A251722 -id:A251722 - cp's OEIS frontend

A048673 Permutation of natural numbers: a(n) = (A003961(n)+1) / 2 [where A003961(n) shifts the prime factorization of n one step towards larger primes].

1, 2, 3, 5, 4, 8, 6, 14, 13, 11, 7, 23, 9, 17, 18, 41, 10, 38, 12, 32, 28, 20, 15, 68, 25, 26, 63, 50, 16, 53, 19, 122, 33, 29, 39, 113, 21, 35, 43, 95, 22, 83, 24, 59, 88, 44, 27, 203, 61, 74, 48, 77, 30, 188, 46, 149, 58, 47, 31, 158, 34, 56, 138, 365, 60, 98, 36, 86, 73

Offset: 1

Antti Karttunen, Jul 14 1999

nonn

Inverse of sequence A064216 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001
From Antti Karttunen, Dec 20 2014: (Start)
Permutation of natural numbers obtained by replacing each prime divisor of n with the next prime and mapping the generated odd numbers back to all natural numbers by adding one and then halving.
Note: there is a 7-cycle almost right in the beginning: (6 8 14 17 10 11 7). (See also comments at A249821. This 7-cycle is endlessly copied in permutations like A250249/A250250.)
The only 3-cycle in range 1 .. 402653184 is (2821 3460 5639).
For 1- and 2-cycles, see A245449.
(End)
The first 5-cycle is (1410, 2783, 2451, 2703, 2803). - Robert Israel, Jan 15 2015
From Michel Marcus, Aug 09 2020: (Start)
(5194, 5356, 6149, 8186, 10709), (46048, 51339, 87915, 102673, 137205) and (175811, 200924, 226175, 246397, 267838) are other 5-cycles.
(10242, 20479, 21413, 29245, 30275, 40354, 48241) is another 7-cycle. (End)
From Antti Karttunen, Feb 10 2021: (Start)
Somewhat artificially, also this permutation can be represented as a binary tree. Each child to the left is obtained by multiplying the parent by 3 and subtracting one, while each child to the right is obtained by applying A253888 to the parent:
                                       1
                                       |
                    ................../ \..................
                   2                                       3
         5......../ \........4                   8......../ \........6
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    14       13         11       7          23       9          17       18
  41 10    38  12     32  28   20 15      68  25   26 63      50  16   53  19
etc.
Each node's (> 1) parent can be obtained with A253889. Sequences A292243, A292244, A292245 and A292246 are constructed from the residues (mod 3) of the vertices encountered on the path from n to the root (1).
(End)

			For n = 6, as 6 = 2 * 3 = prime(1) * prime(2), we have a(6) = ((prime(1+1) * prime(2+1))+1) / 2 = ((3 * 5)+1)/2 = 8.
For n = 12, as 12 = 2^2 * 3, we have a(12) = ((3^2 * 5) + 1)/2 = 23.

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

Inverse: A064216.
Row 1 of A251722, Row 2 of A249822.
One more than A108228, half the terms of A243501.
Fixed points: A048674.
Positions of records: A029744, their values: A246360 (= A007051 interleaved with A057198).
Positions of subrecords: A247283, their values: A247284.
Cf. A246351 (Numbers n such that a(n) < n.)
Cf. A246352 (Numbers n such that a(n) >= n.)
Cf. A246281 (Numbers n such that a(n) <= n.)
Cf. A246282 (Numbers n such that a(n) > n.), A252742 (their char. function)
Cf. A246261 (Numbers n for which a(n) is odd.)
Cf. A246263 (Numbers n for which a(n) is even.)
Cf. A246260 (a(n) reduced modulo 2), A341345 (modulo 3), A341346, A292251 (3-adic valuation), A292252.
Cf. A246342 (Iterates starting from n=12.)
Cf. A246344 (Iterates starting from n=16.)
Cf. also A003602, A003961 (A045965), A108951, A151800, A245449, A249735, A249821, A250471, A250249, A250250.
Cf. A245447 (This permutation "squared", a(a(n)).)
Other permutations whose formulas refer to this sequence: A122111, A243062, A243066, A243500, A243506, A244154, A244319, A245605, A245608, A245610, A245612, A245708, A246265, A246267, A246268, A246363, A249745, A249824, A249826, and also A183209, A254103 that are somewhat similar.
Cf. also prime-shift based binary trees A005940, A163511, A245612 and A244154.
Cf. A253888, A253889, A292243, A292244, A292245 and A292246 for other derived sequences.
Cf. A323893 (Dirichlet inverse), A323894 (sum with it), A336840 (inverse Möbius transform).

