A334860 -id:A334860 - cp's OEIS frontend

A334204 a(n) = A329697(A163511(n)).

0, 0, 0, 1, 0, 2, 1, 1, 0, 3, 2, 2, 1, 2, 1, 2, 0, 4, 3, 3, 2, 3, 2, 4, 1, 3, 2, 3, 1, 3, 2, 2, 0, 5, 4, 4, 3, 4, 3, 6, 2, 4, 3, 5, 2, 5, 4, 4, 1, 4, 3, 4, 2, 4, 3, 4, 1, 4, 3, 3, 2, 3, 2, 2, 0, 6, 5, 5, 4, 5, 4, 8, 3, 5, 4, 7, 3, 7, 6, 6, 2, 5, 4, 6, 3, 6, 5, 6, 2, 6, 5, 5, 4, 5, 4, 4, 1, 5, 4, 5, 3, 5, 4, 6, 2, 5

Offset: 0

Antti Karttunen, Jun 09 2020

nonn

As the underlying sequence A163511 can be represented as a binary tree, so can be this:
                                     0
                                     |
                  ...................0...................
                 0                                       1
       0......../ \........2                   1......../ \........1
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   0       3           2       2           1       2           1       2
  0 4     3 3         2 3     2 4         1 3     2 3         1 3     2 2
etc.
The nodes at the left edge are all zeros, and their right-hand children give positive integers, A000027.
Each left-hand leaning branch stays constant, because A329697(2n) = A329697(n).
The right-hand leaning branches are not necessarily monotonic. For example, a((2^6)-1) = 2 > 1 = a((2^7)-1), because A000040(7) = 17 is a Fermat prime (but A000040(6) = 13 is not), and therefore the latter is only one step away from a power of 2.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000079, A163511, A329697, A334109, A334860, A334867, A334873.

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)));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A334204(n) = A329697(A163511(n));

a(n) = A329697(A163511(n)).
a(n) = A334109(A334860(n)).
a(n) = a(2n) = a(A000265(n)).
For all n >= 0, a(2^n) = 0, a(2^n + 1) = n.

A334866 a(0) = 1, and then after, a(2n) = a(n)^2, a(2n+1) = A334747(a(n)).

Original entry on oeis.org

1, 2, 4, 3, 16, 8, 9, 6, 256, 32, 64, 12, 81, 18, 36, 5, 65536, 512, 1024, 48, 4096, 128, 144, 24, 6561, 162, 324, 27, 1296, 72, 25, 10, 4294967296, 131072, 262144, 768, 1048576, 2048, 2304, 96, 16777216, 8192, 16384, 192, 20736, 288, 576, 20, 43046721, 13122, 26244, 243, 104976, 648, 729, 54, 1679616, 2592, 5184, 108, 625, 50, 100, 15

Offset: 0

Antti Karttunen, Jun 08 2020

This irregular table can be represented as a binary tree. Each child to the left is obtained by squaring the parent, and each child to the right is obtained by applying A334747 to the parent:
                                        1
                                        |
                     ...................2...................
                    4                                       3
         16......../ \........8                   9......../ \........6
         / \                 / \                 / \                 / \
        /   \               /   \               /   \               /   \
       /     \             /     \             /     \             /     \
    256       32         64       12         81       18         36       5
65536  512 1024 48   4096  128 144  24   6561  162 324  27   1296  72   25 10
etc.
This is the mirror image of the tree in A334860.

Index entries for sequences that are permutations of the natural numbers

Cf. A334865 (inverse permutation), A334860 (mirror image).
Composition of permutations A005940 and A225546.
Cf. A000290, A048675, A087808, A334747, A334870.
Cf. A001146 (left edge of the tree), A019565 (right edge), A334110 (the left children of the right edge).

