A048673 -id:A048673 - cp's OEIS frontend

A064216 Replace each p^e with prevprime(p)^e in the prime factorization of odd numbers; inverse of sequence A048673 considered as a permutation of the natural numbers.

1, 2, 3, 5, 4, 7, 11, 6, 13, 17, 10, 19, 9, 8, 23, 29, 14, 15, 31, 22, 37, 41, 12, 43, 25, 26, 47, 21, 34, 53, 59, 20, 33, 61, 38, 67, 71, 18, 35, 73, 16, 79, 39, 46, 83, 55, 58, 51, 89, 28, 97, 101, 30, 103, 107, 62, 109, 57, 44, 65, 49, 74, 27, 113, 82, 127, 85, 24, 131

Offset: 1

Howard A. Landman, Sep 21 2001

a((A003961(n) + 1) / 2) = n and A003961(a(n)) = 2*n - 1 for all n. If the sequence is indexed by odd numbers only, it becomes multiplicative. In this variant sequence, denoted b, even indices don't exist, and we get b(1) = a(1) = 1, b(3) = a(2) = 2, b(5) = 3, b(7) = 5, b(9) = 4 = b(3) * b(3), ... , b(15) = 6 = b(3) * b(5), and so on. This property can also be stated as: a(x) * a(y) = a(((2x - 1) * (2y - 1) + 1) / 2) for x, y > 0. - Reinhard Zumkeller [re-expressed by Peter Munn, May 23 2020]
Not multiplicative in usual sense - but letting m=2n-1=product_j (p_j)^(e_j) then a(n)=a((m+1)/2)=product_j (p_(j-1))^(e_j). - Henry Bottomley, Apr 15 2005
From Antti Karttunen, Jul 25 2016: (Start)
Several permutations that use prime shift operation A064989 in their definition yield a permutation obtained from their odd bisection when composed with this permutation from the right. For example, we have:
A243505(n) = A122111(a(n)).
A243065(n) = A241909(a(n)).
A244153(n) = A156552(a(n)).
A245611(n) = A243071(a(n)).
(End)

			For n=11, the 11th odd number is 2*11 - 1 = 21 = 3^1 * 7^1. Replacing the primes 3 and 7 with the previous primes 2 and 5 gives 2^1 * 5^1 = 10, so a(11) = 10. - _Michael B. Porter_, Jul 25 2016

Odd bisection of A064989 and A252463.
Row 1 of A251721, Row 2 of A249821.
Cf. A048673 (inverse permutation), A048674 (fixed points).
Cf. A246361 (numbers n such that a(n) <= n.)
Cf. A246362 (numbers n such that a(n) > n.)
Cf. A246371 (numbers n such that a(n) < n.)
Cf. A246372 (numbers n such that a(n) >= n.)
Cf. A246373 (primes p such that a(p) >= p.)
Cf. A246374 (primes p such that a(p) < p.)
Cf. A246343 (iterates starting from n=12.)
Cf. A246345 (iterates starting from n=16.)
Cf. A245448 (this permutation "squared", a(a(n)).)
Cf. A253894, A254044, A254045 (binary width, weight and the number of nonleading zeros in base-2 representation of a(n), respectively).
Cf. A285702, A285703 (phi and sigma applied to a(n).)
Here obviously the variant 2, A151799(n) = A007917(n-1), of the prevprime function is used.
Cf. also A003961, A270430, A270431.
Cf. also permutations A122111, A156552, A241909, A243071, A243065, A243505, A244153, A245611, A254116.

