A078898 -id:A078898 - 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).

A249818 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n),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, 42, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 54, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 66, 67, 68, 135, 70, 71, 72, 73, 74, 51, 76, 77, 78, 79, 80, 99, 82, 83, 84, 175, 86, 105

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

a(n) tells which number in square array A246278 is at the same position where n is in array A083221, the sieve of Eratosthenes. 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 A246279 is at the same position where n is in the array A083140, 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: A249817.
There are three different "deep" versions of this permutation, recursing on values of A055396(n) and/or A078898(n), namely: A250246, A250248 and A250250.
Other similar or related permutations: A249816.
Cf. A000040, A005843, A020639, A055396, A078898, A083140, A083221, A246277, A246278, A246279, A249822.
Differs from its inverse A249817 for the first time at n=33, where a(33) = 45, while A249817(33) = 39.

lim = 87; a003961[p_?PrimeQ] := a003961[p] = Prime[PrimePi@ p + 1]; a003961[1] = 1; a003961[n_] :=  a003961[n] = Times @@ (a003961[First@ #]^Last@ # &) /@ FactorInteger@ n; a055396[n_] := PrimePi[FactorInteger[n][[1, 1]]]; a078898 = Block[{nn = 90, spfs}, spfs = Table[FactorInteger[n][[1, 1]], {n, nn}]; Table[Count[Take[spfs, i], spfs[[i]]], {i, nn}]]; a246278 = NestList[Map[a003961, #] &, Table[2 k, {k, lim}], lim]; Table[a246278[[a055396@ n, a078898[[n]]]], {n, 2, lim}]
(* Michael De Vlieger, Jan 04 2016, after Harvey P. Dale at A055396 and A078898 *)

a(1) = 1, a(n) = A246278(A055396(n), A078898(n)).
a(1) = 1, a(n) = A246278(A055396(n), A249822(A055396(n), A246277(n))).
As a composition of other permutations:
a(1) = 1, and for n > 1, a(n) = 1 + A249816(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].

A268674 a(1) = 1, after which, for odd numbers: a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)-1, and for even numbers: a(n) = a(A000265(n)).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 17, 3, 8, 7, 19, 2, 9, 11, 10, 5, 23, 6, 29, 1, 12, 13, 15, 4, 31, 17, 14, 3, 37, 8, 41, 7, 16, 19, 43, 2, 25, 9, 18, 11, 47, 10, 21, 5, 20, 23, 53, 6, 59, 29, 22, 1, 27, 12, 61, 13, 24, 15, 67, 4, 71, 31, 26, 17, 35, 14, 73, 3, 28, 37, 79, 8, 33, 41, 30, 7

Offset: 1

Antti Karttunen, Feb 11 2016

nonn

For odd numbers n > 1, a(n) tells which term is on the immediately preceding row of A083221, in the same column where n itself is.

The sequence offers a left inverse for A250469 that is slightly easier to compute than A250470.

Antti Karttunen, Table of n, a(n) for n = 1..32769
Index entries for sequences generated by sieves

Left inverse of A250469.
Cf. A000040, A000265, A001248, A055396, A078898, A083221.
Cf. also A064989.
Differs from A250470 for the first time at n=42, where a(42)=8, while A250470(42) = 10.

(* b = A250469 *) b[1] = 1; b[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]]]];
a[1] = a[2] = 1; a[n_?EvenQ] := a[n] = a[n/2]; a[n_] := a[n] = For[k = 1, True, k++, If[b[k] == n, Return[k]]];
Array[a, 100] (* Jean-François Alcover, Mar 14 2016 *)

a(1) = 1, after which, a(n) = a(A000265(n)) if n is even, otherwise for odd n, a(n) = A083221(A055396(n)-1, A078898(n)).
Other identities. For all n >= 1:
a(A250469(n)) = n. [This works as a left inverse for sequence A250469.]
a(2n) = a(n). [The even bisection gives the whole sequence back.]
a(2n-1) = A250470(2n-1). [Matches with A250470 on odd numbers.]
a(A000040(n+1)) = A000040(n). [Maps each odd prime to the preceding prime.]
a(A001248(n+1)) = A001248(n). [Maps each square of an odd prime to the square of the preceding prime.]

