A332815 -id:A332815 - cp's OEIS frontend

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.

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

A332817 a(n) = A108548(A163511(n)).

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 6, 5, 16, 27, 18, 25, 12, 15, 10, 7, 32, 81, 54, 125, 36, 75, 50, 49, 24, 45, 30, 35, 20, 21, 14, 13, 64, 243, 162, 625, 108, 375, 250, 343, 72, 225, 150, 245, 100, 147, 98, 169, 48, 135, 90, 175, 60, 105, 70, 91, 40, 63, 42, 65, 28, 39, 26, 11, 128, 729, 486, 3125, 324, 1875, 1250, 2401, 216, 1125, 750, 1715, 500

Offset: 0

Antti Karttunen, Mar 05 2020

This irregular table can be represented as a binary tree. Each child to the left is obtained by doubling the parent, and each child to the right is obtained by applying A332818 to the parent:
                                       1
                                       |
                    ...................2...................
                   4                                       3
         8......../ \........9                   6......../ \........5
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    16       27         18       25         12       15         10       7
  32 81    54  125    36  75   50  49     24  45   30  35     20  21   14 13
etc.
This is the mirror image of the tree in A332815.

Cf. A332811 (inverse permutation).
Cf. A054429, A108548, A163511, A332815 (mirror image).
Cf. A108546 (the right edge of the tree from 2 downward).
Cf. also A332214.

up_to = 26927;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A054429(n) = ((3<<#binary(n\2))-n-1); \\ From A054429
A163511(n) = if(!n,1,A005940(1+A054429(n)));
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]; \\ Antti Karttunen, Mar 05 2020
A108548(n) = { my(f=factor(n)); f[,1] = apply(A108546,apply(primepi,f[,1])); factorback(f); };
A332817(n) = A108548(A163511(n));

a(n) = A108548(A163511(n)).

For n >= 1, a(n) = A332815(A054429(n)).

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

Original entry on oeis.org

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.

A332896 a(1) = 0, and for n > 1, a(n) = 2*a(A332893(n)) + [n == 3 (mod 4)].

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 5, 0, 0, 4, 21, 4, 10, 10, 5, 0, 42, 0, 85, 8, 8, 42, 341, 8, 0, 20, 1, 20, 170, 10, 1365, 0, 40, 84, 11, 0, 682, 170, 21, 16, 2730, 16, 5461, 84, 8, 682, 21845, 16, 0, 0, 85, 40, 10922, 2, 43, 40, 168, 340, 87381, 20, 43690, 2730, 17, 0, 16, 80, 349525, 168, 680, 22, 1398101, 0, 174762, 1364, 1, 340, 32, 42, 5592405, 32, 0, 5460

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

Base-2 expansion of a(n) encodes the steps where numbers of the form 4k+3 are encountered when map x -> A332893(x) is iterated down to 1, starting from x=n. See the binary tree illustrated in A332815.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000975, A108546, A332815, A332893, A332895,

Cf. also A292383.

A332896(n) = if(1==n,n-1,2*A332896(A332893(n)) + (3==(n%4)));

a(1) = 0, and for n > 1, a(n) = 2*a(A332893(n)) + [n == 3 (mod 4)].
Other identities. For n >= 1:
a(2n) = 2*a(n).
a(A108546(n)) = A000975(n-1).

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)).

A332899 a(1) = 0, and for n > 2, a(n) = a(A332893(n)) + A000035(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

a(n) tells how many odd numbers are encountered when map x -> A332893(x) is used to traverse from n to 1, the root of the binary tree A332815. This count includes both the starting n itself if it is odd, but excludes 1 where the iteration ends.

a(n) also gives the index of the largest prime factor (A061395) in A332808(n), which is the inverse permutation of A108548 (see also A108546).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000035, A061395, A080791, A332808, A332811, A332815, A332816, A332893, A332894, A332897, A332898.

Cf. A000079 (after its initial term, gives the positions of 1's).

A332899(n) = if(1==n,0,A332899(A332893(n)) + (n%2));

a(1) = 0, and for n > 1, a(n) = a(A332893(n)) + A000035(n).
a(n) = A000120(A332811(n)).
a(n) = A061395(A332808(n)).
a(n) = A332897(n) + A332898(n).
a(n) <= A332894(n).
For all n > 1, a(n) = 1 + A080791(A332816(n)).

