A283475 -id:A283475 - cp's OEIS frontend

A019565 The squarefree numbers ordered lexicographically by their prime factorization (with factors written in decreasing order). a(n) = Product_{k in I} prime(k+1), where I is the set of indices of nonzero binary digits in n = Sum_{k in I} 2^k.

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 42, 35, 70, 105, 210, 11, 22, 33, 66, 55, 110, 165, 330, 77, 154, 231, 462, 385, 770, 1155, 2310, 13, 26, 39, 78, 65, 130, 195, 390, 91, 182, 273, 546, 455, 910, 1365, 2730, 143, 286, 429, 858, 715, 1430, 2145, 4290

Offset: 0

Marc LeBrun

A permutation of the squarefree numbers A005117. The missing positive numbers are in A013929. - Alois P. Heinz, Sep 06 2014
From Antti Karttunen, Apr 18 & 19 2017: (Start)
Because a(n) toggles the parity of n there are neither fixed points nor any cycles of odd length.
Conjecture: there are no finite cycles of any length. My grounds for this conjecture: any finite cycle in this sequence, if such cycles exist at all, must have at least one member that occurs somewhere in A285319, the terms that seem already to be quite rare. Moreover, any such a number n should satisfy in addition to A019565(n) < n also that A048675^{k}(n) is squarefree, not just for k=0, 1 but for all k >= 0. As there is on average a probability of only 6/(Pi^2) = 0.6079... that any further term encountered on the trajectory of A048675 is squarefree, the total chance that all of them would be squarefree (which is required from the elements of A019565-cycles) is soon minuscule, especially as A048675 is not very tightly bounded (many trajectories seem to skyrocket, at least initially). I am also assuming that usually there is no significant correlation between the binary expansions of n and A048675(n) (apart from their least significant bits), or, for that matter, between their prime factorizations.
See also the slightly stronger conjecture in A285320, which implies that there would neither be any two-way infinite cycles.
If either of the conjectures is false (there are cycles), then certainly neither sequence A285332 nor its inverse A285331 can be a permutation of natural numbers. (End)
The conjecture made in A087207 (see also A288569) implies the two conjectures mentioned above. A further constraint for cycles is that in any A019565-trajectory which starts from a squarefree number (A005117), every other term is of the form 4k+2, while every other term is of the form 6k+3. - Antti Karttunen, Jun 18 2017
The sequence satisfies the exponential function identity, a(x + y) = a(x) * a(y), whenever x and y do not have a 1-bit in the same position, i.e., when A004198(x,y) = 0. See also A283475. - Antti Karttunen, Oct 31 2019
The above identity becomes unconditional if binary exclusive OR, A003987(.,.), is substituted for addition, and A059897(.,.), a multiplicative equivalent of A003987, is substituted for multiplication. This gives us a(A003987(x,y)) = A059897(a(x), a(y)). - Peter Munn, Nov 18 2019
Also the Heinz number of the binary indices of n, where the Heinz number of a sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k), and a number's binary indices (A048793) are the positions of 1's in its reversed binary expansion. - Gus Wiseman, Dec 28 2022

			5 = 2^2+2^0, e_1 = 2, e_2 = 0, prime(2+1) = prime(3) = 5, prime(0+1) = prime(1) = 2, so a(5) = 5*2 = 10.
From _Philippe Deléham_, Jun 03 2015: (Start)
This sequence regarded as a triangle withs rows of lengths 1, 1, 2, 4, 8, 16, ...:
   1;
   2;
   3,  6;
   5, 10, 15, 30;
   7, 14, 21, 42, 35,  70, 105, 210;
  11, 22, 33, 66, 55, 110, 165, 330, 77, 154, 231, 462, 385, 770, 1155, 2310;
  ...
(End)
From _Peter Munn_, Jun 14 2020: (Start)
The initial terms are shown below, equated with the product of their prime factors to exhibit the lexicographic order. We start with 1, since 1 is factored as the empty product and the empty list is first in lexicographic order.
   n     a(n)
   0     1 = .
   1     2 = 2.
   2     3 = 3.
   3     6 = 3*2.
   4     5 = 5.
   5    10 = 5*2.
   6    15 = 5*3.
   7    30 = 5*3*2.
   8     7 = 7.
   9    14 = 7*2.
  10    21 = 7*3.
  11    42 = 7*3*2.
  12    35 = 7*5.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191

