A064216 -id:A064216 - cp's OEIS frontend

A254118 Permutation of natural numbers: a(n) = A249745(1+A254103(n)) - 1.

1, 2, 3, 6, 5, 4, 8, 20, 11, 7, 9, 33, 18, 23, 14, 13, 30, 36, 21, 44, 10, 29, 15, 55, 53, 28, 16, 74, 39, 41, 12, 179, 90, 96, 50, 114, 24, 42, 35, 92, 69, 47, 19, 86, 25, 51, 26, 236, 153, 110, 81, 101, 22, 45, 48, 221, 113, 119, 56, 77, 65, 38, 17, 546, 182

Offset: 1

Antti Karttunen, Feb 05 2015

Antti Karttunen, Table of n, a(n) for n = 1..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A254117.
Other related permutations: A254116, A249745, A254103 (compare to the scatterplot of this one).
Cf. A254120 (= a(2^n)).

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));
A254118(n) = (A254116(n+n+1)-1)/2;
for(n=1, 8191, write("b254118.txt", n, " ", A254118(n)));
(Scheme, two versions)
(define (A254118 n) (+ -1 (A249745 (+ 1 (A254103 n)))))
(define (A254118 n) (/ (+ -1 (A254116 (+ 1 n n))) 2))

from sympy import factorint, prevprime, floor
from operator import mul
from functools import reduce
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 a254116(n): return a064216(a254103(n))
def a(n): return (a254116(2*n + 1) - 1)//2
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

a(n) = A249745(1+A254103(n)) - 1.

a(n) = (A254116((2*n)+1)-1) / 2. [Obtained also from the odd bisection of A254116.]

A266407 Permutation of natural numbers: a(n) = A064989(A263273((2*n)-1)).

Original entry on oeis.org

1, 2, 5, 3, 4, 17, 11, 10, 9, 7, 6, 19, 13, 8, 21, 31, 34, 71, 29, 22, 61, 25, 20, 59, 41, 18, 73, 23, 14, 33, 43, 12, 53, 37, 38, 35, 15, 26, 67, 47, 16, 157, 107, 42, 145, 55, 62, 197, 69, 68, 179, 113, 142, 129, 39, 58, 191, 137, 44, 45, 49, 122, 227, 101, 50, 199, 151, 40, 121, 57, 118, 211, 89, 82, 111, 149, 36, 91, 85

Offset: 1

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..9842
Index entries for sequences that are permutations of the natural numbers

Inverse: A266408.
Cf. A064989, A263273.
Cf. also A064216, A266401, A266403.

A030102(n) = { my(r=[n%3]); while(0M. F. Hasler's Nov 04 2011 code in A030102.
A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3))); \\ Taking of the quotient probably unnecessary.
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)};
A266407 = n -> A064989(A263273((2*n)-1));
for(n=1, 9842, write("b266407.txt", n, " ", A266407(n)));

(define (A266407 n) (A064989 (A263273 (+ n n -1))))

a(n) = A064989(A263273((2*n)-1)).

A285334 a(n) = A046523(A243505(n)).

Original entry on oeis.org

1, 2, 4, 8, 2, 16, 32, 6, 64, 128, 12, 256, 4, 2, 512, 1024, 24, 12, 2048, 48, 4096, 8192, 6, 16384, 8, 96, 32768, 36, 192, 65536, 131072, 12, 72, 262144, 384, 524288, 1048576, 6, 24, 2097152, 2, 4194304, 144, 768, 8388608, 72, 1536, 288, 16777216, 24, 33554432, 67108864, 30, 134217728, 268435456, 3072, 536870912, 576, 48, 216, 16, 6144, 4, 1073741824, 12288

Offset: 1

Antti Karttunen, Apr 24 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024

Cf. A046523, A064216, A122111, A243505, A278221.

(define (A285334 n) (A046523 (A243505 n)))

a(n) = A046523(A243505(n)).

a(n) = A278221(A064216(n)).

A302847 Permutation of natural numbers: a(1) = 1; for n > 1, a(n) = A064989(2+A003961(n-1)).

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 13, 11, 23, 8, 19, 6, 43, 17, 15, 31, 79, 10, 35, 9, 33, 34, 37, 29, 131, 26, 47, 113, 97, 14, 103, 22, 75, 61, 53, 73, 223, 41, 67, 46, 181, 12, 163, 25, 65, 106, 83, 21, 217, 74, 139, 89, 87, 59, 253, 58, 209, 44, 51, 20, 313, 38, 109, 271, 533, 49, 193, 71, 167, 50, 229, 18, 673, 16, 27, 187, 119, 69, 251, 39, 563, 238, 127, 55, 335, 24

Offset: 1

Antti Karttunen, Apr 26 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
David Newman, et al., New sequences from old, Discussion on SeqFan-mailing list, January 2018.
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences computed from indices in prime factorization

Cf. A302848 (inverse).
Cf. A003961, A048673, A064216, A064989.
Cf. also A297165, A302849.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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)};
A302847(n) = if(1==n,n,A064989(2+A003961(n-1)));

a(1) = 1; for n > 1, a(n) = A064989(2+A003961(n-1)).

a(1) = 1; for n > 1, a(n) = A064216(1+A048673(n-1)).

