A332819 -id:A332819 - cp's OEIS frontend

A332893 a(1) = 1, a(2n) = n, a(2n+1) = A332819(2n+1).

1, 1, 2, 2, 3, 3, 5, 4, 4, 5, 13, 6, 7, 7, 6, 8, 11, 9, 17, 10, 10, 11, 29, 12, 9, 13, 8, 14, 19, 15, 37, 16, 26, 17, 15, 18, 23, 19, 14, 20, 31, 21, 41, 22, 12, 23, 53, 24, 25, 25, 22, 26, 43, 27, 39, 28, 34, 29, 61, 30, 47, 31, 20, 32, 21, 33, 73, 34, 58, 35, 89, 36, 59, 37, 18, 38, 65, 39, 97, 40, 16, 41, 101, 42, 33, 43, 38, 44, 67, 45, 35

Offset: 1

Antti Karttunen, Mar 01 2020

nonn

For any node n >= 2 in binary trees like A332815, a(n) gives the parent node of n.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A332815, A332819, A332894.

Cf. also A252463.

A332819(n) = A108548(A064989(A332808(n)));
A332893(n) = if(1==n,n,if(!(n%2),n/2,A332819(n)));

a(1) = 1, after which a(n) = n/2 for even n, and a(n) = A332819(n) for odd n.

A332894 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A332819(2n+1)); also binary width of terms of A332816.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 6, 4, 5, 5, 4, 4, 7, 4, 8, 5, 5, 7, 10, 5, 4, 6, 4, 6, 9, 5, 12, 5, 7, 8, 5, 5, 11, 9, 6, 6, 13, 6, 14, 8, 5, 11, 16, 6, 5, 5, 8, 7, 15, 5, 7, 7, 9, 10, 18, 6, 17, 13, 6, 6, 6, 8, 20, 9, 11, 6, 22, 6, 19, 12, 5, 10, 7, 7, 24, 7, 5, 14, 26, 7, 8, 15, 10, 9, 21, 6, 6, 12, 13, 17, 9, 7, 23, 6, 8, 6, 25, 9, 28, 8, 6

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

a(n) tells how many iterations of A332893 are needed before 1 is reached, i.e., the distance of n from 1 in binary trees like A332815.

Each n > 0 occurs 2^(n-1) times in total.

Antti Karttunen, Table of n, a(n) for n = 1..65537

A left inverse of A000079 and A108546.

Cf. A070939, A252464, A332808, A332815, A332816, A332819, A332893, A332897, A332898, A332899.

A332894(n) = if(1==n,0,if(!(n%2),1+A332894(n/2),1+A332894(A332819(n))));

A332894(n) = if(1==n,0,1+A332894(A332893(n)));

a(n) = A252464(A332808(n)).
a(1) = 0, and for n > 1, a(n) = 1 + a(A332893(n)).
For n >= 1, a(A108546(n)) = n; for all n >= 0, a(2^n) = n.
For n > 1: (Start)
a(n) = 1 + a(n/2) if n is even, and a(n) = 1 + a(A332819(n)), if n is odd.
a(n) = A070939(A332816(n)).
a(n) >= A332899(n).
(End)

A108548 Fully multiplicative with a(prime(j)) = A108546(j), where A108546 is the lexicographically earliest permutation of primes such that after 2 the forms 4k+1 and 4k+3 alternate, and prime(j) is the j-th prime in A000040.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 19, 20, 21, 26, 29, 24, 25, 22, 27, 28, 23, 30, 37, 32, 39, 34, 35, 36, 31, 38, 33, 40, 41, 42, 43, 52, 45, 58, 53, 48, 49, 50, 51, 44, 47, 54, 65, 56, 57, 46, 61, 60, 59, 74, 63, 64, 55, 78, 73, 68, 87, 70, 67, 72

Offset: 1

Reinhard Zumkeller, Jun 10 2005

Multiplicative with a(2^e) = 2^e, else if p is the m-th prime then a(p^e) = q^e where q is the m/2-th prime of the form 4*k + 3 (A002145) for even m and a(p^e) = r^e where r is the (m-1)/2-th prime of the form 4*k + 1 (A002144) for odd m. - David A. Corneth, Apr 25 2022

Permutation of the natural numbers with fixed points A108549: a(A108549(n)) = A108549(n).

Antti Karttunen, Table of n, a(n) for n = 1..26927
Index entries for sequences that are permutations of the natural numbers

Cf. A002144, A002145, A049084, A108546, A108549 (fixed points), A332808 (inverse permutation).
Cf. also A332815, A332817 (this permutation applied to Doudna tree and its mirror image), also A332818, A332819.
Cf. also A267099, A332212 and A348746 for other similar mappings.