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

a(n) = {my(f = factor(2*n-1)); for (k=1, #f~, f[k,1] = precprime(f[k,1]-1)); factorback(f);} \\ Michel Marcus, Mar 17 2016

from sympy import factorint, prevprime
from operator import mul
def a(n):
    f=factorint(2*n - 1)
    return 1 if n==1 else reduce(mul, [prevprime(i)**f[i] for i in f]) # Indranil Ghosh, May 13 2017

(define (A064216 n) (A064989 (- (+ n n) 1))) ;; Antti Karttunen, May 12 2014

a(n) = A064989(2n - 1). - Antti Karttunen, May 12 2014

Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime > 2} ((p^2-p)/(p^2-q(p))) = 0.6621117868..., where q(p) = prevprime(p) (A151799). - Amiram Eldar, Jan 21 2023

More terms from Reinhard Zumkeller, Sep 26 2001

Additional description added by Antti Karttunen, May 12 2014

A245612 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = 3*a(n)-1, a(2n+1) = A254049(a(n)); composition of A048673 and A163511.

Original entry on oeis.org

1, 2, 5, 3, 14, 13, 8, 4, 41, 63, 38, 25, 23, 18, 11, 6, 122, 313, 188, 172, 113, 123, 74, 61, 68, 88, 53, 39, 32, 28, 17, 7, 365, 1563, 938, 1201, 563, 858, 515, 666, 338, 613, 368, 424, 221, 303, 182, 85, 203, 438, 263, 270, 158, 193, 116, 72, 95, 138, 83, 46, 50, 33, 20, 9

Offset: 0

Antti Karttunen, Jul 28 2014

Note the indexing: the domain starts from 0, while the range excludes zero.
From Antti Karttunen, Jul 25 2016: (Start)
This sequence can be represented as a binary tree. Each left hand child is obtained by applying A016789(n-1) when the parent contains n (i.e., multiply by 3, subtract one), and each right hand child is obtained by applying A254049 to the parent's contents:
                                     1
                                     |
                  ...................2...................
                 5                                       3
      14......../ \........13                  8......../ \........4
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  41       63         38       25         23       18         11       6
122 313 188  172   113  123  74  61     68  88   53  39     32  28   17 7
etc.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A245611.

Cf. A016789, A048673, A163511, A243065-A243066, A243505-A243506, A245608, A245610, A244153, A244154, A254049.

Table[(Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ If[n == 0, 1, Prime[#] Product[Prime[m]^(Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]][[-m]]), {m, #}] &[DigitCount[n, 2, 1]]], {n, 0, 63}] (* Michael De Vlieger, Jul 25 2016 *)

(define (A245612 n) (A048673 (A163511 n))) ;; offset 0, a(0) = 1.

a(n) = A048673(A163511(n)).

a(0) = 1, a(1) = 2, a(2n) = 3*a(n)-1, a(2n+1) = A254049(a(n)). - Antti Karttunen, Jul 25 2016

A048674 Fixed points of A048673 and A064216: Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) = (2*n)-1, where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

1, 2, 3, 25, 26, 33, 93, 1034, 970225, 8550146, 325422273, 414690595, 1864797542, 2438037206

Offset: 1

Antti Karttunen, Jul 14 1999

Equally: after 1, numbers n such that, if the prime factorization of 2n-1 = Product_{k >= 1} (p_k)^(c_k) then Product_{k >= 1} (p_{k-1})^(c_k) = n.
Factorization of the initial terms: 1, 2, 3, 5^2, 2*13, 3*11, 3*31, 2*11*47, 5^2*197^2, 2*11*47*8269, 3*11*797*12373, 5*11^2*433*1583, 2*23*59*101*6803, 2*11*53*1201*1741.
The only 3-cycle of permutation A048673 in range 1 .. 402653184 is (2821 3460 5639).
For 2-cycles, take setwise difference of A245449 and this sequence.
Numbers k for which A336853(k) = k-1. - Antti Karttunen, Nov 26 2021

			25 is present, as 2*25 - 1 = 49 = p_4^2, and p_3^2 = 5*5 = 25.
26 is present, as 2*26 - 1 = 51 = 3*17 = p_2 * p_8, and p_1 * p_7 = 2*13 = 26.
Alternatively, as 26 = 2*13 = p_1 * p_7, and ((p_2 * p_8)+1)/2 = ((3*17)+1)/2 = 26 also, thus 26 is present.

