A284003 -id:A284003 - cp's OEIS frontend

A284004 a(n) = A046523(A284003(n)).

1, 2, 6, 2, 30, 6, 2, 6, 210, 30, 6, 30, 2, 6, 30, 6, 2310, 210, 30, 210, 6, 30, 210, 30, 2, 6, 30, 6, 210, 30, 6, 30, 30030, 2310, 210, 2310, 30, 210, 2310, 210, 6, 30, 210, 30, 2310, 210, 30, 210, 2, 6, 30, 6, 210, 30, 6, 30, 2310, 210, 30, 210, 6, 30, 210, 30, 510510, 30030, 2310, 30030, 210, 2310, 30030, 2310, 30, 210, 2310, 210, 30030, 2310, 210, 2310

Offset: 0

Antti Karttunen, Mar 18 2017

nonn

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

Cf. A001222, A002110, A046523, A209281, A284003.

Table[Times @@ MapIndexed[Prime[First@ #2]^#1 &, Reverse@ Sort@ FactorInteger[#][[All, -1]]] - Boole[# == 1] &@ 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 *)

\\ Code for A284003 given under that entry.
A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
A284004(n) = A046523(A284003(n));

(define (A284004 n) (A046523 (A284003 n)))

a(n) = A046523(A284003(n)).

a(n) = A002110(A001222(A284003(n))) = A002110(A209281(n+1)). [Latter so far only conjectured.]

A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 14 1999

nonn

The original motivation for this sequence was to encode the prime factorization of n in the binary representation of a(n), each such representation being unique as long as this map is restricted to A005117 (squarefree numbers, resulting a permutation of nonnegative integers A048672) or any of its subsequence, resulting an injective function like A048623 and A048639.
However, also the restriction to A260443 (not all terms of which are squarefree) results a permutation of nonnegative integers, namely A001477, the identity permutation.
When a polynomial with nonnegative integer coefficients is encoded with the prime factorization of n (e.g., as in A206296, A260443), then a(n) gives the evaluation of that polynomial at x=2.
The primitive completely additive integer sequence that satisfies a(n) = a(A225546(n)), n >= 1. By primitive, we mean that if b is another such sequence, then there is an integer k such that b(n) = k * a(n) for all n >= 1. - Peter Munn, Feb 03 2020
If the binary rank of an integer partition y is given by Sum_i 2^(y_i-1), and the Heinz number is Product_i prime(y_i), then a(n) is the binary rank of the integer partition with Heinz number n. Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices), and the function taking a multiset m to Product_i prime(m_i) is the inverse of A112798 (prime indices). - Gus Wiseman, May 22 2024

			From _Gus Wiseman_, May 22 2024: (Start)
The A018819(7) = 6 cases of binary rank 7 are the following, together with their prime indices:
   30: {1,2,3}
   40: {1,1,1,3}
   54: {1,2,2,2}
   72: {1,1,1,2,2}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
(End)

Antti Karttunen, Table of n, a(n) for n = 1..1024
Hans Havermann, Factorization of the first 10000 terms, in format [[primes], [exponents]]

Row 2 of A104244.
Similar logarithmic functions: A001414, A056239, A090880, A289506, A293447.
Left inverse of the following sequences: A000079, A019565, A038754, A068911, A134683, A260443, A332824.
A003961, A028234, A032742, A055396, A064989, A067029, A225546, A297845 are used to express relationship between terms of this sequence.
Cf. also A048623, A048676, A099884, A277896 and tables A277905, A285325.
Cf. A297108 (Möbius transform), A332813 and A332823 [= a(n) mod 3].
Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000203,A331750), (A005940,A087808), (A007913,A248663), (A007947,A087207), (A097248,A048675), (A206296,A000129), (A248692,A056239), (A283477,A005187), (A284003,A006068), (A285101,A028362), (A285102,A068052), (A293214,A001065), (A318834,A051953), (A319991,A293897), (A319992,A293898), (A320017,A318674), (A329352,A069359), (A332461,A156552), (A332462,A156552), (A332825,A000010) and apparently (A163511,A135529).
See comments/formulas in A277333, A331591, A331740 giving their relationship to this sequence.
The formula section details how the sequence maps the terms of A329050, A329332.
A277892, A322812, A322869, A324573, A324575 give properties of the n-th term of this sequence.
The term k appears A018819(k) times.
The inverse transformation is A019565 (Heinz number of binary indices).
The version for distinct prime indices is A087207.
Numbers k such that a(k) is prime are A277319, counts A372688.
Grouping by image gives A277905.
A014499 lists binary indices of prime numbers.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A000720, A035100, A071814, A096111, A225620, A243055, A304818, A355536, A358136, A372890.
Cf. A087207, A097248.

