A249818 -id:A249818 - cp's OEIS frontend

A250469 a(1) = 1; and for n > 1, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1, where A055396(n) is the index of smallest prime dividing n.

1, 3, 5, 9, 7, 15, 11, 21, 25, 27, 13, 33, 17, 39, 35, 45, 19, 51, 23, 57, 55, 63, 29, 69, 49, 75, 65, 81, 31, 87, 37, 93, 85, 99, 77, 105, 41, 111, 95, 117, 43, 123, 47, 129, 115, 135, 53, 141, 121, 147, 125, 153, 59, 159, 91, 165, 145, 171, 61, 177, 67, 183, 155, 189, 119, 195, 71, 201, 175, 207, 73, 213, 79, 219, 185, 225, 143, 231, 83, 237, 205, 243, 89, 249, 133, 255

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

Permutation of odd numbers.
For n >= 2, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1. In other words, a(n) tells which number is located immediately below n in the sieve of Eratosthenes (see A083140, A083221) in the same column of the sieve that contains n.
A250471(n) = (a(n)+1)/2 is a permutation of natural numbers.
Coincides with A003961 in all terms which are primes. - M. F. Hasler, Sep 17 2016. Note: primes are a proper subset of A280693 which gives all n such that a(n) = A003961(n). - Antti Karttunen, Mar 08 2017

Antti Karttunen, Table of n, a(n) for n = 1..5002

Cf. A000040, A003961, A016945, A046523, A055396, A078898, A083140, A083221, A084967, A249744, A249810, A249820, A249817, A249818, A250471, A266645, A280692, A280693, A283465.

Cf. A250470, A268674 (left inverses, the latter is simpler).

a[1] = 1; a[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[ 1, 1]]; For[k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[ FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1+2 == k2, Return[m2]]]]; Array[a, 100] (* Jean-François Alcover, Mar 08 2016 *)
g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1] &, #] &@ Map[Position[Lookup[s, g@#], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017, Version 10 *)

a(1) = 1, a(n) = A083221(A055396(n)+1, A078898(n)).
a(n) = A249817(A003961(A249818(n))).
Other identities. For all n >= 1:
A250470(a(n)) = A268674(a(n)) = n. [A250470 and A268674 provide left inverses for this function.]
a(2n) = A016945(n-1). [Maps even numbers to the numbers of form 6n+3, in monotone order.]
a(A016945(n-1)) = A084967(n). [Which themselves are mapped to the terms of A084967, etc. Cf. the Example section of A083140.]
a(A000040(n)) = A000040(n+1). [Each prime is mapped to the next prime.]
For all n >= 2, A055396(a(n)) = A055396(n)+1. [A more general rule.]
A046523(a(n)) = A283465(n). - Antti Karttunen, Mar 08 2017

A250246 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n), a(A078898(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, 45, 34, 35, 36, 37, 38, 33, 40, 41, 54, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 42, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 90, 67, 68, 135, 70, 71, 72, 73, 74, 51, 76, 77, 66, 79, 80, 99, 82, 83

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Inverse: A250245.
Other similar permutations: A250243, A250248, A250250, A163511, A252756.
Cf. A003961, A005843, A020639, A055396, A078898, A246278, A250470, A268674, A278524, A302042, A302046.
Differs from the "vanilla version" A249818 for the first time at n=42, where a(42) = 54, while A249818(42) = 42.
Differs from A250250 for the first time at n=73, where a(73) = 73, while A250250(73) = 103.

up_to = 16384;
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
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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))); \\ Antti Karttunen, Apr 01 2018
(Scheme, with memoizing-macro definec from Antti Karttunen's IntSeq-library, three alternative definitions)
(definec (A250246 n) (cond ((<= n 1) n) (else (A246278bi (A055396 n) (A250246 (A078898 n)))))) ;; Code for A246278bi given in A246278
(definec (A250246 n) (cond ((<= n 1) n) ((even? n) (* 2 (A250246 (/ n 2)))) (else (A003961 (A250246 (A250470 n))))))
(define (A250246 n) (A163511 (A252756 n)))

a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).
a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A003961(a(A250470(2n+1))). - Antti Karttunen, Jan 18 2015  - Instead of A250470, one may use A268674 in above formula. - Antti Karttunen, Apr 01 2018
As a composition of related permutations:
a(n) = A163511(A252756(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].
A001221(a(n)) = A302041(n).
A001222(a(n)) = A253557(n).
A008683(a(n)) = A302050(n).
A000005(a(n)) = A302051(n)
A010052(a(n)) = A302052(n), for n >= 1.
A056239(a(n)) = A302039(n).