a048673 = (`div` 2) . (+ 1) . a045965
-- Reinhard Zumkeller, Jul 12 2012

f:= proc(n)
local F,q,t;
  F:= ifactors(n)[2];
  (1 + mul(nextprime(t[1])^t[2], t = F))/2
end proc:
seq(f(n),n=1..1000); # Robert Israel, Jan 15 2015

Table[(Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n, {n, 69}] (* Michael De Vlieger, Dec 18 2014, revised Mar 17 2016 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2; \\ Antti Karttunen, Dec 20 2014

A048673(n) = if(1==n,n,if(n%2,A253888(A048673((n-1)/2)),(3*A048673(n/2))-1)); \\ (Not practical, but demonstrates the construction as a binary tree). - Antti Karttunen, Feb 10 2021

from sympy import factorint, nextprime, prod
def a(n):
    f = factorint(n)
    return 1 if n==1 else (1 + prod(nextprime(i)**f[i] for i in f))//2 # Indranil Ghosh, May 09 2017

(define (A048673 n) (/ (+ 1 (A003961 n)) 2)) ;; Antti Karttunen, Dec 20 2014

From Antti Karttunen, Dec 20 2014: (Start)
a(1) = 1; for n>1: If n = product_{k>=1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k>=1} (p_{k+1})^(c_k)).
a(n) = (A003961(n)+1) / 2.
a(n) = floor((A045965(n)+1)/2).
Other identities. For all n >= 1:
a(n) = A108228(n)+1.
a(n) = A243501(n)/2.
A108951(n) = A181812(a(n)).
a(A246263(A246268(n))) = 2*n.
As a composition of other permutations involving prime-shift operations:
a(n) = A243506(A122111(n)).
a(n) = A243066(A241909(n)).
a(n) = A241909(A243062(n)).
a(n) = A244154(A156552(n)).
a(n) = A245610(A244319(n)).
a(n) = A227413(A246363(n)).
a(n) = A245612(A243071(n)).
a(n) = A245608(A245605(n)).
a(n) = A245610(A244319(n)).
a(n) = A249745(A249824(n)).
For n >= 2, a(n) = A245708(1+A245605(n-1)).
(End)
From Antti Karttunen, Jan 17 2015: (Start)
We also have the following identities:
a(2n) = 3*a(n) - 1. [Thus a(2n+1) = 0 or 1 when reduced modulo 3. See A341346]
a(3n) = 5*a(n) - 2.
a(4n) = 9*a(n) - 4.
a(5n) = 7*a(n) - 3.
a(6n) = 15*a(n) - 7.
a(7n) = 11*a(n) - 5.
a(8n) = 27*a(n) - 13.
a(9n) = 25*a(n) - 12.
and in general:
a(x*y) = (A003961(x) * a(y)) - a(x) + 1, for all x, y >= 1.
(End)
From Antti Karttunen, Feb 10 2021: (Start)
For n > 1, a(2n) = A016789(a(n)-1), a(2n+1) = A253888(a(n)).
a(2^n) = A007051(n) for all n >= 0. [A property shared with A183209 and A254103].
(End)
a(n) = A003602(A003961(n)). - Antti Karttunen, Apr 20 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/4) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 1.0319981... , where nextprime is A151800. - Amiram Eldar, Jan 18 2023

New name and crossrefs to derived sequences added by Antti Karttunen, Dec 20 2014

A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 5, 11, 1, 12, 2, 13, 4, 14, 1, 15, 1, 16, 7, 17, 3, 18, 1, 19, 11, 20, 1, 21, 1, 22, 6, 23, 1, 24, 2, 25, 13, 26, 1, 27, 5, 28, 17, 29, 1, 30, 1, 31, 10, 32, 7, 33, 1, 34, 19, 35, 1, 36, 1, 37, 9, 38, 3, 39, 1, 40, 8, 41, 1, 42

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

If n >= 2, n occurs in column a(n) of A246278.

