A332212 -id:A332212 - cp's OEIS frontend

A332214 Mersenne-prime fixing variant of permutation A163511: a(n) = A332212(A163511(n)).

1, 2, 4, 3, 8, 9, 6, 7, 16, 27, 18, 49, 12, 21, 14, 5, 32, 81, 54, 343, 36, 147, 98, 25, 24, 63, 42, 35, 28, 15, 10, 31, 64, 243, 162, 2401, 108, 1029, 686, 125, 72, 441, 294, 175, 196, 75, 50, 961, 48, 189, 126, 245, 84, 105, 70, 155, 56, 45, 30, 217, 20, 93, 62, 11, 128, 729, 486, 16807, 324, 7203, 4802, 625, 216, 3087, 2058, 875

Offset: 0

Antti Karttunen, Feb 09 2020

Any Mersenne prime (A000668) times any power of 2, i.e., sequence A335431, is fixed by this map (note the indexing), including also all even perfect numbers. It is not currently known whether there are any additional fixed points.

Because a(n) has the same prime signature as A163511(n), it implies that applying A046523 and A052409 to this sequence gives the same results as with A163511, namely, sequences A278531 and A365805. - Antti Karttunen, Oct 09 2023

Cf. A163511, A332211, A332212, A332215 (inverse permutation).
Cf. A278531 [= A046523(a(n))], A290251 [= A001222(a(n))], A365805 [= A052409(a(n))], A366372 [= a(n)-n], A366373 [= gcd(n,a(n))], A366374 (numerator of n/a(n)), A366375 (denominator of n/a(n)), A366376.
Cf. A000043, A000668, A000396, A324200, A335431 (conjectured to give all the fixed points).

PARI
```
A332214(n) = A332212(A163511(n));
```

\\ Needs precomputed data for A332211:
v332211 = readvec("b332211_to.txt"); \\ Prepared with gawk ' { print $2 } ' < b332211.txt > b332211_to.txt
A332211(n) = v332211[n];
A332214(n) = if(!n, 1, my(i=1, p=A332211(i), t=1); while(n>1, if(!(n%2), (t*=p), i++; p=A332211(i)); n >>= 1); (t*p)); \\ Antti Karttunen, Oct 09 2023

a(n) = A332212(A163511(n)).

A364259 a(n) = A331410(A332212(n)).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 2, 2, 2, 0, 1, 2, 2, 1, 3, 1, 3, 1, 2, 2, 3, 2, 3, 2, 2, 0, 2, 1, 3, 2, 4, 2, 3, 1, 1, 3, 4, 1, 3, 3, 3, 1, 4, 2, 2, 2, 3, 3, 2, 2, 3, 3, 1, 2, 2, 2, 4, 0, 3, 2, 1, 1, 4, 3, 4, 2, 4, 4, 3, 2, 3, 3, 2, 1, 4, 1, 4, 3, 2, 4, 4, 1, 3, 3, 4, 3, 3, 3, 3, 1, 5, 4, 3, 2, 3, 2, 3, 2, 4

Offset: 1

Antti Karttunen, Jul 16 2023

nonn

Fully additive because A332212 is fully multiplicative and A331410 is fully additive.

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

Cf. A000079 (positions of 0's), A331410, A332212, A364260.

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

A332211 Lexicographically earliest permutation of primes such that a(n) = 2^n - 1 when n is one of the Mersenne prime exponents (in A000043).

Original entry on oeis.org

2, 3, 7, 5, 31, 11, 127, 13, 17, 19, 23, 29, 8191, 37, 41, 43, 131071, 47, 524287, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 2147483647, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 2305843009213693951, 269, 271, 277, 281, 283, 293, 307, 311, 313

Offset: 1

Antti Karttunen, Feb 09 2020

nonn

Sequence is well-defined also in case there are only a finite number of Mersenne primes.

			For p in A000043: 2, 3, 5, 7, 13, 17, 19, ..., a(p) = (2^p)-1, thus a(2) = 2^2 - 1 = 3, a(3) = 7, a(5) = 31, a(7) = 127, a(13) = 8191, a(17) = 131071, etc., with the rest of positions filled by the least unused prime:
1, 2, 3, 4,  5,  6,   7,  8,  9, 10, 11, 12,   13, 14, 15, 16, 17, ...
2, 3, 7, 5, 31, 11, 127, 13, 17, 19, 23, 29, 8191, 37, 41, 43, 131071, ...

Antti Karttunen, Table of n, a(n) for n = 1..3217

Cf. A000040, A000043, A000668, A332210 (inverse permutation of primes), A332220.

Used to construct permutations A332212, A332214.

up_to = 127;
A332211list(up_to) = { my(v=vector(up_to), xs=Map(), i=1, q); for(n=1,up_to, if(isprime(q=((2^n)-1)), v[n] = q, while(mapisdefined(xs,prime(i)), i++); v[n] = prime(i)); mapput(xs,v[n],n)); (v); };
v332211 = A332211list(up_to);
A332211(n) = v332211[n];
\\ For faster computing of larger values, use precomputed values of A000043:
v000043 = [2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217];
up_to = v000043[#v000043];
A332211list(up_to) = { my(v=vector(up_to), xs=Map(), i=1, q); for(n=1,up_to, if(vecsearch(v000043,n), q = (2^n)-1, while(mapisdefined(xs,prime(i)), i++); q = prime(i)); v[n] = q; mapput(xs,q,n)); (v); };

For all applicable n >= 1, a(A000043(n)) = A000668(n).

A332213 Fully multiplicative with a(p) = A332210(A000720(p)).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 9, 14, 13, 12, 19, 10, 21, 16, 23, 18, 29, 28, 15, 26, 31, 24, 49, 38, 27, 20, 37, 42, 11, 32, 39, 46, 35, 36, 43, 58, 57, 56, 47, 30, 53, 52, 63, 62, 61, 48, 25, 98, 69, 76, 71, 54, 91, 40, 87, 74, 73, 84, 79, 22, 45, 64, 133, 78, 83, 92, 93, 70, 89, 72, 97, 86, 147, 116, 65, 114, 101, 112, 81, 94, 103, 60, 161, 106, 111

Offset: 1

Antti Karttunen, Feb 09 2020

Antti Karttunen, Table of n, a(n) for n = 1..86238
Index entries for sequences that are permutations of the natural numbers

Cf. A000043, A000668, A000720, A332210, A332212 (inverse permutation), A332215.

\\ Needs also code from A332210:
A332213(n) = { my(f=factor(n)); f[,1] = apply(A332210,apply(primepi,f[,1])); factorback(f); };

a(1) = 1, a(p^e) = A332210(A000720(p))^e, a(m*n) = a(m)*a(n).

cp's OEIS Frontend

A332214 Mersenne-prime fixing variant of permutation A163511: a(n) = A332212(A163511(n)).

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A364259 a(n) = A331410(A332212(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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A332211 Lexicographically earliest permutation of primes such that a(n) = 2^n - 1 when n is one of the Mersenne prime exponents (in A000043).

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A332213 Fully multiplicative with a(p) = A332210(A000720(p)).

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Formula

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.