terms = 72;
A111745 = Module[{prs = Prime[Range[2 terms]], m3, m1, min},
     m3 = Select[prs, Mod[#, 4] == 3&];
     m1 = Select[prs, Mod[#, 4] == 1&];
     min = Min[Length[m1], Length[m3]];
     Riffle[Take[m3, min], Take[m1, min]]];
A108546[n_] := If[n == 1, 2, A111745[[n - 1]]];
A049084[n_] := PrimePi[n]*Boole[PrimeQ[n]];
a[n_] := If[n == 1, 1, Module[{p, e}, Product[{p, e} = pe; A108546[A049084[p]]^e, {pe, FactorInteger[n]}]]];
Array[a, terms] (* Jean-François Alcover, Nov 19 2021, using Harvey P. Dale's code for A111745 *)

up_to = 26927; \\ One of the prime fixed points.
A108546list(up_to) = { my(v=vector(up_to), p,q); v[1] = 2; v[2] = 3; v[3] = 5; for(n=4,up_to, p = v[n-2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n];
A108548(n) = { my(f=factor(n)); f[,1] = apply(A108546,apply(primepi,f[,1])); factorback(f); }; \\ Antti Karttunen, Apr 25 2022

Name edited by Antti Karttunen, Apr 25 2022

A332808 Fully multiplicative with a(p) = A332806(A000720(p)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 19, 20, 21, 26, 29, 24, 25, 22, 27, 28, 23, 30, 37, 32, 39, 34, 35, 36, 31, 38, 33, 40, 41, 42, 43, 52, 45, 58, 53, 48, 49, 50, 51, 44, 47, 54, 65, 56, 57, 46, 61, 60, 59, 74, 63, 64, 55, 78, 71, 68, 87, 70, 79, 72, 67, 62, 75, 76, 91, 66, 89, 80, 81, 82, 101

Offset: 1

Antti Karttunen, Feb 27 2020

Antti Karttunen, Table of n, a(n) for n = 1..26927

Cf. A000720, A332806, A108549 (fixed points), A332818, A332819.

Inverse permutation is A108548, from which this differs for the first time at n=67, where a(67) = 71, while A108548(67) = 73.

up_to = 10000;
A332806list(up_to) = { my(v=vector(2), xs=Map(), lista=List([]), p,q,u); v[2] = 3; v[1] = 5; mapput(xs,1,1); mapput(xs,2,2); mapput(xs,3,3);  for(n=4,up_to, p = v[2-(n%2)]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[2-(n%2)] = q; mapput(xs,primepi(q),n)); for(i=1, oo, if(!mapisdefined(xs, i, &u), return(Vec(lista)), listput(lista, prime(u)))); };
v332806 = A332806list(up_to);
A332806(n) = v332806[n];
A332808(n) = { my(f=factor(n)); f[,1] = apply(A332806,apply(primepi,f[,1])); factorback(f); };

A332818 a(n) = A108548(A003961(A332808(n))).

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 13, 27, 25, 21, 17, 45, 11, 39, 35, 81, 19, 75, 29, 63, 65, 51, 37, 135, 49, 33, 125, 117, 23, 105, 41, 243, 85, 57, 91, 225, 31, 87, 55, 189, 43, 195, 53, 153, 175, 111, 61, 405, 169, 147, 95, 99, 47, 375, 119, 351, 145, 69, 73, 315, 59, 123, 325, 729, 77, 255, 89, 171, 185, 273, 97, 675, 67, 93, 245, 261, 221, 165, 101

Offset: 1

Antti Karttunen, Feb 27 2020

Permutation of odd numbers. Preserves prime signature.

Antti Karttunen, Table of n, a(n) for n = 1..26921

Cf. A002144, A002145, A003961, A046523, A108548, A332808.

Cf. A332819 (a left inverse).

A332818(n) = A108548(A003961(A332808(n)));

Fully multiplicative with a(2) = 3, a(A002145(n)) = A002144(n) and a(A002144(n)) = A002145(1+n), for all n >= 1.
a(n) = A108548(A003961(A332808(n))).
A332819(a(n)) = n.
A046523(a(n)) = A046523(n).

A332897 a(1) = 0, a(2) = 1, and for n > 2, a(n) = a(A332893(n)) + [n == 1 (mod 4)].

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 3, 2, 1, 1, 4, 2, 4, 2, 3, 3, 5, 1, 3, 3, 1, 2, 5, 1, 6, 1, 4, 4, 1, 2, 6, 4, 2, 2, 7, 3, 7, 3, 2, 5, 8, 1, 4, 3, 3, 3, 8, 1, 2, 2, 5, 5, 9, 1, 9, 6, 2, 1, 4, 4, 10, 4, 6, 1, 11, 2, 10, 6, 2, 4, 5, 2, 12, 2, 2, 7, 13, 3, 5, 7, 4, 3, 11, 2, 1, 5, 7, 8, 3, 1, 12, 4, 3, 3, 13, 3, 14, 3, 2

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

Starting from x=n, iterate the map x -> A332893(x) which divides even numbers by 2, and for odd n changes every 4k+1 prime in their prime factorization to 4k+3 prime and vice versa (except 3 -> 2), like in A332819. a(n) counts the numbers of the form 4k+1 encountered until 1 has been reached, which is also included in the count when n > 1. This count includes also n itself when it is of the form 4k+1 (A016813) and larger than 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A016813, A332893, A332895, A332898, A332899.

Cf. also A292375.

