A064216 -id:A064216 - cp's OEIS frontend

A246266 Permutation of natural numbers: a(n) = A246262(A064216((2*n)-1)).

1, 2, 3, 6, 8, 5, 4, 11, 9, 15, 20, 7, 12, 26, 17, 32, 16, 21, 35, 18, 10, 22, 39, 31, 42, 47, 14, 53, 55, 25, 28, 13, 38, 40, 67, 41, 34, 72, 29, 81, 46, 24, 62, 87, 52, 93, 45, 23, 96, 98, 63, 57, 60, 70, 77, 75, 19, 111, 112, 78, 116, 36, 85, 69, 121, 44, 74, 127, 56, 131, 135, 50, 89, 137, 30

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

This permutation is induced when A064216 is restricted to odd numbers (equally, when A064989 is restricted to the numbers of form 4k+1) and the resulting numbers are "ranked" with A246262.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A246265.

Cf. also A064216, A064989, A246262, A246268.

(define (A246266 n) (A246262 (A064216 (+ -1 n n))))

a(n) = A246262(A064216((2*n)-1)).

a(n) = A246262(A064989((4*n)-3)).

A253888 a(0) = 1; for n >= 1: a(n) = A048673(1+(2*A064216(n))).

Original entry on oeis.org

1, 3, 4, 6, 7, 13, 18, 15, 9, 63, 39, 28, 43, 12, 10, 27, 31, 16, 19, 138, 88, 123, 45, 25, 78, 48, 30, 81, 24, 73, 55, 105, 22, 36, 108, 72, 438, 111, 21, 37, 303, 33, 148, 42, 93, 87, 103, 213, 54, 91, 58, 298, 171, 34, 363, 165, 172, 198, 102, 49, 69, 163, 76, 46, 115, 228, 333, 288, 61, 135, 319, 90, 130, 75, 52

Offset: 0

Antti Karttunen, Jan 22 2015

nonn

When A048673 is represented as a binary tree, then the node k which contains value n = A048673(k) has as its right child a(n) = A048673(2k+1).

Antti Karttunen, Table of n, a(n) for n = 0..8192

Same sequence sorted into ascending order: A032766.
Also a permutation of A254049.
Cf. A048673, A064216, A253889, A253890, A254051, A254053.

a(0) = 1; for n >= 1: a(n) = A048673(1+(2*A064216(n))).

Also, for n >= 1: a(n) = A254049(1+A064216(n)).

A254116 Permutation of natural numbers: a(n) = A064216(A254103(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 13, 10, 11, 12, 9, 14, 17, 16, 41, 26, 23, 20, 15, 22, 19, 24, 67, 18, 37, 28, 47, 34, 29, 32, 27, 82, 61, 52, 73, 46, 43, 40, 89, 30, 21, 44, 59, 38, 31, 48, 111, 134, 107, 36, 57, 74, 33, 56, 149, 94, 79, 68, 83, 58, 25, 64, 359, 54, 181, 164, 193, 122, 101, 104, 229, 146, 49, 92, 85, 86, 71, 80, 185, 178, 139, 60, 95, 42, 39

Offset: 1

Antti Karttunen, Feb 04 2015

nonn

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

Inverse: A254115.
Fixed points: A254099.
Related permutations: A064216, A254103, A254118.

default(primelimit, 2^30);
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)};
A064216(n) = A064989((2*n)-1);
A254103(n) = { if(0==n,0,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\2)); };
A254116(n) = A064216(A254103(n));
for(n=1, 8192, write("b254116.txt", n, " ", A254116(n)));

from sympy import factorint, prevprime, floor
from operator import mul
def a064216(n):
    f=factorint(2*n - 1)
    return 1 if n==1 else reduce(mul, [prevprime(i)**f[i] for i in f])
def a254103(n):
    if n==0: return 0
    if n%2==0: return 3*a254103(n/2) - 1
    else: return floor((3*(1 + a254103((n - 1)/2)))/2)
def a(n): return a064216(a254103(n)) # Indranil Ghosh, Jun 06 2017

(define (A254116 n) (A064216 (A254103 n)))

a(n) = A064216(A254103(n)).
Other identities. For all n >= 1:
a(n) = a(2n)/2. [Even bisection halved gives back the sequence itself.]
A254118(n) = (a((2*n)+1) - 1)/2. [Likewise, the odd bisection induces A254118.]

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

Original entry on oeis.org

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

A243061 Permutation of natural numbers, a composition of A241909 and A064216: a(n) = A064216(A241909(n)).

Original entry on oeis.org

1, 2, 5, 3, 6, 13, 29, 4, 7, 47, 20, 25, 113, 95, 15, 11, 78, 23, 355, 158, 103, 267, 406, 89, 19, 1247, 17, 1237, 1577, 139, 660, 10, 221, 4363, 67, 38, 8179, 13109, 967, 393, 9266, 515, 21605, 4162, 28, 23601, 19578, 239, 43, 83, 987, 31247

Offset: 1

Antti Karttunen, Jun 02 2014

nonn

This is A241909-conjugate of A243065. Please see the comments at the latter sequence.

Inverse permutation: A243062.

Cf. A064216, A241909, A243065-A243066.

A064216(n) = A064989(n+n-1);
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) };
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
A243061(n) = A064216(A241909(n)); \\ Antti Karttunen, Dec 10 2021

(define (A243061 n) (A064216 (A241909 n)))

a(n) = A064216(A241909(n)).

a(n) = A241909(A243065(A241909(n))).