Fixed points of permutation pair A048673/A064216.
Positions of zeros in A349573.
Subsequence of the following sequences: A245449, A269860, A319630, A349622, A378980 (see also A379216).
This sequence is also obtained as a setwise difference of the following pairs of sequences: A246281 \ A246351, A246352 \ A246282, A246361 \ A246371, A246372 \ A246362.
Cf. A000040, A003961 (A045965), A064989, A285701, A336853.
Cf. also A348514 (fixed points of map A108228, similar to A048673).

A048673 := n -> (A003961(n)+1)/2;
A048674list := proc(upto_n) local b,i; b := [ ]; for i from 1 to upto_n do if(A048673(i) = i) then b := [ op(b), i ]; fi; od: RETURN(b); end;

Join[{1}, Reap[For[n = 1, n < 10^7, n++, ff = FactorInteger[n]; If[Times @@ Power @@@ (NextPrime[ff[[All, 1]]]^ff[[All, 2]]) == 2 n - 1, Print[n]; Sow[n]]]][[2, 1]]] (* Jean-François Alcover, Mar 04 2016 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA048674(n) = ((n+n)==(1+A003961(n))); \\ Antti Karttunen, Nov 26 2021

Entry revised and the names in Maple-code cleaned by Antti Karttunen, Aug 25 2014

Terms a(11) - a(14) added by Antti Karttunen, Sep 11-13 2014

A244154 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = A254049(a(n)), a(2n+1) = 3*a(n)-1; composition of A048673 and A005940.

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 13, 14, 6, 11, 18, 23, 25, 38, 63, 41, 7, 17, 28, 32, 39, 53, 88, 68, 61, 74, 123, 113, 172, 188, 313, 122, 9, 20, 33, 50, 46, 83, 138, 95, 72, 116, 193, 158, 270, 263, 438, 203, 85, 182, 303, 221, 424, 368, 613, 338, 666, 515, 858, 563, 1201, 938, 1563, 365, 10, 26, 43, 59, 60

Offset: 0

Antti Karttunen, Jun 27 2014

Note the indexing: the domain starts from 0, while the range excludes zero.
From Antti Karttunen, May 30 2017: (Start)
This sequence can be represented as a binary tree. Each left hand child is obtained by applying A254049(n) when the parent contains n, and each right hand child is obtained by applying A016789(n-1) (i.e., multiply by 3, subtract one) to the parent's contents:
                                     1
                                     |
                  ...................2...................
                 3                                       5
       4......../ \........8                  13......../ \........14
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   6       11         18       23         25       38         63       41
  7 17   28  32     39  53   88  68     61  74  123  113   172  188  313 122
etc.
This is a mirror image of the tree depicted in A245612.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A244153.

Cf. A005940, A048673, A054429, A243065-A243066, A243505-A243506, A245608, A245610, A245612, A016789, A254049, A285712, A285714, A286613.

(define (A244154 n) (A048673 (A005940 (+ 1 n))))
;; Implementing a new recurrence, with memoization-macro definec:
(definec (A244154 n) (cond ((<= n 1) (+ 1 n)) ((even? n) (A254049 (A244154 (/ n 2)))) (else (+ -1 (* 3 (A244154 (/ (- n 1) 2))))))) ;; Antti Karttunen, May 30 2017

a(n) = A048673(A005940(n+1)).
From Antti Karttunen, May 30 2017: (Start)
a(0) = 1, a(1) = 2, a(2n) = A254049(a(n)), a(2n+1) = 3*a(n)-1.
a(n) = A245612(A054429(n)).
(End)

A254049 Odd bisection of A048673: a(n) = A048673(2*n-1).

Original entry on oeis.org

1, 3, 4, 6, 13, 7, 9, 18, 10, 12, 28, 15, 25, 63, 16, 19, 33, 39, 21, 43, 22, 24, 88, 27, 61, 48, 30, 46, 58, 31, 34, 138, 60, 36, 73, 37, 40, 123, 72, 42, 313, 45, 67, 78, 49, 94, 93, 81, 51, 163, 52, 54, 193, 55, 57, 103, 64, 102, 213, 105, 85, 108, 172, 66, 118, 69, 127, 438, 70, 75, 133, 111, 109, 303

Offset: 1

Antti Karttunen, Jan 24 2015

nonn

Shift the prime factorization of odd numbers one step towards larger primes, add one and divide by two.

			For n = 8, the eighth odd number is 2*8 - 1 = 15 = 3*5 = prime(2) * prime(3). By adding one to both prime indices, we get prime(3) * prime(4) = 5*7 = 35, and (35+1)/2 = 18, thus a(8) = 18. Here prime(n) = A000040(n).

Cf. A032766 (omitting the initial 0, the same sequence sorted into ascending order).
Also a permutation of A253888.
Cf. A000040, A003961, A048673, A249735, A249746, A254050, A254051, A254053.

a(n) = A048673(2*n-1) = (1+A003961(2*n-1)) / 2 = (1+A249735(n)) / 2.

a(n) = A032766(A249746(n)).

A243066 Permutation of natural numbers, the even bisection of A241909 incremented by one and halved; equally, a composition of A241909 and A048673: a(n) = A048673(A241909(n)).

Original entry on oeis.org

1, 2, 5, 3, 14, 13, 41, 4, 8, 63, 122, 25, 365, 313, 38, 6, 1094, 18, 3281, 172, 188, 1563, 9842, 61, 23, 7813, 11, 1201, 29525, 123, 88574, 7, 938, 39063, 113, 39, 265721, 195313, 4688, 666, 797162, 858, 2391485, 8404, 74, 976563, 7174454, 85, 68, 88, 23438, 58825, 21523361, 28

Offset: 1

Antti Karttunen, Jun 01 2014

nonn

For n > 1, 2n is found in A241909 from the position (2*a(n))-1. I.e., A241909((2*a(n))-1) = 2n for all n >= 2.
Or in other words, a(n) gives the position in the odd bisection of A241909 where 2n is located at.
Are there any other fixed points than 1, 2, 18 and 72?

Antti Karttunen, Table of n, a(n) for n = 1..512
Index entries for sequences that are permutations of the natural numbers

Inverse: A243065.

Cf. A048673, A241909, A243505-A243506, A244152-A244154, A243061-A243062.

a(1) = 1, a(n) = (A241909(2*n)+1)/2.
As a composition of related permutations:
a(n) = A048673(A241909(n)).
a(n) = A241909(A243062(A241909(n))).
For all n>=1, a(2^n) = A006254(n).

A291759 Binary encoding of 2-digits in ternary representation of A048673(n).

Original entry on oeis.org

0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 2, 5, 0, 3, 4, 1, 0, 1, 0, 1, 0, 5, 2, 9, 6, 7, 8, 5, 2, 7, 4, 1, 2, 1, 0, 1, 4, 3, 2, 1, 4, 1, 6, 9, 2, 3, 0, 17, 10, 13, 4, 13, 0, 23, 4, 9, 8, 5, 0, 13, 2, 9, 8, 1, 10, 3, 0, 1, 12, 3, 0, 1, 0, 11, 2, 5, 12, 5, 2, 1, 10, 9, 4, 1, 8, 11, 14, 17, 4, 5, 0, 5, 0, 15, 0, 33, 6, 21, 16, 25, 6, 11, 8, 25, 16, 3, 8, 45, 8, 9, 4, 17, 8

Offset: 1

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A048673, A286585, A286586, A289814, A291760, A291763.

a(n) = A289814(A048673(n)).

A243506 Permutation of natural numbers: a(n) = A048673(A122111(n)).