nthprime := proc(n) local i; if(isprime(n)) then for i from 1 to 1000000 do if(ithprime(i) = n) then RETURN(i); fi; od; else RETURN(0); fi; end; # nthprime(2) = 1, nthprime(3) = 2, nthprime(5) = 3, etc. - this is also A049084.
A048675 := proc(n) local s,d; s := 0; for d in ifactors(n)[ 2 ] do s := s + d[ 2 ]*(2^(nthprime(d[ 1 ])-1)); od; RETURN(s); end;
# simpler alternative
f:= n -> add(2^(numtheory:-pi(t[1])-1)*t[2], t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Oct 10 2016

a[1] = 0; a[n_] := Total[ #[[2]]*2^(PrimePi[#[[1]]]-1)& /@ FactorInteger[n] ]; Array[a, 100] (* Jean-François Alcover, Mar 15 2016 *)

a(n) = my(f = factor(n)); sum(k=1, #f~, f[k,2]*2^primepi(f[k,1]))/2; \\ Michel Marcus, Oct 10 2016

\\ The following program reconstructs terms (e.g. for checking purposes) from the factorization file prepared by Hans Havermann:
v048675sigs = readvec("a048675.txt");
A048675(n) = if(n<=2,n-1,my(prsig=v048675sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,ps[i]^es[i])); \\ Antti Karttunen, Feb 02 2020

from sympy import factorint, primepi
def a(n):
    if n==1: return 0
    f=factorint(n)
    return sum([f[i]*2**(primepi(i) - 1) for i in f])
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jun 19 2017

a(1) = 0, a(n) = 1/2 * (e1*2^i1 + e2*2^i2 + ... + ez*2^iz) if n = p_{i1}^e1*p_{i2}^e2*...*p_{iz}^ez, where p_i is the i-th prime. (e.g. p_1 = 2, p_2 = 3).
Totally additive with a(p^e) = e * 2^(PrimePi(p)-1), where PrimePi(n) = A000720(n). [Missing factor e added to the comment by Antti Karttunen, Jul 29 2015]
From Antti Karttunen, Jul 29 2015: (Start)
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A032742(n)). [Where A055396(n) gives the index of the smallest prime dividing n and A032742(n) gives the largest proper divisor of n.]
a(1) = 0; for n > 1, a(n) = (A067029(n) * (2^(A055396(n)-1))) + a(A028234(n)).
Other identities. For all n >= 0:
a(A019565(n)) = n.
a(A260443(n)) = n.
a(A206296(n)) = A000129(n).
a(A005940(n+1)) = A087808(n).
a(A007913(n)) = A248663(n).
a(A007947(n)) = A087207(n).
a(A283477(n)) = A005187(n).
a(A284003(n)) = A006068(n).
a(A285101(n)) = A028362(1+n).
a(A285102(n)) = A068052(n).
Also, it seems that a(A163511(n)) = A135529(n) for n >= 1. (End)
a(1) = 0, a(2n) = 1+a(n), a(2n+1) = 2*a(A064989(2n+1)). - Antti Karttunen, Oct 11 2016
From Peter Munn, Jan 31 2020: (Start)
a(n^2) = a(A003961(n)) = 2 * a(n).
a(A297845(n,k)) = a(n) * a(k).
a(n) = a(A225546(n)).
a(A329332(n,k)) = n * k.
a(A329050(n,k)) = 2^(n+k).
(End)
From Antti Karttunen, Feb 02-25 2020, Feb 01 2021: (Start)
a(n) = Sum_{d|n} A297108(d) = Sum_{d|A225546(n)} A297108(d).
a(n) = a(A097248(n)).
For n >= 2:
A001221(a(n)) = A322812(n), A001222(a(n)) = A277892(n).
A000203(a(n)) = A324573(n), A033879(a(n)) = A324575(n).
For n >= 1, A331750(n) = a(A000203(n)).
For n >= 1, the following chains hold:
A293447(n) >= a(n) >= A331740(n) >= A331591(n).
a(n) >= A087207(n) >= A248663(n).
(End)
a(n) = A087207(A097248(n)). - Flávio V. Fernandes, Jul 16 2025

Entry revised by Antti Karttunen, Jul 29 2015

More linking formulas added by Antti Karttunen, Apr 18 2017

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

A302783 A divisor-or-multiple permutation of natural numbers: a(n) = A052330(A006068(n)).