A353412 The odd part of hybrid shift: a(n) = A000265(A252463(n)).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 5, 1, 1, 5, 7, 3, 11, 7, 3, 1, 13, 9, 17, 5, 5, 11, 19, 3, 9, 13, 1, 7, 23, 15, 29, 1, 7, 17, 15, 9, 31, 19, 11, 5, 37, 21, 41, 11, 3, 23, 43, 3, 25, 25, 13, 13, 47, 27, 21, 7, 17, 29, 53, 15, 59, 31, 5, 1, 33, 33, 61, 17, 19, 35, 67, 9, 71, 37, 9, 19, 35, 39, 73, 5, 1, 41, 79, 21, 39, 43, 23, 11

Offset: 1

Antti Karttunen, Apr 18 2022

nonn

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

Cf. A000005, A252463, A064216, A064989, A320107.

Cf. A000265 (even bisection), A353413 (odd bisection).

A000265(n) = (n>>valuation(n,2));
A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A252463(n) = if(!(n%2),n/2,A064989(n));
A353412(n) = A000265(A252463(n));

from math import prod
from sympy import factorint, prevprime
def A353412(n): return int(bin(prod(1 if p == 2 else prevprime(p)*e for p, e in factorint(n).items()) if n % 2 else n//2)[2:].rstrip('0'),2) # Chai Wah Wu, Apr 18 2022

a(n) = A000265(A252463(n)).
a(2*n) = A000265(n), a(2*n-1) = A353413(n) = A000265(A064216(n)).
For all n >= 1, A000005(a(n)) = A320107(n).

A364063 Expansion of Sum_{k>0} k * x^k / (1 - x^(2*k-1)).

Original entry on oeis.org

1, 3, 4, 5, 8, 7, 8, 14, 10, 11, 18, 13, 17, 22, 16, 17, 26, 26, 20, 30, 22, 23, 42, 25, 30, 38, 28, 38, 42, 31, 32, 55, 44, 35, 50, 37, 38, 65, 50, 41, 63, 43, 56, 62, 46, 58, 66, 62, 50, 81, 52, 53, 100, 55, 56, 78, 58, 74, 94, 74, 68, 86, 80, 65, 90, 67, 82, 124, 70, 71, 98, 86, 92, 117, 76, 77

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A000203, A007503, A064216, A336840.

Cf. A364066, A364085.

a[n_] := DivisorSum[2*n - 1, # + 1 &]/2; Array[a, 100] (* Amiram Eldar, Jul 04 2023*)

PARI
```
a(n) = sumdiv(2*n-1, d, d+1)/2;
```

a(n) = (1/2) * Sum_{d | 2*n-1} (d+1) = A007503(2*n-1)/2.
G.f.: Sum_{k>0} x^k / (1 - x^(2*k-1))^2.
a(n) = A336840(A064216(n)). - Antti Karttunen, Nov 30 2024

A243500 Self-inverse permutation of natural numbers: a(2n) = A003961(A048673(n)), a(2n-1) = 2 * A245448(n).

Original entry on oeis.org

2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 22, 27, 20, 15, 14, 33, 18, 17, 28, 13, 34, 11, 62, 29, 26, 25, 12, 19, 24, 75, 68, 43, 16, 21, 46, 69, 118, 45, 82, 243, 142, 99, 32, 63, 38, 35, 78, 171, 50, 49, 52, 51, 116, 275, 74, 147, 122, 81, 60, 59, 88, 23, 44, 201, 66, 65, 98, 31, 36

Offset: 1

Antti Karttunen, Jul 18 2014

nonn

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

Cf. A003961, A048673, A064989, A243502, A244319, A245448.

a(2n) = A003961(A048673(n)), a(2n-1) = 2 * A245448(n).
a(2n) = A003961(A048673(n)), a(2n-1) = A243502(A064989(2n-1)).
a(2n) = A003961((A003961(n)+1)/2), a(2n-1) = 2 * A064216(A064989(2n-1)).

A302848 Permutation of natural numbers: a(1) = 1; for n > 1, a(n) = 1+A064989(A003961(n)-2).