A250474 Number of times prime(n) occurs as the least prime factor among numbers 1 .. prime(n)^3: a(n) = A078898(A030078(n)).

Original entry on oeis.org

4, 5, 9, 14, 28, 36, 57, 67, 93, 139, 154, 210, 253, 272, 317, 396, 473, 504, 593, 658, 687, 792, 866, 979, 1141, 1229, 1270, 1356, 1397, 1496, 1849, 1947, 2111, 2159, 2457, 2514, 2695, 2880, 3007, 3204, 3398, 3473, 3828, 3904, 4047, 4121, 4583, 5061, 5228, 5309, 5474, 5743, 5832, 6269, 6543, 6816, 7107, 7197, 7488, 7686, 7784, 8295, 9029, 9248, 9354, 9568, 10351

Offset: 1

Antti Karttunen, Nov 23 2014

nonn

Position of the first composite number (which is always 4) on row n of A249821. The fourth column of A249822.
Position of the first nonfixed term on row n of arrays of permutations A251721 and A251722.
According to the definition, this is the number of multiples of prime(n) below prime(n)^3 (and thus, the number of numbers below prime(n)^2) which do not have a smaller factor than prime(n). That is, the numbers remaining below prime(n)^2 after deleting all multiples of primes less than prime(n), as is done by applying the first n-1 steps of the sieve of Eratosthenes (when the first step is elimination of multiples of 2). This explains that the first differences are a(n+1)-a(n) = A050216(n)-1 for n>1, and a(n) = A054272(n)+2. - M. F. Hasler, Dec 31 2014

			prime(1) = 2 occurs as the least prime factor in range [1,8] for four times (all even numbers <= 8), thus a(1) = 4.
prime(2) = 3 occurs as the least prime factor in range [1,27] for five times (when n is: 3, 9, 15, 21, 27), thus a(2) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..5001

One more than A250473. Two more than A054272.
Column 4 of A249822.
Cf. also A250477 (column 6), A250478 (column 8).
Cf. A000040, A000879, A001248, A002110, A005867, A008683, A008836, A020639, A030078, A055396, A078898, A249821, A251721, A251722.

f[n_] := Count[Range[Prime[n]^3], x_ /; Min[First /@ FactorInteger[x]] == Prime@ n]; Array[f, 16] (* Michael De Vlieger, Mar 30 2015 *)

A250474(n) = 3 + primepi(prime(n)^2) - n; \\ Fast implementation.
for(n=1, 5001, write("b250474.txt", n, " ", A250474(n)));
\\ The following program reflects the given sum formula, but is far from the optimal solution:
allocatemem(234567890);
A002110(n) = prod(i=1, n, prime(i));
A020639(n) = if(1==n,n,vecmin(factor(n)[,1]));
A055396(n) = if(1==n,0,primepi(A020639(n)));
A250474(n) = { my(p2 = prime(n)^2); sumdiv(A002110(n-1), d, moebius(d)*(p2\d)); };
for(n=1, 23, print1(A250474(n),", "));

(define (A250474 n) (let loop ((k 2)) (if (not (prime? (A249821bi n k))) k (loop (+ k 1))))) ;; This is even slower. Code for A249821bi given in A249821.

a(n) = 3 + A000879(n) - n = A054272(n) + 2 = A250473(n) + 1.
a(n) = A078898(A030078(n)).
a(1) = 1, a(n) = Sum_{d|A002110(n-1)} moebius(d)*floor(prime(n)^2/d). [Follows when A030078(n), prime(n)^3 is substituted to the similar formula given for A078898(n). Here A002110(n) gives the product of the first n primes. Because the latter is always squarefree, one could use also Liouville's lambda (A008836) instead of Moebius mu (A008683)].
Other identities. For all n >= 1:
A249821(n, a(n)) = 4.