Original entry on oeis.org

1, 2, 6, 3, 24, 12, 4, 8, 120, 60, 20, 40, 5, 10, 30, 15, 840, 420, 140, 280, 35, 70, 210, 105, 7, 14, 42, 21, 168, 84, 28, 56, 7560, 3780, 1260, 2520, 315, 630, 1890, 945, 63, 126, 378, 189, 1512, 756, 252, 504, 9, 18, 54, 27, 216, 108, 36, 72, 1080, 540, 180, 360, 45, 90, 270, 135, 83160, 41580, 13860, 27720, 3465, 6930, 20790, 10395, 693

Offset: 0

Antti Karttunen, Apr 16 2018

nonn

Shares with A064736, A207901, A302781, A302350, etc. a property that a(n) is always either a divisor or a multiple of a(n+1). However, because multiple bits may change simultaneously when moving from A006068(n) to A006068(n+1) [with the restriction that the changing bits are all either toggled on or all toggled off], it means that also here the terms might be divided or multiplied by more than just a single Fermi-Dirac prime (A050376). E.g. a(3) = 3, while a(4) = A050376(1) * A050376(3) * 3 = 2*4*3 = 24. See also comments in A284003.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Cf. A302784 (inverse).
Cf. A006068, A050376, A052330, A302777.
Cf. also A207901 and A284003 (a squarefree analog).

up_to_e = 13;
v050376 = vector(up_to_e);
A050376(n) = v050376[n];
A209229(n) = (n && !bitand(n,n-1));
A302777(n) = A209229(isprimepower(n));
i = 0; for(n=1,oo,if(A302777(n), i++; v050376[i] = n); if(i == up_to_e,break));
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
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
A302783(n) = A052330(A006068(n));

a(n) = A052330(A006068(n)).

a(n) = A207901(A064707(n)).

A303771 Divisor-or-multiple permutation of natural numbers, "Fermi-Dirac piano played with May code": a(n) = A052330(A303767(n)).

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 8, 24, 120, 5, 10, 30, 15, 60, 20, 40, 280, 7, 14, 42, 21, 84, 28, 56, 168, 840, 35, 70, 210, 105, 420, 140, 1260, 9, 18, 54, 27, 108, 36, 72, 216, 1080, 45, 90, 270, 135, 540, 180, 360, 2520, 63, 126, 378, 189, 756, 252, 504, 1512, 7560, 315, 630, 1890, 945, 3780, 41580, 11, 22, 66, 33, 132, 44, 88, 264, 1320, 55, 110, 330, 165, 660, 220

Offset: 0

Antti Karttunen, May 02 2018

nonn

Consider A019565. Imagine that it is an automatic piano that "plays sequences" when an appropriate punched "tape" is fed to it (as its input), i.e., when it is composed from the right with an appropriate sequence p, as A019565(p(n)). The 1-bits in the binary expansion of each p(n) are the "holes" in the tape, and they determine which "tunes" are present on beat n. The "tunes" are actually primes that are multiplied together. Of course only "squarefree music" (sequences containing only squarefree numbers, A005117) is possible to generate this way, thus we call A019565 a "squarefree piano".
There is a more sophisticated instrument, called "Fermi-Dirac piano" (A052330), with which it is possible to produce sequences that may contain any numbers.
If the tape is constructed in such a way that between the successive beats (when moving from p(n) to p(n+1)), either a subset of 0-bits are toggled on (changed to 1's), or a subset of 1-bits are toggled off (changed to 0's), but no both kind of changes occur simultaneously, then when fed as an input to either of these pianos, the resulting sequence "s" (the output) is guaranteed to satisfy the condition that s(n+1) is either a multiple or a divisor of s(n). For example, Gray code A003188 and its inverse A006068 are examples of such tapes, and they produce sequences A302033, A207901 and A284003, A302783.
This divisor-or-multiple permutation is obtained by playing "Fermi-Dirac piano" with the same tape which yields A303760 when "squarefree piano" is played with it. Note that A303760 is not a subsequence of this sequence, as its terms occur in different order than the squarefree terms here.
See also Peter Munn's Apr 11 2018 message on SeqFan-mailing list and comments in A304537.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Michel Marcus, Peter Munn, et al., Discussion on SeqFan-list about similar permutations, April 2018
Index entries for sequences that are permutations of the natural numbers

Cf. A303772 (inverse).
Cf. A019565, A048675, A052330, A303760.
Cf. also A064736, A113552, A207901, A281978, A282291, A302350, A302781, A302783, A303751, A304085, A304531 for similar permutations.

