A322990 -id:A322990 - cp's OEIS frontend

A322988 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(1) = 0 if n is a prime power > 2, f(2) = -1, and f(n) = A322990(n) for all other numbers.

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

Offset: 1

Antti Karttunen, Jan 02 2019

nonn

For all i, j: a(i) = a(j) => A322989(i) = A322989(j).

For all i, j > 2: A305976(i) = A305976(j) => a(i) = a(j).

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

Cf. A289271, A289272, A305976, A322989, A322990.

Cf. A322805, A322822 for analogous constructions for filter sequences.

up_to = 8192;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A289271(n) = { my(v=0,i=0,x=1); for(d=2,oo,if(n==1, return(v)); if(1==gcd(x,d)&&1==omega(d), if(!(n%d)&&1==gcd(d,n/d), v += 2^i; n /= d; x *= d); i++)); }; \\ After Rémy Sigrist's program for A289271.
A289272(n) = { my(m=1, pp=1); while(n>0, pp++; while(!isprimepower(pp)||(gcd(pp,m)>1), pp++); if(n%2, m *= pp); n >>=1); (m); }; \\ Antti Karttunen, Jan 02 2019
A322990(n) = A289272(A289271(n)>>1);
A322988aux(n) = if(2==n,-1,if(isprimepower(n),0,A322990(n)));
v322988 = rgs_transform(vector(up_to,n,A322988aux(n)));
A322988(n) = v322988[n];

A322989 If n is a power of a prime, then a(n) = 0, otherwise a(n) = 1 + a(A322990(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 2, 0, 0, 1, 0, 3, 2, 1, 0, 2, 0, 1, 0, 3, 0, 2, 0, 0, 2, 1, 4, 3, 0, 1, 2, 4, 0, 2, 0, 3, 4, 1, 0, 2, 0, 1, 2, 3, 0, 1, 4, 5, 2, 1, 0, 3, 0, 1, 5, 0, 4, 2, 0, 3, 2, 3, 0, 6, 0, 1, 2, 3, 5, 2, 0, 4, 0, 1, 0, 3, 4, 1, 2, 6, 0, 3, 5, 3, 2, 1, 4, 2, 0, 1, 7, 3, 0, 2, 0, 6, 4

Offset: 1

Antti Karttunen, Jan 01 2019

nonn

For n > 1, a(n) gives the number of edges needed from n to the leftmost branch (where the terms of A000961 are located) in the binary tree illustrated in A289272.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000961, A001221, A010055, A289272, A322990.

Cf. also A322823.

A289271(n) = { my(v=0,i=0,x=1); for(d=2,oo,if(n==1, return(v)); if(1==gcd(x,d)&&1==omega(d), if(!(n%d)&&1==gcd(d,n/d), v += 2^i; n /= d; x *= d); i++)); }; \\ After Rémy Sigrist's program for A289271.
A289272(n) = { my(m=1, pp=1); while(n>0, pp++; while(!isprimepower(pp)||(gcd(pp,m)>1), pp++); if(n%2, m *= pp); n >>=1); (m); };
A322989(n) = if((1==n)||isprimepower(n),0,1+A322989(A322990(n)));
A322990(n) = A289272(A289271(n)>>1);

If A001221(n) <= 1 [when n is in A000961], then a(n) = 0, otherwise a(n) = 1 + a(A322990(n)).

A289271 A bijective binary representation of the prime factorization of a number, shown in decimal (see Comments for precise definition).

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 16, 32, 64, 5, 128, 6, 256, 9, 10, 512, 1024, 17, 2048, 12, 18, 33, 4096, 34, 8192, 65, 16384, 20, 32768, 7, 65536, 131072, 66, 129, 24, 36, 262144, 257, 130, 40, 524288, 11, 1048576, 68, 72, 513, 2097152, 258, 4194304, 1025, 514, 132

Offset: 1

Rémy Sigrist, Jun 30 2017

For n > 0, with prime factorization Product_{i=1..k} p_i ^ e_i (all p_i distinct and all e_i > 0):
- let S_n = A000961 \ { p_i ^ (e_i + j) with i=1..k and j > 0 },
- a(n) = Sum_{i=1..k} 2^#{ s in S_n with 1 < s < p_i ^ e_i }.
In an informal way, we encode the prime powers > 1 that are unitary divisors of n as 1's in binary, while discarding the 0's corresponding to their "proper" multiples.
a(A002110(n)) = 2^n-1 for any n >= 0.
a(A000961(n+1)) = 2^(n-1) for any n > 0.
A000120(a(n)) = A001221(n) for any n > 0 (each prime divisor p of n (alongside the p-adic valuation of n) is encoded as a single 1 bit in the base-2 representation of a(n)).
A000961(2+A007814(a(n))) = A034684(n) for any n > 1 (the least significant bit of a(n) encodes the smallest unitary divisor of n that is larger than 1).
This sequence establishes a bijection between the positive numbers and the nonnegative numbers; see A289272 for the inverse of this sequence.
The numbers 4, 36, 40 and 532 equal their image; are there other such numbers?
This sequence has connections with A034729 (which encodes the divisors of a number, and is not surjective) and A087207 (which encodes the prime divisors of a number, and is not injective).

			For n = 204 = 2^2 * 3 * 17:
- S_204 = A000961 \ { 2^3, 2^4, ..., 3^2, ... }
        = { 1, 2, 3, 4, 5, 7, 11, 13, 17, ... },
- a(204) = 2^#{ 2, 3 } + 2^#{ 2 } + 2^#{ 2, 3, 4, 5, 7, 11, 13 }
         = 2^2 + 2^1 + 2^7
         = 134.
See also the illustration of the first terms in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms
Rémy Sigrist, PARI program for A289271
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A000961, A001221, A002110, A007814, A034684, A034729, A087207, A289272 (inverse), A322988, A322990, A322991, A322992, A322995.

Cf. also A156552, A052331 for similar constructions.

PARI
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See Links section.
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A289271(n) = { my(f = factor(n), pps = vecsort(vector(#f~, i, f[i, 1]^f[i, 2])), s=0, x=1, pp=1, k=-1); for(i=1,#f~, while(pp < pps[i], pp++; while(!isprimepower(pp)||(gcd(pp,x)>1), pp++); k++); s += 2^k; x *= pp); (s); }; \\ Antti Karttunen, Jan 01 2019

A289272 Inverse to A289271.

Original entry on oeis.org

1, 2, 3, 6, 4, 10, 12, 30, 5, 14, 15, 42, 20, 70, 60, 210, 7, 18, 21, 66, 28, 90, 84, 330, 35, 126, 105, 462, 140, 630, 420, 2310, 8, 22, 24, 78, 36, 110, 132, 390, 40, 154, 120, 546, 180, 770, 660, 2730, 56, 198, 168, 858, 252, 990, 924, 4290, 280, 1386, 840

Offset: 0

Rémy Sigrist, Jun 30 2017

a(2^n-1) = A002110(n) for any n >= 0.
a(2^(n-1)) = A000961(n+1) for any n > 0.
A001221(a(n)) = A000120(n) for any n >= 0.
From Antti Karttunen, Jan 01 2019: (Start)
A034684(a(n)) = A000961(1+A001511(n)) for any n >= 1. (See also Rémy Sigrist's comment in A289271).
This sequence can be regarded also as an irregular triangle with rows of lengths 1, 1, 2, 4, 8, 16, ..., that is, it can be represented as a binary tree, where each left hand child contains A322991(k), and each right hand child contains A322992(k), when their parent contains k:
                                     1
                                     |
                  ...................2...................
                 3                                       6
       4......../ \........10                 12......../ \........30
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   5       14         15       42         20       70         60       210
  7 18   21  66     28  90   84  330    35  126 105  462   140  630  420 2310
etc.
The leftmost edge is A000961, the next lefmost is A278568 (after 2: 6, 10, 14, 18, ...), the righmost edge is A002110, the next rightmost A088860 but with 3 instead of 4.
Compare also to trees like A005940 (A163511) and A052330.
(End)