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.

A249824 Permutation of natural numbers: a(n) = A078898(A003961(A003961(2*n))).

Original entry on oeis.org

1, 2, 3, 9, 4, 12, 5, 42, 17, 19, 6, 59, 7, 22, 26, 209, 8, 82, 10, 92, 31, 29, 11, 292, 41, 32, 115, 109, 13, 129, 14, 1042, 40, 39, 48, 409, 15, 49, 45, 459, 16, 152, 18, 142, 180, 52, 20, 1459, 57, 202, 54, 159, 21, 572, 63, 542, 68, 62, 23, 642, 24, 69, 213

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

			a(4) = 9 because of the following. 2n = 2*4 = 8 = 2^3. We replace the prime factor 2 of 8 with the next prime 3 to get 3^3, then replace 3 with 5 to get 5^3 = 125. The smallest prime factor of 125 is 5. 125 is the 9th term of A084967: 5, 25, 35, 55, 65, 85, 95, 115, 125, ..., thus a(4) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..255
Index entries for sequences that are permutations of the natural numbers

Inverse: A249823.
Row 3 of A249822.
Cf. A003961, A048673, A078898, A084967, A243071, A246278, A249734, A249746, A249826, A250475, A275716.
Cf. also A273664, A273669.

t = PositionIndex[FactorInteger[#][[1, 1]] & /@ Range[10^4]]; f[n_] := Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Flatten@ Table[Position[Lookup[t, FactorInteger[#][[1, 1]] ], #] &[f@ f[2 n]], {n, 120}] (* Michael De Vlieger, Jul 25 2016, Version 10 *)

(define (A249824 n) (A078898 (A003961 (A003961 (* 2 n)))))

a(n) = A078898(A003961(A003961(2*n))) = A078898(A003961(A249734(n))).
a(n) = A078898(A246278(3,n)).
As a composition of other permutations:
a(n) = A249746(A048673(n)).
a(n) = A250475(A249826(n)).
a(n) = A275716(A243071(n)).
Other identities. For all n >= 1:
a(2n) = A273669(a(n)) and a(A003961(n)) = A273664(a(n)). -- Antti Karttunen, Aug 07 2016

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.

A302045 a(1) = 0, for n > 1, a(n) = A001511(A078898(n)); Number of instances of the smallest prime factor A020639(n) in nonstandard factorization of n that is based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 1, 4, 2, 1, 1, 2, 1, 1, 3, 3, 2, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 3, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 2, 5, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 2

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

Iterating the map n -> A302044(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 this function (A302045) to those terms gives the corresponding "exponents" of those primes, that is, the count of consecutive occurrences of each prime when iterating the map n -> A302042(n), which gives the same primes with repetitions. Permutation pair A250245/A250246 maps between this non-standard prime factorization of n 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. A001511, A067029, A078898, A250245, A250246, A268674, A302042, A302044.

Cf. also A302035, A302050, A302051.

A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
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)); };
A001511(n) = 1+valuation(n,2);
A302045(n) = if(1==n,0,A001511(A078898(n)));

a(1) = 0, for n > 1, a(n) = A001511(A078898(n)).

For n > 1, a(n) = A250245(A067029(A250246(n))).

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].

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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A249818 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n),A078898(n)).

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A268674 a(1) = 1, after which, for odd numbers: a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)-1, and for even numbers: a(n) = a(A000265(n)).

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A250474 Number of times prime(n) occurs as the least prime factor among numbers 1 .. prime(n)^3: a(n) = A078898(A030078(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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A249824 Permutation of natural numbers: a(n) = A078898(A003961(A003961(2*n))).

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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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