A285703 -id:A285703 - cp's OEIS frontend

A285705 a(n) = 2n - A285703(n) = 2n - A000203(A064216(n)).

1, 1, 2, 2, 3, 4, 2, 4, 4, 2, 4, 4, 13, 13, 6, 2, 10, 12, 6, 4, 4, 2, 18, 4, 19, 10, 6, 24, 4, 6, 2, 22, 18, 6, 10, 4, 2, 37, 30, 6, 51, 4, 30, 16, 6, 20, 4, 24, 8, 44, 4, 2, 34, 4, 2, 16, 4, 36, 34, 36, 65, 10, 86, 14, 4, 4, 26, 76, 6, 2, 10, 48, 50, 55, 10, 2, 56, 36, 6, 16, 42, 6, 70, 4, 37, 46, 6, 98, 16, 6, 2, 4, 58, 76, 100, 10, 2, 52, 4, 2, 16, 60, 54

Offset: 1

Antti Karttunen, Apr 26 2017

Question: Are all terms positive? - Yes, they are, see A286385. (Note added Jul 24 2022).

For listening: fast tempo and percussive instrument, default "modulo 88" pitch mapping, all 10000 terms.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A033879, A064989, A064216, A151799, A285703, A285704, A286385.

Table[2 n - DivisorSigma[1, #] &@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 103}] (* Michael De Vlieger, Apr 26 2017 *)

A064989(n) = { my(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); };
A285705(n) = (n+n - sigma(A064989(n+n-1))); \\ Antti Karttunen, Jul 24 2022

(define (A285705 n) (- (* 2 n) (A285703 n)))

a(n) = 2*n - A285703(n) = 2*n - A000203(A064216(n)).
a(n) = 1 + A286385(A064216(n)). - Antti Karttunen, Jul 24 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.1831523243..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A285704 a(n) = A285703(n) - n = A000203(A064216(n)) - n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 5, 4, 5, 8, 7, 8, 0, 1, 9, 14, 7, 6, 13, 16, 17, 20, 5, 20, 6, 16, 21, 4, 25, 24, 29, 10, 15, 28, 25, 32, 35, 1, 9, 34, -10, 38, 13, 28, 39, 26, 43, 24, 41, 6, 47, 50, 19, 50, 53, 40, 53, 22, 25, 24, -4, 52, -23, 50, 61, 62, 41, -8, 63, 68, 61, 24, 23, 19, 65, 74, 21, 42, 73, 64, 39, 76, 13, 80, 48, 40, 81, -10, 73, 84, 89, 88, 35, 18, -5

Offset: 1

Antti Karttunen, Apr 26 2017

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A001065, A064216, A151799, A285703, A285705.

Table[-n + DivisorSigma[1, #] &@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 95}] (* Michael De Vlieger, Apr 26 2017 *)

Scheme
```
(define (A285704 n) (- (A285703 n) n))
```

a(n) = A285703(n) - n = A000203(A064216(n)) - n.

Sum_{k=1..n} a(k) ~ c * n^2, where c = -1/2 + Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.3168476756..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

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.

Original entry on oeis.org

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

A285702 a(n) = A000010(A064216(n)).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 10, 2, 12, 16, 4, 18, 6, 4, 22, 28, 6, 8, 30, 10, 36, 40, 4, 42, 20, 12, 46, 12, 16, 52, 58, 8, 20, 60, 18, 66, 70, 6, 24, 72, 8, 78, 24, 22, 82, 40, 28, 32, 88, 12, 96, 100, 8, 102, 106, 30, 108, 36, 20, 48, 42, 36, 18, 112, 40, 126, 64, 8, 130, 136, 42, 60, 44, 20, 138, 148, 24, 56, 150, 46, 72, 156, 12, 162, 110, 32, 166, 24, 52, 172, 178

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A064216, A064989, A099774, A151799, A285703.

Odd bisection of the following sequences: A347115, A348045, A349127, A349128.

Table[EulerPhi@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 91}] (* Michael De Vlieger, Apr 26 2017 *)

(define (A285702 n) (A000010 (A064216 n)))

a(n) = A000010(A064216(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.5366875995..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A326041 a(n) = sigma(A064989(n)).

Original entry on oeis.org

1, 1, 3, 1, 4, 3, 6, 1, 7, 4, 8, 3, 12, 6, 12, 1, 14, 7, 18, 4, 18, 8, 20, 3, 13, 12, 15, 6, 24, 12, 30, 1, 24, 14, 24, 7, 32, 18, 36, 4, 38, 18, 42, 8, 28, 20, 44, 3, 31, 13, 42, 12, 48, 15, 32, 6, 54, 24, 54, 12, 60, 30, 42, 1, 48, 24, 62, 14, 60, 24, 68, 7, 72, 32, 39, 18, 48, 36, 74, 4, 31, 38, 80, 18, 56, 42, 72, 8

Offset: 1

Antti Karttunen, Jun 03 2019

The odd bisection is A285703, the even bisection is the sequence itself.

Cf. A000203, A003973, A064989, A151799, A285703.

A326041(n) = if(1==n,n, my(f = factor(n)); prod(i=1, #f~, if(2==f[i,1],1,((precprime(f[i,1]-1)^(1+f[i,2]))-1)/(precprime(f[i,1]-1)-1))));

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) };
A326041(n) = sigma(A064989(n));

Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (q^(e+1)-1)/(q-1), where q = A151799(p).
a(n) = A000203(A064989(n)).
a(2n) = a(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/3) * Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.2722825585..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

cp's OEIS Frontend

A285705 a(n) = 2*n - A285703(n) = 2*n - A000203(A064216(n)).

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A285704 a(n) = A285703(n) - n = A000203(A064216(n)) - n.

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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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A285702 a(n) = A000010(A064216(n)).

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A326041 a(n) = sigma(A064989(n)).

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PARI

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A285705 a(n) = 2n - A285703(n) = 2n - A000203(A064216(n)).