By convention, a(1) = 0 because 1 does not occur in A246278.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Terms of A348717 halved. A305897 is the restricted growth sequence transform.
Positions of terms 1 .. 8 in this sequence are given by the following sequences: A000040, A001248, A006094, A030078, A090076, A251720, A090090, A030514.
Cf. A078898 (has the same role with array A083221 as this sequence has with A246278).
Cf. A000040, A001222, A001358, A003961, A055396, A064989, A064216, A243055, A246272, A249810, A249820, A249735, A252463.
This sequence is also used in the definition of the following permutations: A246274, A246276, A246675, A246677, A246683, A249815, A249817 (A249818), A249823, A249825, A250244, A250245, A250247, A250249.
Also in the definition of arrays A249821, A251721, A251722.
Sum of prime indices of a(n) is A359358(n) + A001222(n) - 1, cf. A326844.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A005940, A241916, A253565, A356958, A358195.

a246277[n_Integer] := Module[{f, p, a064989, a},
  f[x_] := Transpose@FactorInteger[x];
  p[x_] := Which[
    x == 1, 1,
    x == 2, 1,
    True, NextPrime[x, -1]];
  a064989[x_] := Times @@ Power[p /@ First[f[x]], Last[f[x]]];
  a[1] = 0;
  a[x_] := If[EvenQ[x], x/2, NestWhile[a064989, x, OddQ]/2];
a/@Range[n]]; a246277[84] (* Michael De Vlieger, Dec 19 2014 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };

A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2); \\ Antti Karttunen, Apr 30 2022

from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a064989(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
def a(n): return 0 if n==1 else n//2 if n%2==0 else a(a064989(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

;; two different variants, the second one employing memoizing definec-macro)
(define (A246277 n) (if (= 1 n) 0 (let loop ((n n)) (if (even? n) (/ n 2) (loop (A064989 n))))))
(definec (A246277 n) (cond ((= 1 n) 0) ((even? n) (/ n 2)) (else (A246277 (A064989 n)))))

a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)) = a(A064216(n+1)). [Cf. the formula for A252463.]
Instead of the equation for a(2n+1) above, we may write a(A003961(n)) = a(n). - Peter Munn, May 21 2022
Other identities. For all n >= 1, the following holds:
For all w >= 0, a(p_{i} * p_{j} * ... * p_{k}) = a(p_{i+w} * p_{j+w} * ... * p_{k+w}).
For all n >= 2, A001222(a(n)) = A001222(n)-1. [a(n) has one less prime factor than n. Thus each semiprime (A001358) is mapped to some prime (A000040), etc.]
For all n >= 2, a(n) = A078898(A249817(n)).
For semiprimes n = p_i * p_j, j >= i, a(n) = A000040(1+A243055(n)) = p_{1+j-i}.
a(n) = floor(A348717(n)/2). - Antti Karttunen, Apr 30 2022
If n has prime factorization Product_{i=1..k} prime(x_i), then a(n) = Product_{i=2..k} prime(x_i-x_1+1). The opposite version is A358195, prime indices A358172, even bisection A241916. - Gus Wiseman, Dec 29 2022

A083221 Sieve of Eratosthenes arranged as an array and read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

2, 4, 3, 6, 9, 5, 8, 15, 25, 7, 10, 21, 35, 49, 11, 12, 27, 55, 77, 121, 13, 14, 33, 65, 91, 143, 169, 17, 16, 39, 85, 119, 187, 221, 289, 19, 18, 45, 95, 133, 209, 247, 323, 361, 23, 20, 51, 115, 161, 253, 299, 391, 437, 529, 29, 22, 57, 125, 203, 319, 377, 493, 551, 667

Offset: 2

Yasutoshi Kohmoto, Jun 05 2003

This is permutation of natural numbers larger than 1.
From Antti Karttunen, Dec 19 2014: (Start)
If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252460 gives an inverse permutation. See also A249741.
For navigating in this array:
A055396(n) gives the row number of row where n occurs, and A078898(n) gives its column number, both starting their indexing from 1.
A250469(n) gives the number immediately below n, and when n is an odd number >= 3, A250470(n) gives the number immediately above n. If n is a composite, A249744(n) gives the number immediately left of n.
First cube of each row, which is {the initial prime of the row}^3 and also the first number neither a prime or semiprime, occurs on row n at position A250474(n).
(End)
The n-th row contains the numbers whose least prime factor is the n-th prime: A020639(T(n,k)) = A000040(n). - Franklin T. Adams-Watters, Aug 07 2015

			The top left corner of the array:
   2,   4,   6,    8,   10,   12,   14,   16,   18,   20,   22,   24,   26
   3,   9,  15,   21,   27,   33,   39,   45,   51,   57,   63,   69,   75
   5,  25,  35,   55,   65,   85,   95,  115,  125,  145,  155,  175,  185
   7,  49,  77,   91,  119,  133,  161,  203,  217,  259,  287,  301,  329
  11, 121, 143,  187,  209,  253,  319,  341,  407,  451,  473,  517,  583
  13, 169, 221,  247,  299,  377,  403,  481,  533,  559,  611,  689,  767
  17, 289, 323,  391,  493,  527,  629,  697,  731,  799,  901, 1003, 1037
  19, 361, 437,  551,  589,  703,  779,  817,  893, 1007, 1121, 1159, 1273
  23, 529, 667,  713,  851,  943,  989, 1081, 1219, 1357, 1403, 1541, 1633
  29, 841, 899, 1073, 1189, 1247, 1363, 1537, 1711, 1769, 1943, 2059, 2117
  ...

Transpose of A083140.
One more than A249741.
Inverse permutation: A252460.
Column 1: A000040, Column 2: A001248.
Row 1: A005843, Row 2: A016945, Row 3: A084967, Row 4: A084968, Row 5: A084969, Row 6: A084970.
Main diagonal: A083141.
First semiprime in each column occurs at A251717; A251718 & A251719 with additional criteria. A251724 gives the corresponding semiprimes for the latter. See also A251728.
Permutations based on mapping numbers between this array and A246278: A249817, A249818, A250244, A250245, A250247, A250249. See also: A249811, A249814, A249815.
Also used in the definition of the following arrays of permutations: A249821, A251721, A251722.
Cf. A002260, A004736, A004280, A020639, A038179, A055396, A078898, A138511, A249820, A249730, A249735, A249744, A250469, A250470, A250472, A250474.

lim = 11; a = Table[Take[Prime[n] Select[Range[lim^2], GCD[# Prime@ n, Product[Prime@ i, {i, 1, n - 1}]] == 1 &], lim], {n, lim}]; Flatten[Table[a[[i, n - i + 1]], {n, lim}, {i, n}]] (* Michael De Vlieger, Jan 04 2016, after Yasutoshi Kohmoto at A083140 *)

More terms from Hugo Pfoertner, Jun 13 2003

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.

A249746 Permutation of natural numbers: a(n) = A126760(A249735(n)) = A249824(A064216(n)).

Original entry on oeis.org

1, 2, 3, 4, 9, 5, 6, 12, 7, 8, 19, 10, 17, 42, 11, 13, 22, 26, 14, 29, 15, 16, 59, 18, 41, 32, 20, 31, 39, 21, 23, 92, 40, 24, 49, 25, 27, 82, 48, 28, 209, 30, 45, 52, 33, 63, 62, 54, 34, 109, 35, 36, 129, 37, 38, 69, 43, 68, 142, 70, 57, 72, 115, 44, 79, 46, 85, 292, 47, 50, 89, 74, 73, 202, 51, 53, 159, 87, 55, 99, 107, 56, 152, 58, 97, 192, 60

Offset: 1

Antti Karttunen, Nov 23 2014

nonn

Permutation obtained from the odd bisection of A003961 (or from the odd bisection of A048673).

			a(5) = 9 because of the following. 2*A064216(5) = 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(5) = 9.