A285701 a(n) = number of iterations x -> A064216(x) needed to reach a nonprime number when starting from n, a(2) = a(3) = 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

			For n=2, A064216(2) = 2, thus there is exactly one distinct prime that can be reached when iterating A064216 starting from 2, thus a(2) = 1.
For n=19, A064216(19) = 31 (a prime), A064216(31) = 59 (a prime) and A064216(59) = 44 (not a prime), thus there are exactly three distinct primes that are encountered when iterating A064216 starting from 19 before a nonprime is reached, thus a(19) = 3 (the count includes also the starting prime 19).

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

Cf. A000040, A000720, A010051, A048674, A064216, A064989, A245449, A246373, A285700, A285706.

Cf. A005382 (gives positions of terms > 1 from its third term 7 onward).

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)};
A064216(n) = A064989((2*n)-1);
A285701(n) = if(!isprime(n),0,if((2==n)||(3==n),1,1+A285701(A064216(n))));

(definec (A285701 n) (cond ((zero? (A010051 n)) 0) ((or (= 2 n) (= 3 n)) 1) (else (+ 1 (A285701 (A064216 n))))))
;; Another version not requiring A064216 and A064989:
(definec (A285701 n) (cond ((zero? (A010051 n)) 0) ((or (= 2 n) (= 3 n)) 1) ((zero? (A010051 (+ n n -1))) 1) (else (+ 1 (A285701 (A000040 (+ -1 (A000720 (+ n n -1)))))))))

If A010051(n) = 0 [when n is a nonprime], a(n) = 0, otherwise a(n) = 1 + a(A064216(n)), with a(2) = a(3) = 1.

A285706 a(n) = number of iterations x -> A064216(x) needed to reach a nonprime number when starting from prime(n), a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

Length (or size for the closed cycles: [2] and [3]) of the complete "slipping Cunningham chain of the second kind" starting with prime(n). That is, at the end of every step, the next prime q = 2p-1 "slips" by one step towards smaller primes as A064989(q).

After n = 1, 2 (primes 2 & 3) differs from A181715 for the first time at n=22, where a(22) = 2, while A181715(22) = 3, prime(22) = 79.

			See examples in A285701.

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

Cf. A000040, A064216, A064989, A181715, A246373, A285701.

Cf. A137288 (gives the positions of terms > 1 after its two initial terms).

Table[If[n <= 2, 1, -1 + Length@ NestWhileList[Apply[Times, FactorInteger[2 # - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e] &, Prime@ n, PrimeQ@ # &]], {n, 120}] (* Michael De Vlieger, Apr 26 2017 *)

A285706(n) = A285701(prime(n)); \\ The rest of code in A285701.

(define (A285706 n) (A285701 (A000040 n)))

a(n) = A285701(A000040(n)).

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

A349358 Dirichlet inverse of A064216, which is A064989(2n-1), where A064989 is fully multiplicative with a(2) = 1 and a(p) = prevprime(p) for odd primes p.

Original entry on oeis.org

1, -2, -3, -1, -4, 5, -11, 6, -4, -1, -10, 3, -9, 36, 1, -24, -14, 25, -31, 38, 29, -1, -12, -29, -9, 10, 4, -11, -34, 53, -59, 62, 27, -5, 50, -41, -71, 106, 19, -83, -16, -125, -39, 98, 51, -7, -58, 184, 32, 112, -13, -15, -30, -84, -27, -170, 77, 79, -44, -109, -49, 162, 184, -84, -10, 31, -85, 192, -59, -75, -86

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A064216, A064989, A349359, A349398.

Cf. also A323893, A349125.

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); };
A064216(n) = A064989((2*n)-1);
memoA349358 = Map();
A349358(n) = if(1==n,1,my(v); if(mapisdefined(memoA349358,n,&v), v, v = -sumdiv(n,d,if(dA064216(n/d)*A349358(d),0)); mapput(memoA349358,n,v); (v)));

a(1) = 1; a(n) = -Sum_{d|n, d < n} A064216(n/d) * a(d).

a(n) = A349359(n) - A064216(n).

A269853 Permutation of natural numbers: a(1) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A064216(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 10, 9, 12, 15, 14, 19, 16, 13, 22, 31, 20, 27, 18, 21, 24, 63, 30, 23, 28, 17, 38, 43, 32, 35, 26, 29, 44, 39, 62, 87, 40, 37, 54, 79, 36, 75, 42, 25, 48, 159, 126, 127, 60, 61, 46, 51, 56, 55, 34, 53, 76, 123, 86, 107, 64, 41, 70, 71, 52, 247, 58, 125, 88, 143, 78, 251, 124, 45, 174, 59, 80, 287, 74, 33

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

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

Inverse: A269854.

Cf. A003961, A064216, A064989, A243071, A246376.

a[1] = 1; a[n_] := a[n] = If[EvenQ@ n, 2 a[n/2], 1 + 2 a[Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[(n - 1) - 1]]]; Table[a@ n, {n, 83}] (* Michael De Vlieger, Mar 23 2016 *)

a(1) = 1, after which for even n, a(n) = 2*a(n/2), for odd n, a(n) = 1 + 2*a(A064989(n-2)).

cp's OEIS Frontend

A246266 Permutation of natural numbers: a(n) = A246262(A064216((2*n)-1)).

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A253888 a(0) = 1; for n >= 1: a(n) = A048673(1+(2*A064216(n))).

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A254116 Permutation of natural numbers: a(n) = A064216(A254103(n)).

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

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A243061 Permutation of natural numbers, a composition of A241909 and A064216: a(n) = A064216(A241909(n)).

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A285701 a(n) = number of iterations x -> A064216(x) needed to reach a nonprime number when starting from n, a(2) = a(3) = 1.

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A285706 a(n) = number of iterations x -> A064216(x) needed to reach a nonprime number when starting from prime(n), a(1) = a(2) = 1.

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

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

A269853 Permutation of natural numbers: a(1) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A064216(n)).