A334747(n) = { my(c=core(n), m=n); forprime(p=2, , if(c % p, m*=p; break, m/=p)); m; }; \\ From A334747
A334866(n) = if(!n,1,if(!(n%2),A334866(n/2)^2,A334747(A334866((n-1)/2))));

a(0) = 1, and then after, a(2n) = a(n)^2, a(2n+1) = A334747(a(n)).
a(n) = A225546(A005940(1+n)).
For all n >= 0, A048675(a(n)) = A087808(n).

A334747 Let p be the smallest prime not dividing the squarefree part of n. Multiply n by p and divide by the product of all smaller primes.

Original entry on oeis.org

2, 3, 6, 8, 10, 5, 14, 12, 18, 15, 22, 24, 26, 21, 30, 32, 34, 27, 38, 40, 42, 33, 46, 20, 50, 39, 54, 56, 58, 7, 62, 48, 66, 51, 70, 72, 74, 57, 78, 60, 82, 35, 86, 88, 90, 69, 94, 96, 98, 75, 102, 104, 106, 45, 110, 84, 114, 87, 118, 120, 122, 93, 126, 128, 130, 55

Offset: 1

Peter Munn, May 09 2020

A bijection from the positive integers to the nonsquares, A000037.
A003159 (which has asymptotic density 2/3) lists index n such that a(n) = 2n. The sequence maps the terms of A003159 1:1 onto A036554, defining a bijection between them.
Similarly, bijections are defined from A007417 to A325424, from A325424 to A145204\{0}, and from the first in each of the following pairs to the nonsquare integers in the second: (A145204\{0}, A036668), (A036668, A007417), (A036554, A003159), (A332820, A332821), (A332821, A332822), (A332822, A332820). Note that many of these are between sets where membership depends on whether a number's squarefree part divides by 2 and/or 3.
Starting from 1, and iterating the sequence as a(1) = 2, a(2) = 3, a(3) = 6, a(6) = 5, a(5) = 10, etc., runs through the squarefree numbers in the order they appear in A019565. - Antti Karttunen, Jun 08 2020

			168 = 42*4 has squarefree part 42 (and square part 4). The smallest prime absent from 42 = 2*3*7 is 5 and the product of all smaller primes is 2*3 = 6. So a(168) = 168*5/6 = 140.

Permutation of A000037.
Row 2, and therefore column 2, of A331590. Cf. A334748 (row 3).
A007913, A034386, A053669, A225546 are used in formulas defining the sequence.
The formula section details how the sequence maps the terms of A002110, A003961, A019565; and how f(a(n)) relates to f(n) for f = A008833, A048675, A267116; making use of A003986.
Subsequences: A016825 (odd bisection), A036554, A329575.
Bijections are defined that relate to A003159, A007417, A036668, A145204, A325424, A332820, A332821, A332822.
Cf. also binary trees A334860, A334866 and A334870 (a left inverse).

a(n) = {my(c=core(n), m=n); forprime(p=2, , if(c % p, m*=p; break, m/=p)); m;} \\ Michel Marcus, May 22 2020

a(n) = n * m / A034386(m-1), where m = A053669(A007913(n)).
a(n) = A331590(2, n) = A225546(2 * A225546(n)).
a(A019565(n)) = A019565(n+1).
a(k * m^2) = a(k) * m^2.
a(A003961(n)) = 2 * A003961(n).
a(2 * A003961(n)) = A003961(a(n)).
a(A002110(n)) = prime(n+1).
A048675(a(n)) = A048675(n) + 1.
A008833(a(n)) = A008833(n).
A267116(a(n)) = A267116(n) OR 1, where OR denotes the bitwise operation A003986.
a(A003159(n)) = A036554(n) = 2 * A003159(n).
A334870(a(n)) = n. - Antti Karttunen, Jun 08 2020

A334870 If n is a square, a(n) = A000196(n), and for nonsquare n, let p be the smallest prime dividing the squarefree part of n. Divide n by p and multiply by the product of all smaller primes.

