356, 1402, 1540, 1632,
  28, 840, 898, 1072, 1188, 1246, 1362, 1536, 1710, 1768, 1942, 2058, 2116,
...

Inverse: A249742.
Transpose: A114881.
Row 1: A005408, Column 1: A006093, Main diagonal: A249743.
Cf. also A002260, A004736, A038722, A083221.

(define (A249741 n) (- (A083221 (+ n 1)) 1))
;; Alternative version:
(define (A249741 n) (- (A083221bi (A002260 n) (A004736 n)) 1))

a(n) = A083221(n+1) - 1.
As a composition of related permutations:
a(n) = A114881(A038722(n)).
a(n) = A249811(A135764(n)).

A250249 Permutation of natural numbers: a(1) = 1, a(n) = A083221(a(A055396(n)), a(A246277(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 54, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 42, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 78, 67, 68, 111, 70, 71, 72, 103, 74, 51, 76, 77, 126, 79, 80, 45, 82

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

This is a "doubly-recursed" version of A249817.
For primes p_n, a(p_n) = p_{a(n)}.
The first 7-cycle occurs at: (33 39 63 57 99 81 45), which is mirrored by the cycle (66 78 126 114 198 162 90) with terms double the size and also by the cycle (137 167 307 269 523 419 197), consisting of primes (p_33, p_39, p_63, ...).

			For n = 42 = 2*3*7, we see that it occurs as the 21st term on the top row of A246278 (A055396(42) = 1 and A246277(42) = 21), recursing on both yields a(1) = 1, a(21) = 27, thus we find A083221(1,27), the 27th term on A083221's topmost row (also A005843) which is 54, thus a(42) = 54.
Examples for cases where n is a prime:
a(3709) = a(p_518) = p_{a(518)} = A000040(1162) = 9397.
a(3719) = a(p_519) = p_{a(519)} = A000040(1839) = 15767.

Antti Karttunen, Table of n, a(n) for n = 1..3718
Index entries for sequences that are permutations of the natural numbers

Inverse: A250250.
Fixed points: A250251, their complement: A249729.
Cf. A000035, A000040, A055396, A078442, A083221, A049076.
Differs from A250250 for the first time at n=33, where a(33) = 39, while A250250(33) = 45.
Differs from the "vanilla version" A249817 for the first time at n=42, where a(42) = 54, while A249817(42) = 42.

a(1) = 1, a(n) = A083221(a(A055396(n)), a(A246277(n))).
Other identities. For all n >= 1:
a(2n) = 2*a(n), or equally, a(n) = a(2n)/2. [The even bisection halved gives the sequence back].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].
A000035(n) = A000035(a(n)). [Preserves the parity].

A255408 Permutation of natural numbers: a(n) = A083221(A255128(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 25, 20, 21, 22, 19, 24, 23, 26, 27, 28, 29, 30, 49, 32, 33, 34, 35, 36, 31, 38, 39, 40, 37, 42, 41, 44, 45, 46, 43, 48, 55, 50, 51, 52, 47, 54, 121, 56, 57, 58, 77, 60, 53, 62, 63, 64, 65, 66, 59, 68, 69, 70, 61, 72, 169, 74, 75, 76, 67, 78, 85, 80, 81, 82, 71, 84, 91, 86, 87

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

a(n) tells which number in array A083221 (the sieve of Eratosthenes) is at the same position where n is in Ludic array A255127. As both arrays have A005843 (even numbers) and A016945 as their two topmost rows, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A083140 is at the same position where n is in the array A255129, as they are the transposes of above two arrays.

			A255127(3,2) = 19 and A083221(3,2) = 25, thus a(19) = 25.
A255127(8,1) = 23 and A083221(8,1) = 19, thus a(23) = 19.
A255127(9,1) = 25 and A083221(9,1) = 23, thus a(25) = 23.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A255407.
Cf. A003309, A008578, A083221, A255127, A255128.
Similar permutations: A249817.

(define (A255408 n) (A083221 (A255128 n)))

a(n) = A083221(A255128(n)).
Other identities. For all n >= 1:
a(2n) = 2n. [Fixes even numbers.]
a(3n) = 3n. [Fixes multiples of three.]
a(A003309(n)) = A008578(n). [Maps Ludic numbers to noncomposites.]

cp's OEIS Frontend

A249817 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),A246277(n)).

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A250245 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),a(A246277(n))).

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A302042 A032742 analog for a nonstandard factorization process based on the sieve of Eratosthenes (A083221).

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A249821 Square array of permutations: A(row,col) = A246277(A083221(row,col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ... .

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A302041 An omega analog for a nonstandard factorization based on the sieve of Eratosthenes (A083221).

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A302044 A028234 analog for factorization process based on the sieve of Eratosthenes (A083221).

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A302051 An analog of A000005 for nonstandard factorization based on the sieve of Eratosthenes (A083221).

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A249741 Sieve of Eratosthenes minus one: a(n) = A083221(n+1) - 1.

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