default(parisizemax,2^31);
up_to_e = 16;
up_to = (1 + 2^up_to_e);
v050376 = vector(2+up_to_e);
A050376(n) = v050376[n];
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == 2+up_to_e,break));
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A053669(n) = forprime(p=2, , if (n % p, return(p))); \\ From A053669
v303760 = vector(up_to);
m_inverses = Map();
prev=1; for(n=1,up_to,fordiv(prev,d,if(!mapisdefined(m_inverses,d),v303760[n] = d;mapput(m_inverses,d,n);break)); if(!v303760[n], apu = prev; while(mapisdefined(m_inverses,try = prev*A053669(apu)), apu *= A053669(apu)); v303760[n] = try; mapput(m_inverses,try,n)); prev = v303760[n]);
A303760(n) = v303760[n+1];
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A303771(n) = A052330(A048675(A303760(n)));

a(n) = A052330(A303767(n)) = A052330(A048675(A303760(n))). [See comments].

Name amended by Antti Karttunen, May 16 2018

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

A304537 Suspected divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A304533(n)).

Original entry on oeis.org

1, 2, 6, 3, 15, 5, 65, 13, 26, 182, 7, 14, 42, 21, 105, 35, 455, 91, 910, 10, 30, 210, 70, 2730, 39, 78, 546, 273, 1365, 195, 7995, 41, 82, 246, 123, 615, 205, 2665, 533, 1066, 11726, 11, 22, 66, 33, 165, 55, 715, 143, 286, 2002, 77, 154, 462, 231, 1155, 385, 5005, 1001, 10010, 110, 330, 2310, 770, 30030, 429, 858, 6006, 3003, 15015, 2145, 87945, 451, 902

Offset: 0

Antti Karttunen, May 15 2018

nonn

Each a(n) is always either a divisor or a multiple of a(n+1).
Consider A052330. Imagine that it is an automatic piano that "plays sequences" when an appropriate punched "tape" is fed to it (as its input), i.e., when it is composed from the right with an appropriate sequence p, as A019565(p(n)). The 1-bits in the binary expansion of each p(n) are the "holes" in the tape, and they determine which "tunes" are present on beat n. The "tunes" are actually "Fermi-Dirac primes" (A050376) that are multiplied together.
If the tape is constructed in such a way that between the successive beats (when moving from p(n) to p(n+1)), either a subset of 0-bits are toggled on (changed to 1's), or a subset of 1-bits are toggled off (changed to 0's), but no both kind of changes occur simultaneously, then when fed as an input to this piano, the resulting sequence "s" (the output) is guaranteed to satisfy the condition that s(n+1) is either a multiple or a divisor of s(n). Furthermore, if the given sequence p is itself a permutation of natural numbers, then also the produced sequence is. For example, Gray code A003188 and its inverse A006068 are such sequences, and when given as an "input tape" for A052330, they produce permutations A207901 and A302783.
There is a simpler instrument, called "squarefree piano" (A019565), with which it is possible to produce similar divisor-or-multiple sequences, but that contain only squarefree numbers. Given A003188 or A006068 as an input tape for it produces correspondingly sequences A302033 and A284003.
This sequence is obtained by playing "squarefree piano" with the same tape which yields A304531 when "Fermi-Dirac piano" is played with it. However, in this case the sequence A304531 is produced by a greedy algorithm, and thus its tape (A304533) is actually a back-formation, obtained from the "music" (A304531) by applying "tape-recorder" (A052331) to it. Note that this in not a subsequence of A304531, as the terms occur in different order than the squarefree terms of A304531.
See also Peter Munn's Apr 11 2018 message on SeqFan-mailing list.

Antti Karttunen, Table of n, a(n) for n = 0..8447
Michel Marcus, Peter Munn, et al, Discussion on SeqFan-list, April 2018

Cf. A019565, A304531, A304533.

Cf. also A303760, A303771.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A304533(n) = A052331(A304531(1+n));
A304537(n) = A019565(A304533(n));

a(n) = A019565(A304533(n)) = A019565(A052331(A304531(1+n))).

cp's OEIS Frontend

A284004 a(n) = A046523(A284003(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

Formula

A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

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

A302783 A divisor-or-multiple permutation of natural numbers: a(n) = A052330(A006068(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A303771 Divisor-or-multiple permutation of natural numbers, "Fermi-Dirac piano played with May code": a(n) = A052330(A303767(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A304537 Suspected divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A304533(n)).

Views

Author

Keywords