Row 1 of A285321.
Equivalent sequences for k-th-power-free numbers: A101278 (k=3), A101942 (k=4), A101943 (k=5), A054842 (k=10).
Cf. A007088, A030308, A000040, A013929, A005117, A103785, A103786, A110765, A064273, A246353, A283475, A283477, A285319, A285331, A285332, A288569, A293442.
Cf. A109162 (iterates).
Cf. also A048675 (a left inverse), A087207, A097248, A260443, A054841.
Cf. A285315 (numbers for which a(n) < n), A285316 (for which a(n) > n).
Cf. A276076, A276086 (analogous sequences for factorial and primorial bases), A334110 (terms squared).
For partial sums see A288570.
A003961, A003987, A004198, A059897, A089913, A331590, A334747 are used to express relationships between sequence terms.
Column 1 of A329332.
Even bisection (which contains the odd terms): A332382.
A160102 composed with A052330, and subsequence of the latter.
Related to A000079 via A225546, to A057335 via A122111, to A008578 via A336322.
Least prime index of a(n) is A001511.
Greatest prime index of a(n) is A029837 or A070939.
Taking prime indices gives A048793, reverse A272020, row sums A029931.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A000120, A000720, A005940, A066099, A358137, A358170.

a019565 n = product $ zipWith (^) a000040_list (a030308_row n)
-- Reinhard Zumkeller, Apr 27 2013

a:= proc(n) local i, m, r; m:=n; r:=1;
      for i while m>0 do if irem(m,2,'m')=1
        then r:=r*ithprime(i) fi od; r
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 06 2014

Do[m=1;o=1;k1=k;While[ k1>0, k2=Mod[k1, 2];If[k2\[Equal]1, m=m*Prime[o]];k1=(k1-k2)/ 2;o=o+1];Print[m], {k, 0, 55}] (* Lei Zhou, Feb 15 2005 *)
Table[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2], {n, 0, 55}]  (* Michael De Vlieger, Aug 27 2016 *)
b[0] := {1}; b[n_] := Flatten[{ b[n - 1], b[n - 1] * Prime[n] }];
  a = b[6] (* Fred Daniel Kline, Jun 26 2017 *)

a(n)=factorback(vecextract(primes(logint(n+!n,2)+1),n))  \\ M. F. Hasler, Mar 26 2011, updated Aug 22 2014, updated Mar 01 2018

from operator import mul
from functools import reduce
from sympy import prime
def A019565(n):
    return reduce(mul,(prime(i+1) for i,v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1
# Chai Wah Wu, Dec 25 2014

(define (A019565 n) (let loop ((n n) (i 1) (p 1)) (cond ((zero? n) p) ((odd? n) (loop (/ (- n 1) 2) (+ 1 i) (* p (A000040 i)))) (else (loop (/ n 2) (+ 1 i) p))))) ;; (Requires only the implementation of A000040 for prime numbers.) - Antti Karttunen, Apr 20 2017

G.f.: Product_{k>=0} (1 + prime(k+1)*x^2^k), where prime(k)=A000040(k). - Ralf Stephan, Jun 20 2003
a(n) = f(n, 1, 1) with f(x, y, z) = if x > 0 then f(floor(x/2), y*prime(z)^(x mod 2), z+1) else y. - Reinhard Zumkeller, Mar 13 2010
For all n >= 0: A048675(a(n)) = n; A013928(a(n)) = A064273(n). - Antti Karttunen, Jul 29 2015
a(n) = a(2^x)*a(2^y)*a(2^z)*... = prime(x+1)*prime(y+1)*prime(z+1)*..., where n = 2^x + 2^y + 2^z + ... - Benedict W. J. Irwin, Jul 24 2016
From Antti Karttunen, Apr 18 2017 and Jun 18 2017: (Start)
a(n) = A097248(A260443(n)), a(A005187(n)) = A283475(n), A108951(a(n)) = A283477(n).
A055396(a(n)) = A001511(n), a(A087207(n)) = A007947(n). (End)
a(2^n - 1) = A002110(n). - Michael De Vlieger, Jul 05 2017
a(n) = A225546(A000079(n)). - Peter Munn, Oct 31 2019
From Peter Munn, Mar 04 2022: (Start)
a(2n) = A003961(a(n)); a(2n+1) = 2*a(2n).
a(x XOR y) = A059897(a(x), a(y)) = A089913(a(x), a(y)), where XOR denotes bitwise exclusive OR (A003987).
a(n+1) = A334747(a(n)).
a(x+y) = A331590(a(x), a(y)).
a(n) = A336322(A008578(n+1)).
(End)