			A289271(1) = 0, hence a(0) = 1.
A289271(2) = 1, hence a(1) = 2.
A289271(3) = 2, hence a(2) = 3.
A289271(4) = 4, hence a(4) = 4.
A289271(5) = 8, hence a(8) = 5.
A289271(6) = 3, hence a(3) = 6.
A289271(7) = 16, hence a(16) = 7.
A289271(8) = 32, hence a(32) = 8.
A289271(9) = 64, hence a(64) = 9.
A289271(10) = 5, hence a(5) = 10.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A289272
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A000961, A001221, A002110, A034684, A088860, A278568, A289271 (inverse), A322989, A322990, A322991, A322992.

Cf. also A005940, A163511, A052330.

PARI
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See Links section.
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A289272(n) = { my(m=1, pp=1); while(n>0, pp++; while(!isprimepower(pp)||(gcd(pp,m)>1), pp++); if(n%2, m *= pp); n >>=1); (m); }; \\ Antti Karttunen, Jan 01 2019

A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 6, 7, 3, 8, 3, 9, 9, 10, 3, 11, 3, 12, 12, 13, 3, 14, 14, 15, 15, 16, 3, 17, 3, 18, 18, 19, 19, 20, 3, 21, 21, 22, 3, 23, 3, 24, 24, 25, 3, 26, 26, 27, 27, 28, 3, 29, 29, 30, 30, 31, 3, 32, 3, 33, 33, 34, 34, 35, 3, 36, 36, 37, 3, 38, 3, 39, 39, 40, 40, 41, 3, 42, 42, 43, 3, 44, 44, 45, 45, 46, 3, 47, 47, 48, 48, 49, 49, 50, 3, 51, 51, 52, 3, 53, 3, 54, 54

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

This sequence is a restricted growth sequence transform of a function f which is defined as f(n) = A004526(n), unless n is an odd prime, in which case f(n) = -1, which is a constant not in range of A004526. See the Crossrefs section for a list of similar sequences.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A039636(i) = A039636(j).
For all i, j: a(i) = a(j) <=> A323161(i+1) = A323161(j+1).
The shifted version of this filter, A323161, has a remarkable ability to find many sequences related to primes and prime chains. - Antti Karttunen, Jan 06 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A004526, A039636, A305801, A323161.
A list of few similarly constructed sequences follows, where each sequence is an rgs-transform of such function f, for which the value of f(n) is the n-th term of the sequence whose A-number follows after a parenthesis, unless n is of the form ..., in which case f(n) is given a constant value outside of the range of that sequence:
  A322809 (A004526, unless an odd prime) [This sequence],
  A322589 (A007913, unless an odd prime),
  A322591 (A007947, unless an odd prime),
  A322805 (A252463, unless an odd prime),
  A323082 (A300840, unless an odd prime),
  A322822 (A300840, unless n > 2 and a Fermi-Dirac prime, A050376),
  A322988 (A322990, unless a prime power > 2),
  A323078 (A097246, unless an odd prime),
  A322808 (A097246, unless a squarefree number > 2),
  A322816 (A048675, unless an odd prime),
  A322807 (A285330, unless an odd prime),
  A322814 (A286621, unless an odd prime),
  A322824 (A242424, unless an odd prime),
  A322973 (A006370, unless an odd prime),
  A322974 (A049820, unless n > 1 and n is in A046642),
  A323079 (A060681, unless an odd prime),
  A322587 (A295887, unless an odd prime),
  A322588 (A291751, unless an odd prime),
  A322592 (A289625, unless an odd prime),
  A323369 (A323368, unless an odd prime),
  A323371 (A295886, unless an odd prime),
  A323374 (A323373, unless an odd prime),
  A323401 (A323372, unless an odd prime),
  A323405 (A323404, unless an odd prime).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A322809aux(n) = if((n>2)&&isprime(n),-1,(n>>1));
v322809 = rgs_transform(vector(up_to,n,A322809aux(n)));
A322809(n) = v322809[n];

a(n) = A323161(n+1) - 1.

A322991 a(n) = A289272(2*A289271(n)).

Original entry on oeis.org

1, 3, 4, 5, 7, 12, 8, 9, 11, 15, 13, 20, 16, 21, 28, 17, 19, 24, 23, 35, 36, 33, 25, 44, 27, 39, 29, 40, 31, 60, 32, 37, 52, 48, 56, 45, 41, 51, 68, 63, 43, 84, 47, 55, 77, 57, 49, 76, 53, 69, 92, 65, 59, 75, 91, 72, 100, 87, 61, 140, 64, 93, 88, 67, 112, 132, 71, 80, 108, 105, 73, 99, 79, 96, 116, 85, 104, 156, 81, 119, 83, 111, 89

Offset: 1

Antti Karttunen, Jan 01 2019

nonn

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

Permutation of A042965 (apart from zero).
Cf. A289271, A289272, A322990, A322992.
Cf. also A003963, A300841.

A289271(n) = { my(v=0,i=0,x=1); for(d=2,oo,if(n==1, return(v)); if(1==gcd(x,d)&&1==omega(d), if(!(n%d)&&1==gcd(d,n/d), v += 2^i; n /= d; x *= d); i++)); }; \\ After Rémy Sigrist's program for A289271.
A289272(n) = { my(m=1, pp=1); while(n>0, pp++; while(!isprimepower(pp)||(gcd(pp,m)>1), pp++); if(n%2, m *= pp); n >>=1); (m); };
A322991(n) = A289272(2*A289271(n));

a(n) = A289272(2*A289271(n)).

A322992 a(n) = A289272(1+(2*A289271(n))).

Original entry on oeis.org

2, 6, 10, 14, 18, 30, 22, 26, 34, 42, 38, 70, 46, 66, 90, 50, 54, 78, 58, 126, 110, 102, 62, 130, 74, 114, 82, 154, 86, 210, 94, 98, 170, 138, 198, 182, 106, 150, 190, 234, 118, 330, 122, 238, 306, 174, 134, 230, 142, 186, 270, 266, 146, 222, 342, 286, 290, 246, 158, 630, 162, 258, 374, 166, 414, 390, 178, 322, 310, 462, 194, 442, 202

Offset: 1

Antti Karttunen, Jan 01 2019

nonn

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

Permutation of A016825.

Cf. A289271, A289272, A322990, A322991.

A322992(n) = A289272(1+(2*A289271(n)));
A289271(n) = { my(v=0,i=0,x=1); for(d=2,oo,if(n==1, return(v)); if(1==gcd(x,d)&&1==omega(d), if(!(n%d)&&1==gcd(d,n/d), v += 2^i; n /= d; x *= d); i++)); }; \\ After Rémy Sigrist's program for A289271.
A289272(n) = { my(m=1, pp=1); while(n>0, pp++; while(!isprimepower(pp)||(gcd(pp,m)>1), pp++); if(n%2, m *= pp); n >>=1); (m); };

a(n) = A289272(1+(2*A289271(n))).

cp's OEIS Frontend

A322988 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(1) = 0 if n is a prime power > 2, f(2) = -1, and f(n) = A322990(n) for all other numbers.

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A322989 If n is a power of a prime, then a(n) = 0, otherwise a(n) = 1 + a(A322990(n)).

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A289271 A bijective binary representation of the prime factorization of a number, shown in decimal (see Comments for precise definition).

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PARI

A289272 Inverse to A289271.

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PARI

A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

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A322991 a(n) = A289272(2*A289271(n)).

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Formula

A322992 a(n) = A289272(1+(2*A289271(n))).

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Formula