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

A255407 Permutation of natural numbers: a(n) = A255127(A252460(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 23, 20, 21, 22, 25, 24, 19, 26, 27, 28, 29, 30, 37, 32, 33, 34, 35, 36, 41, 38, 39, 40, 43, 42, 47, 44, 45, 46, 53, 48, 31, 50, 51, 52, 61, 54, 49, 56, 57, 58, 67, 60, 71, 62, 63, 64, 65, 66, 77, 68, 69, 70, 83, 72, 89, 74, 75, 76, 59, 78, 91, 80, 81

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

a(n) tells which number in Ludic array A255127 is at the same position where n is in array A083221, the sieve of Eratosthenes. 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 A255129 is at the same position where n is in the array A083140, as they are the transposes of above two arrays.

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

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

Cf. A001248, A003309, A007310, A008578, A083221, A084967, A252460, A254100, A255413, A255127.
Inverse: A255408.
Similar permutations: A249818.

a(n) = A255127(A252460(n)).
Other identities. For all n >= 1:
a(2n) = 2n. [Fixes even numbers.]
a(3n) = 3n. [Fixes multiples of three.]
a(A008578(n)) = A003309(n). [Maps noncomposites to Ludic numbers.]
a(A001248(n)) = A254100(n). [Maps squares of primes to "postludic numbers".]
a(A084967(n)) = a(5*A007310(n)) = A007310((5*n)-3) = A255413(n). [Maps A084967 to A255413.]
(And similarly between other columns and rows of A083221 and A255127.)

A249822 Square array of permutations: A(row,col) = A078898(A246278(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, 9, 3, 2, 1, 7, 8, 4, 14, 3, 2, 1, 8, 6, 12, 4, 28, 3, 2, 1, 9, 14, 5, 21, 4, 36, 3, 2, 1, 10, 13, 42, 5, 33, 4, 57, 3, 2, 1, 11, 11, 17, 92, 5, 45, 4, 67, 3, 2, 1, 12, 7, 19, 33, 305, 5, 63, 4, 93, 3, 2, 1, 13, 23, 6, 25, 39, 455, 5, 80, 4, 139, 3, 2, 1, 14, 9, 59, 6, 43, 61, 944, 5, 116, 4, 154, 3, 2, 1, 15, 17, 7, 144, 6, 52, 70, 1238, 5, 148, 4, 210, 3, 2, 1

Offset: 1

Antti Karttunen, Nov 06 2014

			The top left corner of the array:
1, 2, 3,  4,  5,   6,   7,    8,    9,   10,  11,   12,  13,   14,   15, ...
1, 2, 3,  5,  4,   8,   6,   14,   13,   11,   7,   23,   9,   17,   18, ...
1, 2, 3,  9,  4,  12,   5,   42,   17,   19,   6,   59,   7,   22,   26, ...
1, 2, 3, 14,  4,  21,   5,   92,   33,   25,   6,  144,   7,   32,   39, ...
1, 2, 3, 28,  4,  33,   5,  305,   39,   43,   6,  360,   7,   48,   50, ...
1, 2, 3, 36,  4,  45,   5,  455,   61,   52,   6,  597,   7,   63,   68, ...
1, 2, 3, 57,  4,  63,   5,  944,   70,   76,   6, 1053,   7,   95,   84, ...
1, 2, 3, 67,  4,  80,   5, 1238,   96,   99,   6, 1502,   7,  106,  121, ...
...

Inverse permutations can be found from table A249821.
Row k+1 is a right-to-left composition of the first k rows of A251722.
Row 1: A000027 (an identity permutation), Row 2: A048673, Row 3: A249824, Row 4: A249826.
Column 4: A250474, Column 6: A250477, Column 8: A250478.
Cf. A002260, A004736, A078898, A246278, A249818.

(define (A249822 n) (A249822bi (A002260 n) (A004736 n)))
(define (A249822bi row col) (A078898 (A246278bi row col))) ;; Code for A246278bi given in A246278.

A249820 a(1) = 0 and for n > 1: a(n) = A249810(n) - A078898(n) = A078898(A003961(n)) - A078898(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 2, 0, -1, 0, 6, 0, 4, 0, 1, 0, -4, 0, 11, 0, -4, 4, 3, 0, 3, 0, 25, -1, -7, 0, 20, 0, -7, -1, 12, 0, 7, 0, -2, 4, -8, 0, 44, 0, 0, -2, 0, 0, 36, 0, 22, -2, -13, 0, 23, 0, -12, 8, 90, 0, 0, 0, -5, -2, 4, 0, 77, 0, -16, 4, -3, 0, 4, 0, 55, 28, -19, 0, 41, 0, -19, -4, 15, 0, 43, 0, -2, -3, -20, 0, 155, 0, 12, 5, 24, 0

Offset: 1

Antti Karttunen, Dec 08 2014

sign

a(n) tells how many columns off A003961(n) is from the column where n is in square array A083221 (Cf. A083140, the sieve of Eratosthenes. The column index of n in that table is given by A078898(n)).

			For n = 8 = 2*2*2, A003961(8) = 27 (3*3*3), and while 8 is on row 1 and column 4 of A083221, 27 on the next row is in column 5, thus a(8) = 5 - 4 = 1.
For n = 10 = 2*5, A003961(10) = 21 (3*7), and while 10 is on row 1 and column 5 of A083221, 21 on the next row is in column 4, thus a(10) = 4 - 5 = -1.

Antti Karttunen, Table of n, a(n) for n = 1..3071

Cf. A003961, A078898, A083221, A083140, A246277, A249810, A249817, A249818, A249821, A249822, A251721, A251722.

(define (A249820 n) (if (= 1 n) 0 (- (A249810 n) (A078898 n))))

a(n) = A249810(n) - A078898(n) = A078898(A003961(n)) - A078898(n).

a(k) = 0 when k is a prime or square of prime, among some other numbers.

A250250 Permutation of natural numbers: a(1) = 1, a(n) = A246278(a(A055396(n)),a(A078898(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, 45, 34, 35, 36, 37, 38, 33, 40, 41, 54, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 42, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 90, 67, 68, 135, 70, 71, 72, 103, 74, 51, 76, 77, 66, 79, 80, 99, 82, 83

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

This is a "doubly-recursed" version of A249818.

Antti Karttunen, Table of n, a(n) for n = 1..3432
Index entries for sequences that are permutations of the natural numbers

Inverse: A250249.
Fixed points: A250251, their complement: A249729.
Cf. A000035, A000040, A055396, A078442, A078898, A049076, A246278.
See also other (somewhat) similar permutations: A245821, A057505.
Differs from the "vanilla version" A249818 for the first time at n=42, where a(42) = 54, while A249818(42) = 42.

a(1) = 1, a(n) = A246278(a(A055396(n)), a(A078898(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].

A266645 Permutation of natural numbers: a(n) = A064989(A250469(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 14, 13, 22, 15, 12, 17, 26, 19, 34, 21, 20, 23, 38, 25, 18, 33, 16, 29, 46, 31, 58, 39, 28, 35, 30, 37, 62, 51, 44, 41, 74, 43, 82, 57, 24, 47, 86, 49, 50, 27, 52, 53, 94, 55, 42, 69, 68, 59, 106, 61, 118, 87, 40, 65, 66, 67, 122, 45, 76, 71, 134, 73, 142, 93, 36, 77, 70, 79, 146, 111, 32, 83, 158, 85, 78, 123

Offset: 1

Antti Karttunen, Jan 02 2016

nonn

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

Inverse: A266646.
Cf. A000035, A003961, A020639, A055396, A064989, A250469.
Related permutations: A266403, A266416, A249817, A249818.

f[n_] := Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[ Function[{m, n}, f[Lookup[s, g[n] + 1][[m]] - Boole[n == 1]]][#1, First@ #2] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 09 2017, Version 10 *)

a(n) = A064989(A250469(n)).
a(n) = A064989(A249817(A003961(A249818(n)))).
As a composition of related permutations:
a(n) = A266416(A266403(n)).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
A020639(a(n)) = A020639(n). [More generally, it preserves the smallest prime dividing n.]
A055396(a(n)) = A055396(n).

A255553 Permutation of natural numbers: a(n) = A255551(A252460(n)).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 9, 8, 5, 10, 13, 12, 15, 14, 11, 16, 21, 18, 25, 20, 17, 22, 31, 24, 19, 26, 23, 28, 33, 30, 37, 32, 29, 34, 39, 36, 43, 38, 35, 40, 49, 42, 51, 44, 41, 46, 63, 48, 27, 50, 47, 52, 67, 54, 61, 56, 53, 58, 69, 60, 73, 62, 59, 64, 81, 66, 75, 68, 65, 70, 79, 72, 87, 74, 71, 76, 57, 78, 93, 80, 77, 82, 99, 84, 103, 86, 83, 88, 105, 90

Offset: 1

Antti Karttunen, Feb 26 2015

nonn

a(n) tells which number in array A255551, constructed from Lucky sieve, is at the same position where n is in array A083221, constructed from the sieve of Eratosthenes. As both arrays have A005843 (even numbers) as their topmost row, this permutation fixes all of them.

Antti Karttunen, Table of n, a(n) for n = 1..6840
Index entries for sequences that are permutations of the natural numbers

Inverse: A255554.
Similar or related permutations: A255407, A255408, A249817, A249818, A252460, A255551.
Cf. A000040, A000959, A001248, A083141, A219178, A255550.

(define (A255553 n) (A255551 (A252460 n)))

a(n) = A255551(A252460(n)).
Other identities:
a(2n) = 2n. [Fixes even numbers.]
For all n >= 1, a(A083141(n)) = A255550(n).
For all n >= 2, a(A000040(n)) = A000959(n).
For all n >= 2, a(A001248(n)) = A219178(n).

A266646 Permutation of natural numbers: a(n) = A250470(A003961(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 16, 13, 12, 15, 28, 17, 26, 19, 22, 21, 14, 23, 46, 25, 18, 51, 34, 29, 36, 31, 82, 27, 20, 35, 76, 37, 24, 33, 64, 41, 56, 43, 40, 69, 30, 47, 136, 49, 50, 39, 52, 53, 126, 55, 100, 45, 32, 59, 106, 61, 38, 111, 244, 65, 66, 67, 58, 57, 78, 71, 226, 73, 42, 99, 70, 77, 86, 79, 190, 249, 44, 83, 166, 85

Offset: 1

Antti Karttunen, Jan 02 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..767
Index entries for sequences that are permutations of the natural numbers

Inverse: A266645.
Cf. A000035, A020639, A055396, A003961, A250470.
Related permutations: A249817, A249818, A266403, A266415.

(define (A266646 n) (A250470 (A003961 n)))

a(n) = A250470(A003961(n)).
As a composition of related permutations:
a(n) = A266403(A266415(n)).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
A020639(a(n)) = A020639(n). [More generally, it preserves the smallest prime dividing n.]
A055396(a(n)) = A055396(n).

cp's OEIS Frontend

A250469 a(1) = 1; and for n > 1, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1, where A055396(n) is the index of smallest prime dividing n.

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A250246 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).

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PARI

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

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A255407 Permutation of natural numbers: a(n) = A255127(A252460(n)).

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A249822 Square array of permutations: A(row,col) = A078898(A246278(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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A249820 a(1) = 0 and for n > 1: a(n) = A249810(n) - A078898(n) = A078898(A003961(n)) - A078898(n).

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A250250 Permutation of natural numbers: a(1) = 1, a(n) = A246278(a(A055396(n)),a(A078898(n))).

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A266645 Permutation of natural numbers: a(n) = A064989(A250469(n)).

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A255553 Permutation of natural numbers: a(n) = A255551(A252460(n)).

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