Definition corrected by Klaus-R. Löffler, Aug 20 2014

New name from Peter Munn, Jun 14 2020

A283477 If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 180, 360, 210, 420, 1260, 2520, 6300, 12600, 37800, 75600, 2310, 4620, 13860, 27720, 69300, 138600, 415800, 831600, 485100, 970200, 2910600, 5821200, 14553000, 29106000, 87318000, 174636000, 30030, 60060, 180180, 360360, 900900, 1801800, 5405400, 10810800, 6306300, 12612600, 37837800, 75675600

Offset: 0

Antti Karttunen, Mar 16 2017

nonn

a(n) = Product of distinct primorials larger than one, obtained as Product_{i} A002110(1+i), where i ranges over the zero-based positions of the 1-bits present in the binary representation of n.
This sequence can be represented as a binary tree. Each child to the left is obtained as A283980(k), and each child to the right is obtained as 2*A283980(k), when their parent contains k:
                                      1
                                      |
                   ...................2....................
                  6                                       12
       30......../ \........60                 180......../ \......360
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
  210       420      1260       2520     6300       12600   37800       75600
etc.

Cf. A129912 (same terms, but sorted into ascending order).
Cf. A000120, A001221, A001222, A002110, A005187, A005361, A006068, A007814, A007913, A019565, A029931, A030308, A046660, A048675, A051903, A056169, A065120, A070939, A072411, A090880, A097248, A108951, A124757, A248663, A276075, A276085, A283475, A283483, A283980, A283984, A283985, A284001, A284002, A284003, A284005, A324287, A324289, A324341, A324342, A324343.
Cf. A005940, A052330, A322827, A323505 for other similar trees.
Cf. also A260443.

Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]], {n, 0, 43}] (* Michael De Vlieger, Mar 18 2017 *)

A283477(n) = prod(i=0,exponent(n),if(bittest(n,i),vecprod(primes(1+i)),1)) \\ Edited by M. F. Hasler, Nov 11 2019

from sympy import prime, primerange, factorint
from operator import mul
from functools import reduce
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a108951(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # after Chai Wah Wu
def a(n): return a108951(a019565(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 22 2017

from sympy import primorial
from math import prod
def A283477(n): return prod(primorial(i) for i, b in enumerate(bin(n)[:1:-1],1) if b =='1') # Chai Wah Wu, Dec 08 2022

(define (A283477 n) (A108951 (A019565 n)))
;; Recursive "binary tree" implementation, using memoization-macro definec:
(definec (A283477 n) (cond ((zero? n) 1) ((even? n) (A283980 (A283477 (/ n 2)))) (else (* 2 (A283980 (A283477 (/ (- n 1) 2)))))))

a(0) = 1; a(2n) = A283980(a(n)), a(2n+1) = 2*A283980(a(n)).
Other identities. For all n >= 0 (or for n >= 1):
a(2n+1) = 2*a(2n).
a(n) = A108951(A019565(n)).
A097248(a(n)) = A283475(n).
A007814(a(n)) = A051903(a(n)) = A000120(n).
A001221(a(n)) = A070939(n).
A001222(a(n)) = A029931(n).
A048675(a(n)) = A005187(n).
A248663(a(n)) = A006068(n).
A090880(a(n)) = A283483(n).
A276075(a(n)) = A283984(n).
A276085(a(n)) = A283985(n).
A046660(a(n)) = A124757(n).
A056169(a(n)) = A065120(n). [seems to be]
A005361(a(n)) = A284001(n).
A072411(a(n)) = A284002(n).
A007913(a(n)) = A284003(n).
A000005(a(n)) = A284005(n).
A324286(a(n)) = A324287(n).
A276086(a(n)) = A324289(n).
A267263(a(n)) = A324341(n).
A276150(a(n)) = A324342(n). [subsequences in the latter are converging towards this sequence]
G.f.: Product_{k>=0} (1 + prime(k + 1)# * x^(2^k)), where prime()# = A002110. - Ilya Gutkovskiy, Aug 19 2019

More formulas and the binary tree illustration added by Antti Karttunen, Mar 19 2017

