A269848 -id:A269848 - cp's OEIS frontend

A269847 Permutation of natural numbers: a(1) = 1, for n > 1, if n is an odd prime, a(n) = A003961(a(A000720(n))), otherwise a(n) = 2*a(n-A000720(n)).

1, 2, 3, 4, 5, 6, 9, 8, 10, 12, 7, 18, 15, 16, 20, 24, 25, 14, 27, 36, 30, 32, 21, 40, 48, 50, 28, 54, 45, 72, 11, 60, 64, 42, 80, 96, 75, 100, 56, 108, 35, 90, 81, 144, 22, 120, 63, 128, 84, 160, 192, 150, 135, 200, 112, 216, 70, 180, 49, 162, 33, 288, 44, 240, 126, 256, 125, 168, 320, 384, 225, 300, 105, 270, 400

Offset: 1

Antti Karttunen, Mar 06 2016

nonn

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

Inverse: A269848.
Cf. A000040, A000720, A003961, A005843, A007097, A010051, A026233, A065090, A065091.
Related or similar permutations: A071574, A163511, A246681, A257730, A269857.

a(1) = 1, and for n > 1, if n is an odd prime, a(n) = A003961(a(A000720(n))), otherwise [when n is 2 or composite] a(n) = 2*a(n-A000720(n)).
a(1) = 1; if n is an odd prime, a(n) = A003961(a(A026233(n))), else a(n) = A005843(a(A026233(n))).
Declarative definition:
a(1)=1, a(A065091(n)) = A003961(a(n+1)), a(A065090(n+1)) = 2*a(n).
As a composition of other permutations:
a(n) = A163511(A071574(n)).
Other identities. For all n >= 1:
a(A007097(n)) = A000040(n). [Maps the terms of primeth recurrence to primes.]

A269858 Permutation of natural numbers: a(1) = 1, a(2n) = A065090(1+a(n)), a(2n+1) = A000040(a(A268674(2n+1))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 8, 7, 9, 31, 10, 127, 18, 13, 14, 709, 12, 5381, 15, 19, 45, 52711, 16, 17, 165, 23, 27, 648391, 21, 9737333, 22, 29, 856, 41, 20, 174440041, 6185, 61, 24, 3657500101, 28, 88362852307, 63, 43, 58644, 2428095424619, 25, 59, 26, 37, 212, 75063692618249, 34, 67, 39, 47

Offset: 1

Antti Karttunen, Mar 06 2016

nonn

Terms for prime positions copied from A007097.

Index entries for sequences that are permutations of the natural numbers

Inverse: A269857.
Cf. A000040, A007097, A065090, A268674.
Related or similar permutations: A237739, A252756, A269848.

a(1) = 1, a(2) = 2, for n > 2, if n is even, a(n) = A002808(a(n/2)-1), and for odd n, a(n) = A000040(a(A268674(n))).
As a composition of other permutations:
a(n) = A237739(A252756(n)).
Other identities. For all n >= 1:
a(A000040(n)) = A007097(n). [Maps primes to the primeth recurrence.]

A371985 For n a power of 2, a(n) = n. Otherwise a(n) is the smallest novel multiple of a(n - 2^m), where 2^m is the greatest power of 2 not exceeding n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 12, 20, 15, 18, 27, 16, 11, 14, 21, 24, 25, 30, 36, 40, 28, 50, 48, 60, 45, 54, 81, 32, 13, 22, 33, 44, 35, 42, 63, 56, 49, 70, 72, 80, 75, 90, 108, 96, 55, 84, 105, 120, 100, 150, 144, 160, 112, 200, 192, 180, 135, 162, 243, 64, 17

Offset: 1

David James Sycamore, Apr 15 2024

nonn

Reminiscent of the Doudna sequence A005940; also of A052330 and A269848.
All powers of 2 (a(n) = n) are assigned first in order to avoid the second part of the definition giving a(n) = 2^k for some n which is not a power of 2 (see Example for a(12) = 20).
It follows from the definition that all powers of 2, all primes and all multiples of all primes are terms so this sequence is a permutation of the positive integers (A000027), with primes in order.
Each prime power appears before any of its multiples, meaning that this sequence has "property S" as defined in A368900.

			a(3) = 3, because 2 is the greatest power of 2 not exceeding 3 and 3-2 = 1, so a(3) = 3, the least novel multiple of a(1) = 1.
a(12) is the smallest novel multiple of a(12-8) = a(4) = 4, and at this point in the sequence 4,8,12 are all prior terms and a(16) = 16 is already taken, so a(12) = 20.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
David A. Corneth, PARI program
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Fan style binary tree showing a(n), n = 1..8192, with a color function showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue or purple. Purple represents powerful numbers that are not prime powers.
Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A005940, A052330, A269848, A368900.

nn = 10; c[] := False; m[] := 1; a[1] = 1; c[1] = True;
Do[If[i == 0,
   k = 2^j + i,
   (While[Set[k, m[#] #]; Or[c[k], IntegerQ@ Log2[k]], m[#]++]) &@ a[i]];
  Set[{a[2^j + i], c[k]}, {k, True}], {j, nn}, {i, 0, 2^j - 1}];
Array[a, 2^(nn + 1) - 1] (* Michael De Vlieger, Apr 15 2024 *)

PARI
```
\\ See PARI link
```

a(2^k + 1) = prime(k+1).

cp's OEIS Frontend

A269847 Permutation of natural numbers: a(1) = 1, for n > 1, if n is an odd prime, a(n) = A003961(a(A000720(n))), otherwise a(n) = 2*a(n-A000720(n)).

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A269858 Permutation of natural numbers: a(1) = 1, a(2n) = A065090(1+a(n)), a(2n+1) = A000040(a(A268674(2n+1))).

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A371985 For n a power of 2, a(n) = n. Otherwise a(n) is the smallest novel multiple of a(n - 2^m), where 2^m is the greatest power of 2 not exceeding n.

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