A332808 -id:A332808 - cp's OEIS frontend

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

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).

A332819 a(n) = A108548(A064989(A332808(n))).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 27 2020

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

A left inverse of A332818.

Cf. A002144, A002145, A064989, A108548, A332808.

A332819(n) = A108548(A064989(A332808(n)));

Fully multiplicative with a(2) = 1, a(3) = 2, a(A002144(n)) = A002145(n), and a(A002145(1+n)) = A002144(n) for all n >= 1.
a(n) = A108548(A064989(A332808(n))).
a(A332818(n)) = 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)

A332811 a(n) = A243071(A332808(n)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 63, 12, 31, 30, 13, 8, 127, 10, 255, 28, 29, 126, 1023, 24, 11, 62, 9, 60, 511, 26, 4095, 16, 125, 254, 27, 20, 2047, 510, 61, 56, 8191, 58, 16383, 252, 25, 2046, 65535, 48, 23, 22, 253, 124, 32767, 18, 123, 120, 509, 1022, 262143, 52, 131071, 8190, 57, 32, 59, 250, 1048575, 508, 2045, 54, 4194303, 40, 524287, 4094, 21

Offset: 1

Antti Karttunen, Mar 05 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Cf. A332817 (inverse permutation).
Cf. A054429, A243071, A332808, A332816, A332895, A332896, A332899.
Cf. also A332215.

up_to = 26927;
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)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
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); };
A332811(n) = A243071(A332808(n));

a(n) = A243071(A332808(n)).
For n > 1, a(n) = A054429(A332816(n)).
a(n) = A332895(n) + A332896(n).
a(n) = A332895(n) OR A332896(n) = A332895(n) XOR A332896(n).
A000120(a(n)) = A332899(n).

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

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.

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)).

A332806 Permutation of primes, inverse of A108546.

Original entry on oeis.org

2, 3, 5, 7, 13, 11, 17, 19, 29, 23, 37, 31, 41, 43, 53, 47, 61, 59, 71, 79, 67, 89, 101, 73, 83, 97, 107, 113, 103, 109, 131, 139, 127, 151, 137, 163, 149, 173, 181, 157, 193, 167, 199, 179, 191, 223, 229, 239, 251, 197, 211, 263, 227, 271, 233, 281, 241, 293, 257, 269, 311, 277, 317, 337, 283, 307, 349, 313, 359, 331, 347, 373, 383, 353

Offset: 1

Antti Karttunen, Feb 27 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..19962

Cf. A000040, A108547, A332805, A332808.

Differs from its inverse A108546 for the first time at n=19, where a(19) = 71, while A108546(19) = 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];

a(n) = A000040(A332805(n)).

a(A000720(A108547(n))) = A108547(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)

cp's OEIS Frontend

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

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A332819 a(n) = A108548(A064989(A332808(n))).

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

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A332811 a(n) = A243071(A332808(n)).

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

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

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A332806 Permutation of primes, inverse of A108546.

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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.