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

Inverse: A249745.
Row 2 of A251722.
Cf. A003961, A007310, A048673, A084967, A126760, A249735, A064216, A249824.
Cf. also A273664, A273669.

t = PositionIndex[FactorInteger[#][[1, 1]] & /@ Range[10^6]]; f[n_] := Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Flatten@ Map[Position[Lookup[t, FactorInteger[#][[1, 1]] ], #] &[f@ f[2 #]] &, Table[Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1], {n, 87}]] (* Michael De Vlieger, Jul 25 2016, Version 10 *)

(define (A249746 n) (define (Ainv_of_A007310off0 n) (+ (* 2 (floor->exact (/ n 6))) (/ (- (modulo n 6) 1) 4))) (+ 1 (Ainv_of_A007310off0 (A003961 (+ n n -1)))))

a(n) = 1 + f(A003961(2n - 1)), where f(n) = 2*floor[n/6] + ((n mod 6)-1)/4. [Here 1 + f(A007310(n)) = n.]
a(n) = A126760(A249735(n)). - Antti Karttunen, Jul 25 2016
As a composition of related permutations:
a(n) = A249824(A064216(n)).
Other identities. For all n >= 1:
A249735(n) = A007310(a(n)).
a(3n-1) = A273669(a(n)) and a(A254049(n)) = A273664(a(n)). - Antti Karttunen, Aug 07 2016

A251721 Square array of permutations: A(row,col) = A249822(row, A249821(row+1, 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, 5, 3, 2, 1, 4, 4, 3, 2, 1, 7, 6, 4, 3, 2, 1, 11, 7, 5, 4, 3, 2, 1, 6, 9, 6, 5, 4, 3, 2, 1, 13, 10, 7, 6, 5, 4, 3, 2, 1, 17, 5, 8, 7, 6, 5, 4, 3, 2, 1, 10, 12, 10, 8, 7, 6, 5, 4, 3, 2, 1, 19, 15, 11, 9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 13, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 8, 16, 14, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 19, 15, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Antti Karttunen, Dec 07 2014

These are the "first differences" between permutations of array A249821, in a sense that by composing the first k rows of this array [from left to right, as in a(n) = row_1(row_2(...(row_k(n))))], one obtains row k+1 of A249821.

On row n, the first A250473(n) terms are fixed, and the first non-fixed term comes at A250474(n).

			The top left corner of the array:
1, 2, 3, 5, 4, 7, 11, 6, 13, 17, 10, 19, 9, 8, 23, 29, 14, 15, 31, 22, ...
1, 2, 3, 4, 6, 7, 9, 10, 5, 12, 15, 8, 16, 19, 21, 22, 13, 24, 11, 27, ...
1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 9, 16, 18, 20, 21, 23, 24, ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, ...
...

Inverse permutations can be found from array A251722.
Row 1: A064216, Row 2: A249745, Row 3: A250475.
Cf. A000027, A002260, A004736, A078898, A083221, A246277, A246278, A249821, A249822, A250473, A250474.

(define (A251721bi row col) (A249822bi row (A249821bi (+ row 1) col)))
(define (A251721 n) (A251721bi (A002260 n) (A004736 n)))
;; Code for A249821bi and A249822bi given in A249821, A249822.

A(row,col) = A249822(row, A249821(row+1, col)).

A(row,col) = A078898(A246278(row, A246277(A083221(row+1, col)))).

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.

cp's OEIS Frontend

A048673 Permutation of natural numbers: a(n) = (A003961(n)+1) / 2 [where A003961(n) shifts the prime factorization of n one step towards larger primes].

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A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)).

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A083221 Sieve of Eratosthenes arranged as an array and read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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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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A249746 Permutation of natural numbers: a(n) = A126760(A249735(n)) = A249824(A064216(n)).

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A251721 Square array of permutations: A(row,col) = A249822(row, A249821(row+1, 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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