Original entry on oeis.org

1, 2, 3, 6, 4, 12, 5, 10, 20, 18, 8, 42, 7, 30, 15, 74, 14, 72, 11, 60, 48, 32, 9, 86, 44, 26, 75, 90, 24, 102, 16, 240, 21, 22, 19, 212, 23, 62, 80, 92, 38, 158, 13, 58, 168, 40, 27, 320, 66, 70, 59, 150, 35, 368, 84, 160, 110, 56, 54, 312, 34, 108, 111, 720, 45, 192, 39, 122, 78, 228, 68, 662, 36, 50, 33, 112, 87, 134, 17, 328, 416, 114, 47, 300, 128

Offset: 1

Antti Karttunen, Apr 26 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
David Newman, et al., New sequences from old, Discussion on SeqFan-mailing list, January 2018.
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences computed from indices in prime factorization

Cf. A302847 (inverse).

Cf. A003961, A048673, A064216, A064989, A302850.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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)};
A302848(n) = if(1==n,n,1+A064989(A003961(n)-2));

a(1) = 1; for n > 1, a(n) = 1+A064989(A003961(n)-2).

a(1) = 1; for n > 1, a(n) = 1+A064216(A048673(n)-1).

A305424 Permutation of natural numbers: a(n) = A305422(2*n-1).

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 11, 8, 16, 13, 5, 22, 19, 12, 14, 25, 50, 29, 31, 28, 37, 38, 24, 41, 9, 32, 26, 47, 44, 55, 59, 10, 20, 61, 21, 118, 67, 88, 110, 53, 69, 18, 64, 73, 94, 87, 43, 52, 91, 100, 58, 97, 56, 15, 103, 62, 82, 109, 115, 48, 23, 74, 76, 49, 98, 117, 113, 152, 131, 46, 148, 137, 143, 164, 218, 27, 96, 227, 145, 230, 89, 182, 200

Offset: 1

Antti Karttunen, Jun 08 2018

nonn

Odd bisection of A305422 and A305425.

Cf. A305423 (inverse).
Cf. A014580, A091225, A304522, A305425.
Cf. also A064216.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A305424(n) = A305422(n+n-1);

a(n) = A305422(2*n-1).

A349122 Inverse Möbius transform of A349128, where A349128(n) = phi(A064989(n)), A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and phi is Euler totient function.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 5, 4, 4, 6, 7, 6, 11, 10, 6, 5, 13, 8, 17, 9, 10, 14, 19, 8, 9, 22, 8, 15, 23, 12, 29, 6, 14, 26, 15, 12, 31, 34, 22, 12, 37, 20, 41, 21, 12, 38, 43, 10, 25, 18, 26, 33, 47, 16, 21, 20, 34, 46, 53, 18, 59, 58, 20, 7, 33, 28, 61, 39, 38, 30, 67, 16, 71, 62, 18, 51, 35, 44, 73, 15, 16, 74, 79, 30, 39

Offset: 1

Antti Karttunen, Nov 13 2021

Multiplicative because A349128 is.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000010, A003961, A064216, A064989, A151799, A285702, A349127, A349128, A349138.

f[p_, e_] := NextPrime[p, -1]^e; f[2, e_] := e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)

A349128(n) = { my(f = factor(n), q); prod(i=1, #f~, if(2==f[i,1], 1, q = precprime(f[i,1]-1); (q-1)*(q^(f[i,2]-1)))); };
A349122(n) = sumdiv(n,d,A349128(d));

from sympy import prevprime, factorint, prod
def f(p, e):
    return e+1 if p == 2 else prevprime(p)**e
def a(n):
    return prod(f(p, e) for p, e in factorint(n).items()) # Sebastian Karlsson, Nov 15 2021

a(n) = Sum_{d|n} A349128(d).
For all n >= 1, a(A003961(n)) = n, a(2*n-1) = A064216(n).
From Sebastian Karlsson, Nov 15 2021: (Start)
a(2*n-1) = A064989(2*n-1).
Multiplicative with a(2^e) = e + 1 and a(p^e) = prevprime(p)^e for odd primes p. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (4/9) * Product_{p prime > 2} ((p^2-p)/(p^2-prevprime(p))) = 0.2942719052..., where prevprime is A151799. - Amiram Eldar, Dec 24 2022

cp's OEIS Frontend

A254118 Permutation of natural numbers: a(n) = A249745(1+A254103(n)) - 1.

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A266407 Permutation of natural numbers: a(n) = A064989(A263273((2*n)-1)).

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A285334 a(n) = A046523(A243505(n)).

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A302847 Permutation of natural numbers: a(1) = 1; for n > 1, a(n) = A064989(2+A003961(n-1)).

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A353412 The odd part of hybrid shift: a(n) = A000265(A252463(n)).

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A364063 Expansion of Sum_{k>0} k * x^k / (1 - x^(2*k-1)).

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A243500 Self-inverse permutation of natural numbers: a(2n) = A003961(A048673(n)), a(2n-1) = 2 * A245448(n).

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A302848 Permutation of natural numbers: a(1) = 1; for n > 1, a(n) = 1+A064989(A003961(n)-2).

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A305424 Permutation of natural numbers: a(n) = A305422(2*n-1).

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