Original entry on oeis.org

1, 2, 5, 3, 14, 8, 41, 4, 13, 23, 122, 11, 365, 68, 38, 6, 1094, 18, 3281, 32, 113, 203, 9842, 17, 63, 608, 25, 95, 29525, 53, 88574, 7, 338, 1823, 188, 28, 265721, 5468, 1013, 50, 797162, 158, 2391485, 284, 74, 16403, 7174454, 20, 313, 88, 3038, 851, 21523361, 39, 563, 149, 9113, 49208, 64570082, 83, 193710245, 147623, 221, 9

Offset: 1

Antti Karttunen, Jun 25 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Indranil Ghosh, Python program for computing the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse: A243505.
Cf. A000040, A000079, A006254, A007051.
Related or similar permutations: A048673, A122111, A243065-A243066, A244981-A244982, A244983-A244984, A244153-A244154.

(define (A243506 n) (A048673 (A122111 n)))

a(n) = A048673(A122111(n)).
a(n) = A243066(A241916(n)).
For all n >= 1, a(A000040(n)) = A007051(n) and a(A000079(n)) = A006254(n).

A292251 The 3-adic valuation of A048673(n).

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0

Offset: 1

Antti Karttunen, Sep 12 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

One less than the even bisection of A292252.
Cf. A007814, A007949, A048673, A292250.
Cf. also A292241, A292261.

IntegerExponent[#, 3] & /@ Table[(Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n, {n, 120}] (* Michael De Vlieger, Sep 12 2017 *)

a(n) = A007814(1+A292250(n)).
a(n) = A007949(A048673(n)).
a(n) = A007949(3*A048673(n)) - 1.
a(n) = A292252(2n)-1.

A336840 Inverse Möbius transform of A048673.

Original entry on oeis.org

1, 3, 4, 8, 5, 14, 7, 22, 17, 18, 8, 42, 10, 26, 26, 63, 11, 65, 13, 55, 38, 30, 16, 124, 30, 38, 80, 81, 17, 100, 20, 185, 44, 42, 50, 206, 22, 50, 56, 164, 23, 148, 25, 94, 127, 62, 28, 368, 68, 117, 62, 120, 31, 316, 58, 244, 74, 66, 32, 318, 35, 78, 189, 550, 74, 172, 37, 133, 92, 196, 38, 626, 41, 86, 174, 159

Offset: 1

Antti Karttunen, Aug 07 2020

nonn

Arithmetic mean of the number of divisors (A000005) and prime-shifted sigma (A003973), thus a(n) is the average between the number of and the sum of divisors of A003961(n).

The local minima occur on primes p, where p/2 < a(p) <= (p+1).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000005, A003961, A003973, A007503, A048673, A113415, A336838, A336839, A336841, A349371, A349384, A349385, A349910 [= a(n)-A048673(n)], A364063, A378520 (Dirichlet inverse).

A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A336840(n) = sumdiv(n,d,A048673(d));

A336840(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(numdiv(n)+sigma(factorback(f))); };

a(n) = Sum_{d|n} A048673(d).
a(n) = (1/2) * (A000005(n) + A003973(n)).
a(n) = A113415(A003961(n)). - Antti Karttunen, Jun 01 2022
a(n) = A349371(A003961(n)) = A364063(A048673(n)). - Antti Karttunen, Nov 30 2024

cp's OEIS Frontend

A064216 Replace each p^e with prevprime(p)^e in the prime factorization of odd numbers; inverse of sequence A048673 considered as a permutation of the natural numbers.

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A245612 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = 3*a(n)-1, a(2n+1) = A254049(a(n)); composition of A048673 and A163511.

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A048674 Fixed points of A048673 and A064216: Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) = (2*n)-1, where p_k indicates the k-th prime, A000040(k).

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A244154 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = A254049(a(n)), a(2n+1) = 3*a(n)-1; composition of A048673 and A005940.

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A254049 Odd bisection of A048673: a(n) = A048673(2*n-1).

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A243066 Permutation of natural numbers, the even bisection of A241909 incremented by one and halved; equally, a composition of A241909 and A048673: a(n) = A048673(A241909(n)).

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A291759 Binary encoding of 2-digits in ternary representation of A048673(n).

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A243506 Permutation of natural numbers: a(n) = A048673(A122111(n)).

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