A332816 a(n) = A156552(A332808(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 32, 11, 16, 17, 10, 15, 64, 13, 128, 19, 18, 65, 512, 23, 12, 33, 14, 35, 256, 21, 2048, 31, 66, 129, 20, 27, 1024, 257, 34, 39, 4096, 37, 8192, 131, 22, 1025, 32768, 47, 24, 25, 130, 67, 16384, 29, 68, 71, 258, 513, 131072, 43, 65536, 4097, 38, 63, 36, 133, 524288, 259, 1026, 41, 2097152, 55, 262144, 2049, 26, 515

Offset: 1

Antti Karttunen, Feb 28 2020

nonn

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

Cf. A332815 (inverse permutation).

Cf. A070939, A156552, A332808, A332894, A332899.

up_to = 26927;
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
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); };
A332816(n) = A156552(A332808(n));

a(n) = A156552(A332808(n)).
For n > 1, A070939(a(n)) = A332894(n).
For n >= 1: (Start)
A080791(a(n)) = A332899(n)-1.
Among many identities given in A156552 that apply here as well we have for example the following ones:
A000120(a(n)) = A001222(n).
A069010(a(n)) = A001221(n).
A106737(a(n)) = A000005(n).
(End)

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)

A332895 a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A332893(n)) + [n == 1 (mod 4)].

Original entry on oeis.org

0, 1, 2, 2, 5, 4, 10, 4, 5, 10, 42, 8, 21, 20, 8, 8, 85, 10, 170, 20, 21, 84, 682, 16, 11, 42, 8, 40, 341, 16, 2730, 16, 85, 170, 16, 20, 1365, 340, 40, 40, 5461, 42, 10922, 168, 17, 1364, 43690, 32, 23, 22, 168, 84, 21845, 16, 80, 80, 341, 682, 174762, 32, 87381, 5460, 40, 32, 43, 170, 699050, 340, 1365, 32, 2796202

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000975, A108546, A332815, A332893, A332896, A332897.

Cf. also A292385.

A332895(n) = if(n<=2,n-1,2*A332895(A332893(n)) + (1==(n%4)));

a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A332893(n)) + [n == 1 (mod 4)].
For n > 1, a(2n) = 2*a(n).
For n >= 1, a(A108546(n)) = A000975(n); A000120(a(n)) = A332897(n).

A332901 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A332896(i)) = A278222(A332896(j)) for all i, j.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 1, 2, 4, 2, 3, 3, 3, 1, 4, 1, 5, 2, 2, 4, 6, 2, 1, 3, 2, 3, 5, 3, 7, 1, 3, 4, 8, 1, 6, 5, 4, 2, 7, 2, 9, 4, 2, 6, 10, 2, 1, 1, 5, 3, 9, 2, 11, 3, 4, 5, 12, 3, 10, 7, 3, 1, 2, 3, 13, 4, 5, 8, 14, 1, 12, 6, 2, 5, 2, 4, 15, 2, 1, 7, 16, 2, 3, 9, 6, 4, 13, 2, 17, 6, 6, 10, 18, 2, 14, 1, 4, 1, 15, 5, 19, 3, 3

Offset: 1

Antti Karttunen, Mar 04 2020

nonn

Restricted growth sequence transform of A278222(A332896(n)).
This is a variant of A292583: Instead of runs of numbers of the form 4k+3 encountered on trajectories of the standard Doudna-tree (A005940), this relates to the corresponding trajectories in A332815-tree. See comments in A292583.
For all i, j:
  a(i) = a(j) => A053866(i) = A053866(j),
  a(i) = a(j) => A332898(i) = A332898(j).

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

Cf. A053866, A278222, A332896, A332898, A332900.
Cf. A028982 (positions of ones).
Cf. also A292583.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
v332901 = rgs_transform(vector(up_to,n,A278222(A332896(n))));
A332901(n) = v332901[n];

cp's OEIS Frontend

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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A332817 a(n) = A108548(A163511(n)).

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

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A332896 a(1) = 0, and for n > 1, a(n) = 2*a(A332893(n)) + [n == 3 (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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A332899 a(1) = 0, and for n > 2, a(n) = a(A332893(n)) + A000035(n).

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A332816 a(n) = A156552(A332808(n)).

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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 4k+1 and 4k+3 alternate, and prime(j) is the j-th prime in A000040.