A332897(n) = if(n<=2,n-1,A332897(A332893(n)) + (1==(n%4)));

a(1) = 0, a(2) = 1, and for n > 2, a(n) = a(A332893(n)) + [n == 1 (mod 4)].

a(n) = A000120(A332895(n)).

A332898 a(1) = 0, and for n > 1, a(n) = a(A332893(n)) + [n == 3 (mod 4)].

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 0, 1, 3, 1, 2, 2, 2, 0, 3, 0, 4, 1, 1, 3, 5, 1, 0, 2, 1, 2, 4, 2, 6, 0, 2, 3, 3, 0, 5, 4, 3, 1, 6, 1, 7, 3, 1, 5, 8, 1, 0, 0, 4, 2, 7, 1, 4, 2, 3, 4, 9, 2, 8, 6, 2, 0, 1, 2, 10, 3, 4, 3, 11, 0, 9, 5, 1, 4, 1, 3, 12, 1, 0, 6, 13, 1, 2, 7, 5, 3, 10, 1, 4, 5, 5, 8, 5, 1, 11, 0, 3, 0, 12, 4, 14, 2, 2

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

Starting from x=n, iterate the map x -> A332893(x) which divides even numbers by 2, and for odd n, changes every 4k+1 prime in the prime factorization to 4k+3 prime and vice versa (except 3 --> 2), like in A332819. a(n) counts the numbers of the form 4k+3 encountered until 1 has been reached. The count includes also n itself if it is of the form 4k+3 (A004767).

In other words, locate the node which contains n in binary tree A332815 and traverse from that node towards the root, counting all numbers of the form 4k+3 that occur on the path.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000120, A004767, A332815, A332893, A332896, A332897, A332899.
Cf. A028982 (positions of zeros).
Cf. also A292377.

A332898(n) = if(1==n,0,A332898(A332893(n)) + (3==(n%4)));

a(1) = 0, and for n > 1, a(n) = a(A332893(n)) + [n == 3 (mod 4)].

a(n) = A000120(A332896(n)).

A348747 Fully multiplicative with a(2) = 1, a(3) = 2, a(5) = 3, a(A002144(1+n)) = A002144(n) and a(A002145(1+n)) = a(A002145(1+n)) for all n >= 1, where A002144 and A002145 give the primes of the form 4k+1 and 4k+3 respectively.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 7, 1, 4, 3, 11, 2, 5, 7, 6, 1, 13, 4, 19, 3, 14, 11, 23, 2, 9, 5, 8, 7, 17, 6, 31, 1, 22, 13, 21, 4, 29, 19, 10, 3, 37, 14, 43, 11, 12, 23, 47, 2, 49, 9, 26, 5, 41, 8, 33, 7, 38, 17, 59, 6, 53, 31, 28, 1, 15, 22, 67, 13, 46, 21, 71, 4, 61, 29, 18, 19, 77, 10, 79, 3, 16, 37, 83, 14, 39, 43, 34, 11, 73

Offset: 1

Antti Karttunen, Nov 02 2021

Left inverse of A348746.
Cf. A000265, A000720, A002144, A002145, A348745.
Cf. also A064989, A332819 for similar maps.

A348747(n) = { my(f=factor(n)); for(k=1,#f~, if(f[k,1]<=3, f[k,1]--, if(5==f[k,1], f[k,1]=3, if(1==(f[k,1]%4), forstep(i=primepi(f[k,1])-1,0,-1,if(1==(prime(i)%4), f[k,1]=prime(i); break)))))); factorback(f); };

Fully multiplicative with a(p) = A348745(A000720(p)).
a(A348746(n)) = n.
a(2n) = a(A000265(n)) = a(n).

cp's OEIS Frontend

A332893 a(1) = 1, a(2n) = n, a(2n+1) = A332819(2n+1).

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A332894 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A332819(2n+1)); also binary width of terms of A332816.

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A108548 Fully multiplicative with a(prime(j)) = A108546(j), where A108546 is the lexicographically earliest permutation of primes such that after 2 the forms 4*k+1 and 4*k+3 alternate, and prime(j) is the j-th prime in A000040.

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A332808 Fully multiplicative with a(p) = A332806(A000720(p)).

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A332818 a(n) = A108548(A003961(A332808(n))).

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A332897 a(1) = 0, a(2) = 1, and for n > 2, a(n) = a(A332893(n)) + [n == 1 (mod 4)].

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A332898 a(1) = 0, and for n > 1, a(n) = a(A332893(n)) + [n == 3 (mod 4)].

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A348747 Fully multiplicative with a(2) = 1, a(3) = 2, a(5) = 3, a(A002144(1+n)) = A002144(n) and a(A002145(1+n)) = a(A002145(1+n)) for all n >= 1, where A002144 and A002145 give the primes of the form 4k+1 and 4k+3 respectively.

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A108548 Fully multiplicative with a(prime(j)) = A108546(j), where A108546 is the lexicographically earliest permutation of primes such that after 2 the forms 4k+1 and 4k+3 alternate, and prime(j) is the j-th prime in A000040.