cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A332215 Mersenne-prime fixing variant of A243071: a(n) = A243071(A332213(n)).

Original entry on oeis.org

0, 1, 3, 2, 15, 6, 7, 4, 5, 30, 63, 12, 255, 14, 29, 8, 511, 10, 1023, 60, 13, 126, 2047, 24, 23, 510, 9, 28, 4095, 58, 31, 16, 125, 1022, 27, 20, 16383, 2046, 509, 120, 32767, 26, 65535, 252, 57, 4094, 262143, 48, 11, 46, 1021, 1020, 1048575, 18, 119, 56, 2045, 8190, 2097151, 116, 4194303, 62, 25, 32, 503, 250, 8388607, 2044, 4093, 54, 16777215, 40
Offset: 1

Views

Author

Antti Karttunen, Feb 09 2020

Keywords

Comments

Any Mersenne prime (A000668) times any power of 2 (i.e., 2^k, for k>=0) is fixed by this sequence, including also all even perfect numbers.
From Antti Karttunen, Jul 10 2020: (Start)
This is a "tuned variant" of A243071, and has many of the same properties.
For example, for n > 1, A007814(a(n)) = A007814(n) - A209229(n), that is, this map preserves the 2-adic valuation of n, except when n is a power of two, in which cases that value is decremented by one, and in particular, a(2^k * n) = 2^k * a(n) for all n > 1. Also, like A243071, this bijection maps primes to the terms of A000225 (binary repunits). However, the "tuning" (A332213) has a specific effect that each Mersenne prime (A000668) is mapped to itself. Therefore the terms of A335431 are fixed by this map. Furthermore, I conjecture that there are no other fixed points. For the starters, see the proof in A335879, which shows that at least none of the terms of A335882 are fixed.
(End)

Links

Crossrefs

Cf. A243071, A332210, A332213, A332214 (inverse permutation), A335431 (conjectured to be all the fixed points), A335879.
Cf. A000043, A000668, A000396, A324200.

Programs

Formula

a(n) = A243071(A332213(n)).
For all n >= 1, a(A335431(n)) = A335431(n), a(A335882(n)) = A335879(n). - Antti Karttunen, Jul 10 2020

A332212 Fully multiplicative with a(p) = A332211(A000720(p)).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 9, 14, 31, 12, 11, 10, 21, 16, 127, 18, 13, 28, 15, 62, 17, 24, 49, 22, 27, 20, 19, 42, 23, 32, 93, 254, 35, 36, 29, 26, 33, 56, 8191, 30, 37, 124, 63, 34, 41, 48, 25, 98, 381, 44, 43, 54, 217, 40, 39, 38, 131071, 84, 47, 46, 45, 64, 77, 186, 524287, 508, 51, 70, 53, 72, 59, 58, 147, 52, 155, 66, 61, 112, 81, 16382, 67, 60
Offset: 1

Views

Author

Antti Karttunen, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A000043, A000668, A000720, A332211, A332213 (inverse permutation), A332214.

Programs

  • PARI
    \\ Needs also code from A332211:
A332212(n) = { my(f=factor(n)); f[,1] = apply(A332211,apply(primepi,f[,1])); factorback(f); };

Formula

a(1) = 1, a(p^e) = A332211(A000720(p))^e, a(m*n) = a(m)*a(n).

A332210 Permutation of primes, inverse of A332211.

Original entry on oeis.org

2, 3, 7, 5, 13, 19, 23, 29, 31, 37, 11, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 17, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383
Offset: 1

Views

Author

Antti Karttunen, Feb 09 2020

Keywords

Comments

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

Links

Crossrefs

Cf. A000040, A000043, A000668, A000720, A059305, A332211, A332220.
Used to construct permutations A332213, A332215.

Programs

  • PARI
    up_to = 127;
A332210list(up_to) = { my(lista=List([]), xs=Map(), i=1, q, u); for(n=1,up_to, if(!isprime(q=((2^n)-1)), while(mapisdefined(xs,prime(i)), i++); q = prime(i)); mapput(xs,q,n)); for(i=1,oo,if(!mapisdefined(xs,prime(i),&u),return(Vec(lista)),listput(lista,prime(u)))); };
\\ For computing a larger number of terms, use the precomputed values of A000043:
v000043 = [2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279, 2203,2281,3217,4253,4423,9689,9941,11213,19937, 21701,23209,44497,86243,110503,132049,216091, 756839,859433,1257787,1398269,2976221,3021377, 6972593,13466917,20996011,24036583,25964951, 30402457,32582657,37156667,42643801,43112609];
A332210list(up_to) = { my(lista=List([]), xs=Map(), m000043 = Map(), i=1, q, u); for(k=1,#v000043,mapput(m000043,v000043[k],k)); for(n=1,min(up_to,v000043[#v000043]), if(mapisdefined(m000043,n), q = (2^n)-1, while(mapisdefined(xs,prime(i)), i++); q = prime(i)); mapput(xs,q,n)); for(i=1,oo,if(!mapisdefined(xs,prime(i),&u),return(Vec(lista)),listput(lista,prime(u)))); };
v332210 = A332210list(up_to);
A332210(n) = v332210[n];

Formula

For all applicable n >= 1, a(A059305(n)) = a(A000720(A000668(n))) = A000040(A000043(n)).
Showing 1-3 of 3 results.

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