Four more linking formulas added by Antti Karttunen, Feb 25 2019

A097248 a(n) is the eventual stable point reached when iterating k -> A097246(k), starting from k = n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 6, 5, 10, 11, 5, 13, 14, 15, 5, 17, 10, 19, 15, 21, 22, 23, 10, 7, 26, 15, 21, 29, 30, 31, 10, 33, 34, 35, 15, 37, 38, 39, 30, 41, 42, 43, 33, 7, 46, 47, 15, 11, 14, 51, 39, 53, 30, 55, 42, 57, 58, 59, 7, 61, 62, 35, 15, 65, 66, 67, 51, 69, 70, 71, 30, 73, 74, 21

Offset: 1

Reinhard Zumkeller, Aug 03 2004

nonn

a(n) = r(n,m) with m such that r(n,m)=r(n,m+1), where r(n,k) = A097246(r(n,k-1)), r(n,0)=n. (The original definition.)
A097248(n) = r(n,a(n)).
From Antti Karttunen, Nov 15 2016: (Start)
The above remark could be interpreted to mean that A097249(n) <= a(n).
All terms are squarefree, and the squarefree numbers are the fixed points.
These are also fixed points eventually reached when iterating A277886.
(End)

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

Range of values is A005117.
Cf. A008683, A019565, A048675, A097246, A097247, A097249, A277905, A332824.
A003961, A225546, A277885, A277886, A331590 are used to express relationship between terms of this sequence.
The formula section also details how the sequence maps the terms of A007913, A260443, A329050, A329332.
See comments/formulas in A283475, A283478, A331751 giving their relationship to this sequence.

Table[FixedPoint[Times @@ Map[#1^#2 & @@ # &, Partition[#, 2, 2] &@ Flatten[FactorInteger[#] /. {p_, e_} /; e >= 2 :> {If[OddQ@ e, {p, 1}, {1, 1}], {NextPrime@ p, Floor[e/2]}}]] &, n], {n, 75}] (* Michael De Vlieger, Mar 18 2017 *)

A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
\\ Antti Karttunen, Mar 18 2017

from sympy import factorint, nextprime
from operator import mul
def a097246(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(nextprime(i)**int(f[i]/2))*(i**(f[i]%2)) for i in f])
def a(n):
    k=a097246(n)
    while k!=n:
        n=k
        k=a097246(k)
    return k # Indranil Ghosh, May 15 2017

;; with memoization-macro definec
;; Two implementations:
(definec (A097248 n) (if (not (zero? (A008683 n))) n (A097248 (A097246 n))))
(definec (A097248 n) (if (zero? (A277885 n)) n (A097248 (A277886 n))))
;; Antti Karttunen, Nov 15 2016

a(A005117(n)) = A005117(n).
From Antti Karttunen, Nov 15 2016: (Start)
If A008683(n) <> 0 [when n is squarefree], a(n) = n, otherwise a(n) = a(A097246(n)).
If A277885(n) = 0, a(n) = n, otherwise a(n) = a(A277886(n)).
A007913(a(n)) = a(n).
a(A007913(n)) = A007913(n).
A048675(a(n)) = A048675(n).
a(A260443(n)) = A019565(n).
(End)
From Peter Munn, Feb 06 2020: (Start)
a(1) = 1; a(p) = p, for prime p; a(m*k) = A331590(a(m), a(k)).
a(A331590(m,k)) = A331590(a(m), a(k)).
a(n^2) = a(A003961(n)) = A003961(a(n)).
a(A225546(n)) = a(n).
a(n) = A225546(2^A048675(n)) = A019565(A048675(n)).
a(A329050(n,k)) = prime(n+k-1) = A000040(n+k-1).
a(A329332(n,k)) = A019565(n * k).
Equivalently, a(A019565(n)^k) = A019565(n * k).
(End)
From Antti Karttunen, Feb 22-25 & Mar 01 2020: (Start)
a(A019565(x)*A019565(y)) = A019565(x+y).
a(A332461(n)) = A332462(n).
a(A332824(n)) = A019565(n).
a(A277905(n,k)) = A277905(n,1) = A019565(n), for all n >= 1, and 1 <= k <= A018819(n).
(End)

Name changed and the original definition moved to the Comments section by Antti Karttunen, Nov 15 2016

A302033 a(n) = A019565(A003188(n)).