Original entry on oeis.org

1, 1, 2, 2, 6, 3, 30, 4, 3, 5, 210, 8, 2310, 7, 10, 4, 30030, 9, 510510, 24, 14, 11, 9699690, 12, 5, 13, 18, 120, 223092870, 15, 6469693230, 16, 22, 17, 42, 6, 200560490130, 19, 26, 20, 7420738134810, 21, 304250263527210, 840, 54, 23, 13082761331670030, 32, 7, 25, 34, 9240, 614889782588491410, 27, 66, 28, 38, 29

Offset: 1

Antti Karttunen, Jun 08 2020

nonn

Each natural numbers occurs exactly twice in this sequence.
In binary trees like A334860 and A334866, for n > 2, a(n) gives the parent node of node n.
For nonsquare numbers, n, with squarefree part A019565(k) and square part m, a(n) is the number with squarefree part A019565(k-1) and square part m. - Peter Munn, Jul 14 2020

Antti Karttunen, Table of n, a(n) for n = 1..1024

A left inverse of A334747.

Cf. A000196, A019565, A225546, A252463, A334860, A334866, A334868, A334869, A334871, A334872.

Array[If[IntegerQ[#2], #2, #1/#2*Product[Prime@i, {i, PrimePi@#2 - 1}] & @@ {#1, FactorInteger[#2 /. (c_ : 1)*a_^(b_ : 0) :> (c*a^b)^2][[1, 1]]}] & @@ {#, Sqrt[#]} &, 58] (* Michael De Vlieger, Jun 26 2020 *)

A334870(n) = if(issquare(n),sqrtint(n),my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));

a(A334747(n)) = n.
a(A000040(n)) = A002110(n-1).
a(n^2) = n.
a(n) = A225546(A252463(A225546(n))). - Peter Munn, Jun 08 2020

A334110 The squares of squarefree numbers (A062503), ordered lexicographically according to their prime factors. a(n) = Product_{k in I} prime(k+1)^2, where I are the indices of nonzero binary digits in n = Sum_{k in I} 2^k.

Original entry on oeis.org

1, 4, 9, 36, 25, 100, 225, 900, 49, 196, 441, 1764, 1225, 4900, 11025, 44100, 121, 484, 1089, 4356, 3025, 12100, 27225, 108900, 5929, 23716, 53361, 213444, 148225, 592900, 1334025, 5336100, 169, 676, 1521, 6084, 4225, 16900, 38025, 152100, 8281, 33124, 74529, 298116, 207025, 828100, 1863225, 7452900, 20449, 81796, 184041

Offset: 0

Antti Karttunen and Peter Munn, May 01 2020

nonn

For the lexicographic ordering, the prime factors must be written in nonincreasing order. If we write the factors in nondecreasing order, we get a lexicographically ordered set with an order type that is greater than a natural number index - the resulting sequence does not include all qualifying numbers. (Note also that the symbols used for the lexicographic order are the prime numbers, not their digits.)
a(n) is the n-th power of 4 in the monoid defined in A331590.
Conjecture: a(n) is the position of the first occurrence of n in A334109.

			The initial terms are shown below, equated with the product of their prime factors to exhibit the lexicographic ordering. The list starts with 1, since 1 is factored as the empty product and the empty list is first in lexicographic order.
    1 = .
    4 = 2*2.
    9 = 3*3.
   36 = 3*3*2*2.
   25 = 5*5.
  100 = 5*5*2*2.
  225 = 5*5*3*3.
  900 = 5*5*3*3*2*2.
   49 = 7*7.
  196 = 7*7*2*2.
  441 = 7*7*3*3.

Cf. A000079, A019565 (square roots), A048675, A097248, A225546, A267116, A332382, A334109 (a left inverse).
Column 2 of A329332. Permutation of A062503.
After 1, the right children of the leftmost edge of A334860, or respectively, the left children of the rightmost edge of A334866.
Subsequences: A001248, A061742, A166329.
Subsequence of A052330.
A003961, A003987, A059897, A331590 are used to express relationship between terms of this sequence.

Array[If[# == 0, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[3^#]]] &, 51, 0] (* Michael De Vlieger, May 26 2020 *)

A334110(n) = { my(p=2,m=1); while(n, if(n%2, m *= p^2); n >>= 1; p = nextprime(1+p)); (m); };

a(n) = A019565(n)^2.
For n >= 1, a(A000079(n-1)) = A001248(n).
For all n >= 0, A334109(a(n)) = n.
a(n+k) = A331590(a(n), a(k)).
a(n XOR k) = A059897(a(n), a(k)), where XOR denotes bitwise exclusive-or, A003987.
a(n) = A225546(3^n).
a(2n) = A003961(a(n)).
a(2n+1) = 4 * a(2n).
a(2^k-1) = A061742(k).
A267116(a(n)) = 2.
A048675(a(n)) = 2n.
A097248(a(n)) = A332382(n) = A019565(2n).

A334109 a(n) = A329697(A225546(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 4, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 8, 4, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 4, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 8, 2, 5, 0, 0, 0, 1, 0

Offset: 1

Antti Karttunen, Apr 29 2020

nonn

Conjecture: Each k >= 0 occurs for the first time at A334110(k) = A019565(k)^2. Note that each k must occur first time on square n, because of the identity a(n) = a(A008833(n)). However, is there any reason to exclude squares with prime exponents > 2 from the candidates? See also comments in A334204.

Antti Karttunen, Table of n, a(n) for n = 1..10200
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001248, A005117 (positions of zeros), A334110.
Cf. A003961, A008833, A019565, A059895, A225546, A329697, A331590, A331736, A334204, A334860.
Cf. also A334107, A334108.

Map[-1 + Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] &, Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 105] ] (* Michael De Vlieger, May 26 2020 *)

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A334109(n) = { my(f=factor(n),pis=apply(primepi,f[,1]),es=f[,2]); sum(k=1,#f~,(2^(pis[k]-1))*A329697(A019565(es[k]))); };

Additive with a(prime(i)^j) = A000079(i-1) * A329697(A019565(j)), a(m*n) = a(m)+a(n) if gcd(m,n) = 1.
Alternatively, additive with a(prime(i)^(2^k)) = 2^(i-1) * A329697(prime(k+1)), a(m*n) = a(m)+a(n) if A059895(m,n) = 1. - Peter Munn, May 04 2020
a(n) = A329697(A225546(n)) = A329697(A331736(n)).
a(n) = a(A008833(n)).
For all n >= 0, a(A334110(n)) = n, a(A334860(n)) = A334204(n).
a(A331590(m,k)) = a(m) + a(k); a(A003961(n)) = 2*a(n). - Peter Munn, Apr 30 2020

A334871 Number of steps needed to reach 1 when starting from n and iterating with A334870.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 8, 3, 3, 5, 16, 4, 32, 9, 6, 3, 64, 4, 128, 6, 10, 17, 256, 5, 5, 33, 5, 10, 512, 7, 1024, 4, 18, 65, 12, 4, 2048, 129, 34, 7, 4096, 11, 8192, 18, 7, 257, 16384, 5, 9, 6, 66, 34, 32768, 6, 20, 11, 130, 513, 65536, 8, 131072, 1025, 11, 4, 36, 19, 262144, 66, 258, 13, 524288, 5, 1048576, 2049, 7, 130

Offset: 1

Antti Karttunen, Jun 08 2020

nonn

Distance of n from the root (1) in binary trees like A334860 and A334866.
Each n > 0 occurs 2^(n-1) times.
a(n) is the size of the inner lining of the integer partition with Heinz number A225546(n), which is also the size of the largest hook of the same partition. (After Gus Wiseman's Apr 02 2019 comment in A252464).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A007913, A008833, A070939, A225546, A252464, A299090, A334859, A334860, A334865, A334866, A334869, A334870, A334872.

A334870(n) = if(issquare(n),sqrtint(n),my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
A334871(n) = { my(s=0); while(n>1,s++; n = A334870(n)); (s); };

\\ Much faster:
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A334871(n) = { my(s=0); while(n>1, if(issquare(n), s++; n = sqrtint(n), s += A048675(core(n)); n /= core(n))); (s); };

a(1) = 0; for n > 1, a(n) = 1 + a(A334870(n)).
a(n) = A252464(A225546(n)).
a(n) = A048675(A007913(n)) + a(A008833(n)).
For n > 1, a(n) = 1 + A048675(A007913(n)) + a(A000188(n)).
For n > 1, a(n) = A070939(A334859(n)) = A070939(A334865(n)).
For all n >= 1, a(n) >= A299090(n).
For all n >= 1, a(n) >= A334872(n).

A334859 a(n) = A243071(A225546(n)).

Original entry on oeis.org

0, 1, 2, 3, 8, 4, 128, 6, 5, 16, 32768, 12, 2147483648, 256, 32, 7, 9223372036854775808, 10, 170141183460469231731687303715884105728, 48, 512, 65536, 57896044618658097711785492504343953926634992332820282019728792003956564819968, 24, 17, 4294967296, 20, 768

Offset: 1

Antti Karttunen, Jun 08 2020

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation of A334860. Composition of permutations A225546 and A243071, and also of A054429 and A334865.

Cf. A299090, A334871.

a(n) = A243071(A225546(n)).
a(n) = A054429(A334865(n)).
For n >= 1, A000120(a(n)) = A299090(n).
For n > 1, A070939(a(n)) = A334871(n).

A335428 Prime exponent of the first Fermi-Dirac factor (number of form p^(2^k), A050376) reached, when starting from n and iterating with A334870, with a(1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 2, 1, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Jun 26 2020

nonn

			For n=27, when iterating with A334870, we obtain the path 27 -> 18 -> 9, with that 9 being the first prime power encountered that has an exponent of the form 2^k, as 9 = 3^2, thus a(27) = 2. See the binary tree A334860 or A334866 for how such paths go.
For n=900, when iterating with A334870 we obtain the path 900 -> 30 -> 15 -> 10 -> 5, and 5^1 is finally a prime power with an exponent that is two's power, thus a(900) = 1. Note that 900 is the first such position of 1 in this sequence that is not listed in A333634.

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

Cf. A050376, A209229, A302777, A334860, A334866, A334870, A334872.

A209229(n) = (n && !bitand(n,n-1));
A302777(n) = A209229(isprimepower(n));
A334870(n) = if(issquare(n),sqrtint(n),my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
A335428(n) = { while(n>1 && !A302777(n), n = A334870(n)); isprimepower(n); };

\\ Faster, A209229 and A302777 like in above:
A335428(n) = if(1==n,0, while(!A302777(n), if(issquarefree(n), return(1)); if(issquare(n), n = sqrtint(n), n /= core(n))); isprimepower(n));

cp's OEIS Frontend

A334204 a(n) = A329697(A163511(n)).

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A334866 a(0) = 1, and then after, a(2n) = a(n)^2, a(2n+1) = A334747(a(n)).

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A334747 Let p be the smallest prime not dividing the squarefree part of n. Multiply n by p and divide by the product of all smaller primes.

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A334870 If n is a square, a(n) = A000196(n), and for nonsquare n, let p be the smallest prime dividing the squarefree part of n. Divide n by p and multiply by the product of all smaller primes.

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A334110 The squares of squarefree numbers (A062503), ordered lexicographically according to their prime factors. a(n) = Product_{k in I} prime(k+1)^2, where I are the indices of nonzero binary digits in n = Sum_{k in I} 2^k.

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A334109 a(n) = A329697(A225546(n)).

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A334871 Number of steps needed to reach 1 when starting from n and iterating with A334870.

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A334859 a(n) = A243071(A225546(n)).

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