Original entry on oeis.org

1, 2, 6, 3, 15, 30, 10, 5, 35, 70, 210, 105, 21, 42, 14, 7, 77, 154, 462, 231, 1155, 2310, 770, 385, 55, 110, 330, 165, 33, 66, 22, 11, 143, 286, 858, 429, 2145, 4290, 1430, 715, 5005, 10010, 30030, 15015, 3003, 6006, 2002, 1001, 91, 182, 546, 273, 1365, 2730, 910, 455, 65, 130, 390, 195, 39, 78, 26, 13, 221, 442, 1326, 663, 3315, 6630, 2210, 1105

Offset: 0

Antti Karttunen & Peter Munn, Apr 16 2018

A squarefree analog of A207901 (and the subsequence consisting of its squarefree terms): Each term is either a divisor or a multiple of the next one, and the terms differ by a single prime factor. Compare also to A284003.

For all n >= 0, max(a(n + 1), a(n)) / min(a(n + 1), a(n)) = A094290(n + 1) = prime(valuation(n + 1, 2) + 1) = A000040(A001511(n + 1)). [See Russ Cox's Dec 04 2010 comment in A007814.] - David A. Corneth & Antti Karttunen, Apr 16 2018

Antti Karttunen, Table of n, a(n) for n = 0..8191

A permutation of A005117. Subsequence of A207901.
Cf. A302054 (gives the sum of prime divisors).
Cf. A000040, A001511, A003188, A019565, A034947, A059897, A064706, A094290.
Cf. also A277811, A283475, A284003.

Array[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[BitXor[#, Floor[#/2]], 2] &, 72, 0] (* Michael De Vlieger, Apr 27 2018 *)

A003188(n) = bitxor(n, n>>1);
A019565(n) = {my(j); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A302033(n) = A019565(A003188(n));

first(n) = {my(pr = primes(1 + logint(n, 2)), ex = vector(#pr, i, 1), res = vector(n)); res[1] = 1; for(i = 1, n-1, v = valuation(i, 2); res[i + 1] = res[i] * pr[v++] ^ ex[v]; ex[v]*=-1); res}

a(n) = A019565(A003188(n)).
a(n) = A284003(A064706(n)).
a(n+1) = A059897(a(n), A094290(n+1)). - Peter Munn, Apr 01 2019

A284003 a(n) = A007913(A283477(n)) = A019565(A006068(n)).

Original entry on oeis.org

1, 2, 6, 3, 30, 15, 5, 10, 210, 105, 35, 70, 7, 14, 42, 21, 2310, 1155, 385, 770, 77, 154, 462, 231, 11, 22, 66, 33, 330, 165, 55, 110, 30030, 15015, 5005, 10010, 1001, 2002, 6006, 3003, 143, 286, 858, 429, 4290, 2145, 715, 1430, 13, 26, 78, 39, 390, 195, 65, 130, 2730, 1365, 455, 910, 91, 182, 546, 273, 510510, 255255, 85085, 170170, 17017

Offset: 0

Antti Karttunen, Mar 18 2017

A squarefree analog of A302783. Each term is either a divisor or a multiple of the next one. In contrast to A302033 at each step the previous term can be multiplied (or divided), not just by a single prime, but possibly by a product of several distinct ones, A019565(A000975(k)). E.g., a(3) = 3, a(4) = 2*5*a(3) = 30. - Antti Karttunen, Apr 17 2018

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

A permutation of A005117.
Cf. A000975, A001222, A006068, A007913, A019565, A046523, A048675, A064707, A209281, A283477, A284004.
Cf. also A277811, A283475, A302033, A302783.

Table[Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Times @@ (p^Mod[e, 2])] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]]], {n, 0, 52}] (* Michael De Vlieger, Mar 18 2017 *)

A007913(n) = core(n);
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From Charles R Greathouse IV, Jun 28 2015
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A283477(n) = A108951(A019565(n));
A284003(n) = A007913(A283477(n));

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A284003(n) = A019565(A006068(n)); \\ (and use A019565 from above) - Antti Karttunen, Apr 16 2018

(define (A284003 n) (A007913 (A283477 n)))

a(n) = A007913(A283477(n)).
Other identities. For all n >= 0:
A048675(a(n)) = A006068(n).
A046523(a(n)) = A284004(n).
It seems that A001222(a(n)) = A209281(n).
a(n) = A019565(A006068(n)) = A302033(A064707(n)). - Antti Karttunen, Apr 16 2018

Name amended with a second formula by Antti Karttunen, Apr 16 2018

A323247 a(n) = A005187(A156552(n)).

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 15, 11, 10, 16, 31, 19, 63, 32, 18, 26, 127, 23, 255, 35, 34, 64, 511, 42, 22, 128, 25, 67, 1023, 39, 2047, 57, 66, 256, 38, 50, 4095, 512, 130, 74, 8191, 71, 16383, 131, 41, 1024, 32767, 89, 46, 47, 258, 259, 65535, 54, 70, 138, 514, 2048, 131071, 82, 262143, 4096, 73, 120, 134, 135, 524287, 515

Offset: 1

Antti Karttunen, Jan 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A001222, A005187, A006530, A156552, A252464, A253560, A323243, A323248.

Cf. also A283475.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A323247(n) = A005187(A156552(n));

a(n) = A005187(A156552(n)).
a(n) = A323243(n) + A323248(n).
For n >= 2, a(A253560(n)) = a(n*A006530(n)) = a(n) + A000225(A001222(n)).
For n >= 1 and k >= 1, a(n*A000040(k+A000720(A006530(n)))) = a(n) + A000225(k+A001222(n)).

A280700 Binary weight of terms of A005187: a(n) = A000120(A005187(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 16 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A000120, A001221, A001222, A005187, A279357, A280705, A283475.

Table[DigitCount[2 n - DigitCount[2 n, 2, 1], 2, 1], {n, 0, 120}] (* Michael De Vlieger, Mar 18 2017 *)

b(n) = if(n<1, 0, b(n\2) + n%2);
for(n=0, 150, print1(b(2*n - b(2*n)), ", ")) \\ Indranil Ghosh, Mar 21 2017

def A(n): return bin(2*n - bin(2*n)[2:].count("1"))[2:].count("1")
print([A(n) for n in range(151)]) # Indranil Ghosh, Mar 21 2017

(define (A280700 n) (A000120 (A005187 n)))

a(n) = A000120(A005187(n)).

a(n) = A001221(A283475(n)) = A001222(A283475(n)) = A001222(A280705(n)).

A283478 a(n) = A097248(A108951(n)).

Original entry on oeis.org

1, 2, 6, 3, 30, 5, 210, 6, 15, 7, 2310, 10, 30030, 11, 21, 5, 510510, 30, 9699690, 14, 33, 13, 223092870, 15, 105, 17, 14, 22, 6469693230, 42, 200560490130, 10, 39, 19, 165, 7, 7420738134810, 23, 51, 21, 304250263527210, 66, 13082761331670030, 26, 70, 29, 614889782588491410, 30, 1155, 210, 57, 34, 32589158477190044730, 21, 195, 33, 69, 31

Offset: 1

Antti Karttunen, Mar 16 2017

nonn

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

Cf. A019565, A097248, A108951, A283475.

Table[FixedPoint[Times @@ Map[#1^#2 & @@ # &, Partition[#, 2, 2] &@ Flatten[FactorInteger[#] /. {p_, e_} /; e >= 2 :> {If[OddQ@ e, {p, 1}, {1, 1}], {NextPrime@ p, Floor[e/2]}}]] &, #] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]], {n, 58}] (* Michael De Vlieger, Mar 18 2017 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From Charles R Greathouse IV, Jun 28 2015
A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
A283478(n) = A097248(A108951(n));

from sympy import primerange, factorint, nextprime
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a108951(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
def a097246(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(nextprime(i)**int(f[i]/2))*(i**(f[i]%2)) for i in f])
def a097248(n):
    k=a097246(n)
    while k!=n:
        n=k
        k=a097246(k)
    return k
def a(n): return a097248(a108951(n)) # Indranil Ghosh, May 15 2017

(define (A283478 n) (A097248 (A108951 n)))

a(n) = A097248(A108951(n)).
Other identities:
For all n >= 0, a(A019565(n)) = A283475(n).

cp's OEIS Frontend

A280705 a(n) = A002110(A280700(n)) = A046523(A283475(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A019565 The squarefree numbers ordered lexicographically by their prime factorization (with factors written in decreasing order). a(n) = Product_{k in I} prime(k+1), where I is the set of indices of nonzero binary digits in n = Sum_{k in I} 2^k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Scheme

Formula

Extensions

A283477 If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

Scheme

Formula

Extensions

A097248 a(n) is the eventual stable point reached when iterating k -> A097246(k), starting from k = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

Formula

Extensions

A302033 a(n) = A019565(A003188(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A284003 a(n) = A007913(A283477(n)) = A019565(A006068(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI