A283477 -id:A283477 - cp's OEIS frontend

A276150 Sum of digits when n is written in primorial base (A049345); minimal number of primorials (A002110) that add to n.

0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 4

Offset: 0

Antti Karttunen, Aug 22 2016

The sum of digits of n in primorial base is odd if n is 1 or 2 (mod 4) and even if n is 0 or 3 (mod 4). Proof: primorials are 1 or 2 (mod 4) and a(n) can be constructed via the greedy algorithm. So if n = 4k + r where 0 <= r < 4, 4k needs an even number of primorials and r needs hammingweight(r) = A000120(r) primorials. Q.E.D. - David A. Corneth, Feb 27 2019

			For n=24, which is "400" in primorial base (as 24 = 4*(3*2*1) + 0*(2*1) + 0*1, see A049345), the sum of digits is 4, thus a(24) = 4.

Antti Karttunen, Table of n, a(n) for n = 0..30030
Index entries for sequences related to primorial base

Cf. A000120, A001222, A002110, A049345, A053589, A235168, A260188, A267263, A276084, A276086, A276151, A277022, A278226, A283477, A319713, A319715 (inverse Möbius transform), A321683, A324342, A324382, A324383, A324386, A324387, A371091, A373605, A373606, A373607.
Cf. A333426 [k such that a(k)|k], A339215 [numbers not of the form x+a(x) for any x], A358977 [k such that gcd(k, a(k)) = 1].
Cf. A338835, A366693.
Cf. A014601, A042963 (positions of even and odd terms), A343048 (positions of records).
Differs from analogous A034968 for the first time at n=24.

nn = 120; b = MixedRadix[Reverse@ Prime@ NestWhileList[# + 1 &, 1, Times @@ Prime@ Range[# + 1] <= nn &]]; Table[Total@ IntegerDigits[n, b], {n, 0, nn}] (* Version 10.2, or *)
nn = 120; f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Total@ f@ n, {n, 0, 120}] (* Michael De Vlieger, Aug 26 2016 *)

A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); }; \\ Antti Karttunen, Feb 27 2019

from sympy import prime, primefactors
def Omega(n): return 0 if n==1 else Omega(n//primefactors(n)[0]) + 1
def a276086(n):
    i=0
    m=pr=1
    while n>0:
        i+=1
        N=prime(i)*pr
        if n%N!=0:
            m*=(prime(i)**((n%N)/pr))
            n-=n%N
        pr=N
    return m
def a(n): return Omega(a276086(n))
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 23 2017

a(n) = 1 + a(A276151(n)) = 1 + a(n-A002110(A276084(n))), a(0) = 0.
or for n >= 1: a(n) = 1 + a(n-A260188(n)).
Other identities and observations. For all n >= 0:
a(n) = A001222(A276086(n)) = A001222(A278226(n)).
a(n) >= A371091(n) >= A267263(n).
From Antti Karttunen, Feb 27 2019: (Start)
a(n) = A000120(A277022(n)).
a(A283477(n)) = A324342(n).
(End)
a(n) = A373606(n) + A373607(n). - Antti Karttunen, Jun 19 2024

A052330 Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k; then the numbers b_k*S_k are the next 2^k terms.

Original entry on oeis.org

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

Offset: 0

Christian G. Bower, Dec 15 1999

Inverse of sequence A064358 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001
This sequence is not exactly a permutation because it has offset 0 but doesn't contain 0. A052331 is its exact inverse, which has offset 1 and contains 0. See also A064358.
Are there any other values of n besides 4 and 36 with a(n) = n? - Thomas Ordowski, Apr 01 2005
4 = 100 = 4^1 * 3^0 * 2^0, 36 = 100100 = 9^1 * 7^0 * 5^0 * 4^1 * 3^0 * 2^0. - Thomas Ordowski, May 26 2005
Ordering of positive integers by increasing "Fermi-Dirac representation", which is a representation of the "Fermi-Dirac factorization", term implying that each prime power with a power of two as exponent may appear at most once in the "Fermi-Dirac factorization" of n. (Cf. comment in A050376; see also the OEIS Wiki page.) - Daniel Forgues, Feb 11 2011
The subsequence consisting of the squarefree terms is A019565. - Peter Munn, Mar 28 2018
Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH-number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k). A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. Then a(n) is the number whose binary indices are the parts of the strict integer partition with FDH-number n. - Gus Wiseman, Aug 19 2019
The set of indices of odd-valued terms has asymptotic density 0. In this sense (using the order they appear in this permutation) 100% of numbers are even. - Peter Munn, Aug 26 2019

			Terms following 5 are 10, 15, 30, 20, 40, 60, 120; this is followed by 7 as 6 has already occurred. - _Philippe Deléham_, Jun 03 2015
From _Antti Karttunen_, Apr 13 2018, after also _Philippe Deléham_'s Jun 03 2015 example: (Start)
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 A300841(k), and each right hand child contains 2*A300841(k), when their parent contains k:
                                     1
                                     |
                  ...................2...................
                 3                                       6
       4......../ \........8                  12......../ \........24
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   5       10         15       30         20       40         60      120
  7 14   21  42     28  56   84  168    35  70  105  210   140 280  420 840
  etc.
Compare also to trees like A005940 and A283477, and sequences A207901 and A302783.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..8191 (Terms 0..1023 from T. D. Noe)
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..16383, with a color function representing primes in red, perfect powers of primes in gold, squarefree composites in greens, and numbers neither squarefree nor prime powers in blue or purple. Purple represents powerful numbers that are not prime powers. This is a 15 level version of Karttunen's diagram in the example.
OEIS Wiki, Ordering of positive integers by increasing "Fermi-Dirac representation"
Index entries for sequences that are permutations of the natural numbers

Subsequences: A019565 (squarefree terms), A050376 (the left edge from 2 onward), A336882 (odd terms).
Cf. A050030, A052331 (inverse), A096111, A096113, A096114, A096115, A096116, A096118, A096119, A182979, A207901, A300841, A302023, A302783.
Cf. A000120, A029931, A048793, A064547, A070939, A213925, A299755, A299757, A327041.

a = {1}; Do[a = Join[a, a*Min[Complement[Range[Max[a] + 1], a]]], {n, 1, 6}]; a (* Ivan Neretin, May 09 2015 *)

up_to_e = 13; \\ Good for computing up to n = (2^13)-1
v050376 = vector(up_to_e);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); }; \\ Antti Karttunen, Apr 12 2018

a(0)=1; a(n+2^k)=a(n)*b(k) for n < 2^k, k = 0, 1, ... where b is A050376. - Thomas Ordowski, Mar 04 2005
The binary representation of n, n = Sum_{i=0..1+floor(log_2(n))} n_i * 2^i, n_i in {0,1}, is taken as the "Fermi-Dirac representation" (A182979) of a(n), a(n) = Product_{i=0..1+floor(log_2(n))} (b_i)^(n_i) where b_i is A050376(i), i.e., the i-th "Fermi-Dirac prime" (prime power with exponent being a power of 2). - Daniel Forgues, Feb 12 2011
From Antti Karttunen, Apr 12 & 17 2018: (Start)
a(0) = 1; a(2n) = A300841(a(n)), a(2n+1) = 2*A300841(a(n)).
a(n) = A207901(A006068(n)) = A302783(A003188(n)) = A302781(A302845(n)).
(End)

Entry revised Mar 17 2005 by N. J. A. Sloane, based on comments from several people, especially David Wasserman and Thomas Ordowski

A072411 LCM of exponents in prime factorization of n, a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 3

Offset: 1

Labos Elemer, Jun 17 2002

The sums of the first 10^k terms, for k = 1, 2, ..., are 14, 168, 1779, 17959, 180665, 1808044, 18084622, 180856637, 1808585068, 18085891506, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.8085... . - Amiram Eldar, Sep 10 2022

			n = 288 = 2*2*2*2*2*3*3; lcm(5,2) = 10; Product(5,2) = 10, max(5,2) = 5;
n = 180 = 2*2*3*3*5; lcm(2,2,1) = 2; Product(2,2,1) = 4; max(2,2,1) = 2; it deviates both from maximum of exponents (A051903, for the first time at n=72), and product of exponents (A005361, for the first time at n=36).
For n = 36 = 2*2*3*3 = 2^2 * 3^2 we have a(36) = lcm(2,2) = 2.
For n = 72 = 2*2*2*3*3 = 2^3 * 3^2 we have a(72) = lcm(2,3) = 6.
For n = 144 = 2^4 * 3^2 we have a(144) = lcm(2,4) = 4.
For n = 360 = 2^3 * 3^2 * 5^1 we have a(360) = lcm(1,2,3) = 6.

Cf. A028234, A067029, A072412-A072414, A273058, A284569, A290103.
Similar sequences: A001222 (sum of exponents), A005361 (product), A051903 (maximal exponent), A051904 (minimal exponent), A052409 (gcd of exponents), A267115 (bitwise-and), A267116 (bitwise-or), A268387 (bitwise-xor).
Cf. also A055092, A060131.
Differs from A290107 for the first time at n=144.
After the initial term, differs from A157754 for the first time at n=360.

Table[LCM @@ Last /@ FactorInteger[n], {n, 2, 100}] (* Ray Chandler, Jan 24 2006 *)

a(n) = lcm(factor(n)[,2]); \\ Michel Marcus, Mar 25 2017

from sympy import lcm, factorint
def a(n):
    l=[]
    f=factorint(n)
    for i in f: l+=[f[i],]
    return lcm(l)
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Mar 25 2017

a(1) = 1; for n > 1, a(n) = lcm(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 09 2016
From Antti Karttunen, Aug 22 2017: (Start)
a(n) = A284569(A156552(n)).
a(n) = A290103(A181819(n)).
a(A289625(n)) = A002322(n).
a(A290095(n)) = A055092(n).
a(A275725(n)) = A060131(n).
a(A260443(n)) = A277326(n).
a(A283477(n)) = A284002(n). (End)

a(1) = 1 prepended and the data section filled up to 120 terms by Antti Karttunen, Aug 09 2016

A248663 Binary encoding of the prime factors of the squarefree part of n.

Original entry on oeis.org

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

Offset: 1

Peter Kagey, Jan 11 2015

The binary digits of a(n) encode the prime factorization of A007913(n), where the i-th digit from the right is 1 if and only if prime(i) divides A007913(n), otherwise 0. - Robert Israel, Jan 12 2015
Old name: a(1) = 0; a(A000040(n)) = 2^(n-1), and a(n*m) = a(n) XOR a(m).
XOR is the bitwise exclusive or operation (A003987).
a(k^2) = 0 for a natural number k.
Equivalently, the i-th binary digit from the right is 1 iff prime(i) divides n an odd number of times, otherwise zero. - Ethan Beihl, Oct 15 2016
When a polynomial with nonnegative integer coefficients is encoded with the prime factorization of n (e.g., as in A206296, A260443, with scheme explained in A206284), then A048675(n) gives the evaluation of that polynomial at x=2. This sequence is otherwise similar, except the polynomial is evaluated over the field GF(2), which implies also that all its coefficients are essentially reduced modulo 2. - Antti Karttunen, Dec 11 2015
Squarefree numbers (A005117) give the positions k where a(k) = A048675(k). - Antti Karttunen, Oct 29 2016
From Peter Munn, Jun 07 2021: (Start)
When we encode polynomials with nonnegative integer coefficients as described by Antti Karttunen above, polynomial addition is represented by integer multiplication, multiplication is represented by A297845(.,.), and this sequence represents a surjective semiring homomorphism to polynomials in GF(2)[x] (encoded as described in A048720). The mapping of addition operations by this homomorphism is part of the sequence definition: "a(n*m) = a(n) XOR a(m)". The mapping of multiplication is given by a(A297845(n, k)) = A048720(a(n), a(k)).
In a related way, A329329 defines a representation of a different set of polynomials as positive integers, namely polynomials in GF(2)[x,y].
Let P_n(x,y) denote the polynomial represented, as in A329329, by n >= 1. If 0 is substituted for y in P_n(x,y), we get a polynomial P'_n(x,y) (in which y does not appear, of course) that is equivalent to a polynomial P'_n(x) in GF(2)[x]. a(n) is the integer encoding of P'_n(x) (described in A048720).
Viewed as above, this sequence represents another surjective homomorphism, a homomorphism between polynomial rings, with A329329(.,.)/A059897(.,.) and A048720(.,.)/A003987(.,.) as the respective ring operations.
a(n) can be composed as a(n) = A048675(A007913(n)) and the effect of the A007913(.) component corresponds to different operations on the respective polynomial domains of the two homomorphisms described above. In the first homomorphism, coefficients are reduced modulo 2; in the second, 0 is substituted for y. This is illustrated in the examples.
(End)

			a(3500) = a(2^2 * 5^3 * 7) = a(2) XOR a(2) XOR a(5) XOR a(5) XOR a(5) XOR a(7) = 1 XOR 1 XOR 4 XOR 4 XOR 4 XOR 8 = 0b0100 XOR 0b1000 = 0b1100 = 12.
From _Peter Munn_, Jun 07 2021: (Start)
The examples in the table below illustrate the homomorphisms (between polynomial structures) represented by this sequence.
The staggering of the rows is to show how the mapping n -> A007913(n) -> A048675(A007913(n)) = a(n) relates to the encoded polynomials, as not all encodings are relevant at each stage.
For an explanation of each polynomial encoding, see the sequence referenced in the relevant column heading. (Note also that A007913 generates squarefree numbers, and with these encodings, all squarefree numbers represent equivalent polynomials in N[x] and GF(2)[x,y].)
                     |<-----    encoded polynomials    ----->|
  n  A007913(n) a(n) |         N[x]    GF(2)[x,y]    GF(2)[x]|
                     |Cf.:  A206284       A329329     A048720|
--------------------------------------------------------------
  24                            x+3         x+y+1
          6                     x+1           x+1
                  3                                       x+1
--------------------------------------------------------------
  36                           2x+2          xy+y
          1                       0             0
                  0                                         0
--------------------------------------------------------------
  60                        x^2+x+2       x^2+x+y
         15                   x^2+x         x^2+x
                  6                                     x^2+x
--------------------------------------------------------------
  90                       x^2+2x+1      x^2+xy+1
         10                   x^2+1         x^2+1
                  5                                     x^2+1
--------------------------------------------------------------
This sequence is a left inverse of A019565. A019565(.) maps a(n) to A007913(n) for all n, effectively reversing the second stage of the mapping from n to a(n) shown above. So, with the encodings used here, A019565(.) represents each of two injective homomorphisms that map polynomials in GF(2)[x] to equivalent polynomials in N[x] and GF(2)[x,y] respectively.
(End)

Peter Kagey, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Squarefree Part
Wikipedia, Polynomial ring
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A000040, A000079, A000720, A005117, A020639, A032742, A048672, A055396, A100112, A206284.
A048675 composed with A007913. A007814 composed with A225546.
A homomorphism from A297845/A003991 and from A329329/A059897 to A048720/A003987, mapping A206296 to A168081, A260443 to A264977, A265408 to A265407, A275734 to A275808, A276076 to A276074, A283477 to A006068.
A left inverse of A019565.
Other sequences used to express relationship between terms of this sequence: A003961, A007913, A331590, A334747.
Cf. also A099884, A277330.
A087207 is the analogous sequence with OR.
A277417 gives the positions where coincides with A277333.
A000290 gives the positions of zeros.

import Data.Bits (xor)
a248663 = foldr (xor) 0 . map (\i -> 2^(i - 1)) . a112798_row
-- Peter Kagey, Sep 16 2016

f:= proc(n)
local F,f;
F:= select(t -> t[2]::odd, ifactors(n)[2]);
add(2^(numtheory:-pi(f[1])-1), f = F)
end proc:
seq(f(i),i=1..100); # Robert Israel, Jan 12 2015

a[1] = 0; a[n_] := a[n] = If[PrimeQ@ n, 2^(PrimePi@ n - 1), BitXor[a[#], a[n/#]] &@ FactorInteger[n][[1, 1]]]; Array[a, 66] (* Michael De Vlieger, Sep 16 2016 *)

A248663(n) = vecsum(apply(p -> 2^(primepi(p)-1),factor(core(n))[,1])); \\ Antti Karttunen, Feb 15 2021

from sympy import factorint, primepi
from sympy.ntheory.factor_ import core
def a048675(n):
    f=factorint(n)
    return 0 if n==1 else sum([f[i]*2**(primepi(i) - 1) for i in f])
def a(n): return a048675(core(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 21 2017

require 'prime'
def f(n)
  a = 0
  reverse_primes = Prime.each(n).to_a.reverse
  reverse_primes.each do |prime|
    a <<= 1
    while n % prime == 0
      n /= prime
      a ^= 1
    end
  end
  a
end
(Scheme, with memoizing-macro definec)
(definec (A248663 n) (cond ((= 1 n) 0) ((= 1 (A010051 n)) (A000079 (- (A000720 n) 1))) (else (A003987bi (A248663 (A020639 n)) (A248663 (A032742 n)))))) ;; Where A003987bi computes bitwise-XOR as in A003987.
;; Alternatively:
(definec (A248663 n) (cond ((= 1 n) 0) (else (A003987bi (A000079 (- (A055396 n) 1)) (A248663 (A032742 n))))))
;; Antti Karttunen, Dec 11 2015

a(1) = 0; for n > 1, if n is a prime, a(n) = 2^(A000720(n)-1), otherwise a(A020639(n)) XOR a(A032742(n)). [After the definition.] - Antti Karttunen, Dec 11 2015
For n > 1, this simplifies to: a(n) = 2^(A055396(n)-1) XOR 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. Cf. a similar formula for A048675.]
Other identities and observations. For all n >= 0:
a(n) = A048672(A100112(A007913(n))). - Peter Kagey, Dec 10 2015
From Antti Karttunen, Dec 11 2015, Sep 19 & Oct 27 2016, Feb 15 2021: (Start)
a(n) = a(A007913(n)). [The result depends only on the squarefree part of n.]
a(n) = A048675(A007913(n)).
a(A206296(n)) = A168081(n).
a(A260443(n)) = A264977(n).
a(A265408(n)) = A265407(n).
a(A275734(n)) = A275808(n).
a(A276076(n)) = A276074(n).
a(A283477(n)) = A006068(n).
(End)
From Peter Munn, Jan 09 2021 and Apr 20 2021: (Start)
a(n) = A007814(A225546(n)).
a(A019565(n)) = n; A019565(a(n)) = A007913(n).
a(A003961(n)) = 2 * a(n).
a(A297845(n, k)) = A048720(a(n), a(k)).
a(A329329(n, k)) = A048720(a(n), a(k)).
a(A059897(n, k)) = A003987(a(n), a(k)).
a(A331590(n, k)) = a(n) + a(k).
a(A334747(n)) = a(n) + 1.
(End)

New name from Peter Munn, Nov 01 2023

A129912 Numbers that are products of distinct primorial numbers (see A002110).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 180, 210, 360, 420, 1260, 2310, 2520, 4620, 6300, 12600, 13860, 27720, 30030, 37800, 60060, 69300, 75600, 138600, 180180, 360360, 415800, 485100, 510510, 831600, 900900, 970200, 1021020, 1801800, 2910600, 3063060, 5405400

Offset: 1

Bill McEachen, Jun 05 2007, Jun 06 2007, Jul 06 2007, Aug 07 2007

Conjecture: every odd prime p is either adjacent to a term of A129912 or a prime distance q from some term of A129912, where q < p. - Bill McEachen, Jun 03 2010, edited for clarity in Feb 26 2019

The first 2^20 terms k > 2 of A283477 all satisfy also the condition that the differences k-A151799(k) and A151800(k)-k are always either 1 or prime, like is also conjectured to hold for A002182 (cf. also the conjecture given in A117825). However, for A025487, which is a supersequence of both sequences, this is not always true: 512 is a member of A025487, but A151800(512) = 521, with 521 - 512 = 9, which is a composite number. - Antti Karttunen, Feb 26 2019

			For s = 4 there are 8 (generally 2^(s-1)) such numbers: 210 = 2*3*5*7, 420 = 2^2*3*5*7 = (2*3*5*7)*2, 1260 = 2^2*3^2*5*7 = (2*3*5*7)*(2*3), 6300 = 2^2*3^2*5^2*7 = (2*3*5*7)*(2*3*5), 2520 = 2^3*3^2*5*7 = (2*3*5*7)*(2*3)*2, 12600 = 2^3*3^2*5^2*7 = (2*3*5*7)*(2*3*5)*2, 37800 = 2^3*3^3*5^2*7 = (2*3*5*7)*(2*3*5)*(2*3), 75600 = 2^4*3^3*5^2*7 = (2*3*5*7)*(2*3*5)*(2*3)*2.

CRC Standard Mathematical Tables, 28th Ed., CRC Press

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Bill McEachen, Normalized A129912.
Robert Potter, Primorial Conjecture.
John Sokol, Sokol's Prime Conjecture, 2002.
Wikipedia, Primorial.
Index entries for sequences related to primorial numbers.

Subsequence of A025487. Sequence A283477 sorted into ascending order.

Cf. A002110, A117825, A151799, A151800.

Clear[f]; f[m_] := f[m] = Union[Times @@@ Subsets[FoldList[Times, 1, Prime[Range[m]]]]][[1 ;; 100]]; f[10]; f[m = 11]; While[f[m] != f[m-1], m++]; f[m] (* Jean-François Alcover, Mar 03 2014 *) (* or *)
pr[n_] := Product[Prime[n + 1 - i]^i, {i, n}]; upto[mx_] := Block[{ric, j = 1}, ric[n_, ip_, ex_] := If[n < mx, Block[{p = Prime[ip + 1]}, If[ex == 1, Sow@ n]; ric[n p^ex, ip + 1, ex]; If[ex > 1, ric[n p^(ex - 1), ip + 1, ex - 1]]]]; Sort@ Reap[ Sow[1]; While[pr[j] < mx, ric[2^j, 1, j]; j++]][[2, 1]]];
upto[10^30] (* faster, Giovanni Resta, Apr 02 2017 *)

is(n)=my(o=valuation(n,2),t); if(o<1||n<2, return(n==1)); n>>=o; forprime(p=3,, t=valuation(n,p); n/=p^t; if(t>o || tCharles R Greathouse IV, Oct 22 2015

Apart from 1 and 2, numbers of the form 2^k(1)*3^k(2)*5^k(3)*...*p(s)^k(s), where p(s) is s-th prime, k(i)>0 for i=1..s, k(i)-k(i-1) = 0 or 1 for i=2..s and |{k(1),k(2),..,k(s)}|=k(1). - Vladeta Jovovic, Jun 14 2007

Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + 1/A002110(n)) = 1.8177952875... . - Amiram Eldar, Jun 03 2023

Edited by N. J. A. Sloane, Jun 09 2007, Aug 08 2007
I corrected the Potter link to reflect its relocation. - Bill McEachen, Sep 12 2009
I added link to Wikicommons image. - Bill McEachen, Sep 16 2009
I again corrected the Potter link for its relocation - Bill McEachen, May 30 2013

A329900 Primorial deflation of n: starting from x = n, repeatedly divide x by the largest primorial A002110(k) that divides it, until x is an odd number. Then a(n) = Product prime(k_i), for primorial indices k_1 >= k_2 >= ..., encountered in the process.

Original entry on oeis.org

1, 2, 1, 4, 1, 3, 1, 8, 1, 2, 1, 6, 1, 2, 1, 16, 1, 3, 1, 4, 1, 2, 1, 12, 1, 2, 1, 4, 1, 5, 1, 32, 1, 2, 1, 9, 1, 2, 1, 8, 1, 3, 1, 4, 1, 2, 1, 24, 1, 2, 1, 4, 1, 3, 1, 8, 1, 2, 1, 10, 1, 2, 1, 64, 1, 3, 1, 4, 1, 2, 1, 18, 1, 2, 1, 4, 1, 3, 1, 16, 1, 2, 1, 6, 1, 2, 1, 8, 1, 5, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 3, 1, 8, 1

Offset: 1

Antti Karttunen, Dec 22 2019

nonn

When applied to arbitrary n, the "primorial deflation" (term coined by Matthew Vandermast in A181815) induces the splitting of n to two factors A328478(n)*A328479(n) = n, where we call A328478(n) the non-deflatable component of n (which is essentially discarded), while A328479(n) is the deflatable component. Only if n is in A025487, then the entire n is deflatable, i.e., A328478(n) = 1 and A328479(n) = n.

According to Daniel Suteu, also the ratio (A319626(n) / A319627(n)) can be viewed as a "primorial deflation". That definition coincides with this one when restricted to terms of A025487, as for all k in A025487, A319626(k) = a(k), and A319627(k) = 1. - Antti Karttunen, Dec 29 2019

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

A left inverse of A108951. Coincides with A319626 on A025487.

Cf. A002110, A002182, A111701, A181815, A181817, A181819, A181821, A276084, A304886, A319626, A319627, A328478, A328479, A329889, A329902, A330685, A330686, A330689.

Array[If[OddQ@ #, 1, Times @@ Prime@ # &@ Rest@ NestWhile[Append[#1, {#3, Drop[#, -LengthWhile[Reverse@ #, # == 0 &]] &[#2 - PadRight[ConstantArray[1, #3], Length@ #2]]}] & @@ {#1, #2, LengthWhile[#2, # > 0 &]} & @@ {#, #[[-1, -1]]} &, {{0, TakeWhile[If[# == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #], # > 0 &]}}, And[FreeQ[#[[-1, -1]], 0], Length[#[[-1, -1]] ] != 0] &][[All, 1]] ] &, 105] (* Michael De Vlieger, Dec 28 2019 *)
Array[Times @@ Prime@(TakeWhile[Reap[FixedPointList[Block[{k = 1}, While[Mod[#, Prime@ k] == 0, k++]; Sow[k - 1]; #/Product[Prime@ i, {i, k - 1}]] &, #]][[-1, 1]], # > 0 &]) &, 105] (* Michael De Vlieger, Jan 11 2020 *)

A329900(n) = { my(m=1, pp=1); while(1, forprime(p=2, ,if(n%p, if(2==p, return(m), break), n /= p; pp = p)); m *= pp); (m); };

A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A276084(n) = { for(i=1,oo,if(n%prime(i),return(i-1))); }
A329900(n) = if(n%2,1,prime(A276084(n))*A329900(A111701(n)));

For odd n, a(n) = 1, for even n, a(n) = A000040(A276084(n)) * a(A111701(n)).
For even n, a(n) = A000040(A276084(n)) * a(n/A002110(A276084(n))).
A108951(a(n)) = A328479(n), for n >= 1.
a(A108951(n)) = n, for n >= 1.
a(A328479(n)) = a(n),  for n >= 1.
a(A328478(n)) = 1, for n >= 1.
a(A002110(n)) = A000040(n), for n >= 1.
a(A000142(n)) = A307035(n), for n >= 0.
a(A283477(n)) = A019565(n), for n >= 0.
a(A329886(n)) = A005940(1+n), for n >= 0.
a(A329887(n)) = A163511(n), for n >= 0.
a(A329602(n)) = A329888(n), for n >= 1.
a(A025487(n)) = A181815(n), for n >= 1.
a(A124859(n)) = A181819(n), for n >= 1.
a(A181817(n)) = A025487(n), for n >= 1.
a(A181821(n)) = A122111(n), for n >= 1.
a(A002182(n)) = A329902(n), for n >= 1.
a(A260633(n)) = A329889(n), for n >= 1.
a(A033833(n)) = A330685(n), for n >= 1.
a(A307866(1+n)) = A330686(n), for n >= 1.
a(A330687(n)) = A330689(n), for n >= 1.

A283980 a(n) = A003961(n)*A006519(n).

Original entry on oeis.org

1, 6, 5, 36, 7, 30, 11, 216, 25, 42, 13, 180, 17, 66, 35, 1296, 19, 150, 23, 252, 55, 78, 29, 1080, 49, 102, 125, 396, 31, 210, 37, 7776, 65, 114, 77, 900, 41, 138, 85, 1512, 43, 330, 47, 468, 175, 174, 53, 6480, 121, 294, 95, 612, 59, 750, 91, 2376, 115, 186, 61, 1260, 67, 222, 275, 46656, 119, 390, 71, 684, 145, 462, 73, 5400, 79, 246, 245

Offset: 1

Antti Karttunen, Mar 19 2017

Completely multiplicative since both A003961 and A006519 are. - Andrew Howroyd, Jul 25 2018

			From _Michael De Vlieger_, Dec 29 2019: (Start)
a(1) = 1 since 1 is the empty product.
a(2) = 6 because 2 = 2^1 in form p_k^e; switching p_(k+1) for p, we have 3^1 = 3, and the largest power of 2 dividing 2 is 2^1 = 2; thus 3 * 2 = 6.
a(4) = 36 since 4 = 2^2 -> 4(3^2).
a(6) = 30 since 6 = 2^1 * 3^1 -> 2(3 * 5).
a(12) = 180 since 12 = 2^2 * 3 -> 4(3^2 * 5) = 4(45) = 180.
a(30) = 210 since 30 = 2 * 3 * 5 -> 2(3 * 5 * 7) = 210.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A003961, A006519, A283477.

Array[(Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1])*2^IntegerExponent[#, 2] &, 75] (* Michael De Vlieger, Dec 29 2019 *)

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); if(p==2, 6, nextprime(p+1))^e)} \\ Andrew Howroyd, Jul 25 2018

from sympy import nextprime, factorint
from math import prod
def A283980(n): return prod(nextprime(p)**e if p > 2 else 6**e for p, e in factorint(n).items()) # Chai Wah Wu, Dec 08 2022

(define (A283980 n) (* (A006519 n) (A003961 n)))

a(n) = A003961(n)*A006519(n).
From Michael De Vlieger, Dec 29 2019: (Start)
a(p_k) = p_(k+1) for odd prime p.
a(2^k) = 6^k.
a(p_k#) = p_(k+1)# for p_k# = A002110(k). (End)

A329886 Primorial inflation of Doudna-tree: a(0) = 1, a(1) = 2; for n > 1, if n is even, a(n) = A283980(a(n/2)), and if n is odd, then a(n) = 2*a((n-1)/2).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 36, 8, 210, 60, 180, 24, 900, 72, 216, 16, 2310, 420, 1260, 120, 6300, 360, 1080, 48, 44100, 1800, 5400, 144, 27000, 432, 1296, 32, 30030, 4620, 13860, 840, 69300, 2520, 7560, 240, 485100, 12600, 37800, 720, 189000, 2160, 6480, 96, 5336100, 88200, 264600, 3600, 1323000, 10800, 32400, 288, 9261000

Offset: 0

Antti Karttunen, Dec 24 2019

			This irregular table can be represented as a binary tree. Each child to the left is obtained by applying A283980 to the parent, and each child to the right is obtained by doubling the parent:
                                     1
                                     |
                  ...................2...................
                 6                                       4
      30......../ \........12                 36......../ \........8
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  210      60         180     24          900      72         216      16
etc.
A329887 is the mirror image of the same tree. See also A342000.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Michael De Vlieger, Tree showing levels 0 <= j <= 5
Michael De Vlieger, Tree showing levels 0 <= j <= 7

Permutation of A025487.
Cf. A005940, A054429, A108951, A283980, A329900, A342000.
Cf. also A283477, A322827, A329887, A337376/A337377.

Block[{a}, a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ@ n, (Times @@ Map[Prime[PrimePi@#1 + 1]^#2 & @@ # &, FactorInteger[#]] - Boole[# == 1])*2^IntegerExponent[#, 2] &[a[n/2]], 2 a[(n - 1)/2]]; Array[a, 57, 0]]
(* or, via Doudna *)
Map[Times @@ Flatten@ MapIndexed[ConstantArray[Prime[First[#2]], #1] &, Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &@ Sort[Flatten[ConstantArray[PrimePi@#1, #2] & @@@ FactorInteger[#]], Greater]] &, Nest[Append[#1, Prime[1 + BitLength[#2] - DigitCount[#2, 2, 1]]*#1[[#2 - 2^Floor@ Log2@ #2 + 1]]] & @@ {#, Length@ #} &, {1}, 57] ] (* Michael De Vlieger, Mar 05 2021 *)

A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A329886(n) = if(n<2,1+n,if(!(n%2),A283980(A329886(n/2)),2*A329886(n\2)));

a(0) = 1, a(1) = 2; for n > 1, if n is even, a(n) = A283980(a(n/2)), and if n is odd, then a(n) = 2*a((n-1)/2).
a(n) = A108951(A005940(1+n)).
For n >= 1, a(n) = A329887(A054429(n)).

Name amended by Antti Karttunen, Mar 05 2021

A283475 a(n) = A019565(A005187(n)).

Original entry on oeis.org

1, 2, 6, 5, 30, 7, 21, 42, 210, 11, 33, 66, 165, 330, 154, 231, 2310, 13, 39, 78, 195, 390, 182, 273, 1365, 2730, 286, 429, 1430, 2145, 1001, 2002, 30030, 17, 51, 102, 255, 510, 238, 357, 1785, 3570, 374, 561, 1870, 2805, 1309, 2618, 19635, 39270, 442, 663, 2210, 3315, 1547, 3094, 15470, 23205, 2431, 4862, 12155

Offset: 0

Antti Karttunen, Mar 15 2017

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

Cf. A000040, A000051, A001221, A001222, A004198, A005117, A005187, A019565, A046523, A097248, A108951, A280700, A280705, A283477, A283478.

Cf. A283476 (same sequence sorted into ascending order).

Map[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[#, 2] &, Table[2 n - DigitCount[2 n, 2, 1], {n, 0, 60}]] (* Michael De Vlieger, Mar 16 2017 *)

(define (A283475 n) (A019565 (A005187 n)))

a(n) = A019565(A005187(n)).
a(n) = A097248(A283477(n)) = A097248(A108951(A019565(n))) = A283478(A019565(n)).
Other identities:
If A004198(x,y) = 0, then a(x+y) = A097248(a(x)*a(y)).
For all n >= 1, a(A000051(n)) = A000040(n+2).
For all n >= 0, A001221(a(n)) = A001222(a(n)) = A280700(n).
For all n >= 0, A046523(a(n)) = A280705(n).

A322827 A permutation of A025487: Sequence of least representatives of distinct prime signatures obtained from the run lengths present in the binary expansion of n.

Original entry on oeis.org

1, 2, 6, 4, 36, 30, 12, 8, 216, 180, 210, 900, 72, 60, 24, 16, 1296, 1080, 1260, 5400, 44100, 2310, 6300, 27000, 432, 360, 420, 1800, 144, 120, 48, 32, 7776, 6480, 7560, 32400, 264600, 13860, 37800, 162000, 9261000, 485100, 30030, 5336100, 1323000, 69300, 189000, 810000, 2592, 2160, 2520, 10800, 88200, 4620, 12600

Offset: 0

Antti Karttunen, Jan 16 2019

A101296(a(n)) gives a permutation of natural numbers.

			The sequence can be represented as a binary tree:
                                      1
                                      |
                   ...................2...................
                  6                                       4
       36......../ \........30                 12......../ \........8
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
   216      180         210    900         72       60         24       16
etc.
Both children are multiples of their common parent, see A323503, A323504 and A323507.
The value of a(n) is computed from the binary expansion of n as follows: Starting from the least significant end of the binary expansion of n (A007088), we record the successive run lengths, subtract one from all lengths except the first one, and use the reversed partial sums of these adjusted values as the exponents of successive primes.
For 11, which is "1011" in base 2, we have run lengths [2, 1, 1] when scanned from the right, and when one is subtracted from all except the first, we have [2, 0, 0], partial sums of which is [2, 2, 2], which stays same when reversed, thus a(11) = 2^2 * 3^2 * 5^2 = 900.
For 13, which is "1101" in base 2, we have run lengths [1, 1, 2] when scanned from the right, and when one is subtracted from all except the first, we have [1, 0, 1], partial sums of which is [1, 1, 2], reversed [2, 1, 1], thus a(13) = 2^2 * 3^1 * 5^1 = 60.
Sequence A227183 is based on the same algorithm.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A000079 (right edge), A000400 (left edge, apart from 2), A005811, A046523, A101296, A227183, A322585, A322825, A323503, A323504, A323507.
Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822.
Cf. A005940, A283477, A323505 for other similar trees.

{1}~Join~Array[Times @@ MapIndexed[Prime[First@ #2]^#1 &, Reverse@ Accumulate@ MapIndexed[Length[#1] - Boole[First@ #2 > 1] &, Split@ Reverse@ IntegerDigits[#, 2]]] &, 54] (* Michael De Vlieger, Feb 05 2020 *)

A322827(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vecrev(Vec(o)), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));

a(n) = A046523(a(n)) = A046523(A322825(n)).
A001221(a(n)) = A005811(n).
A001222(a(n)) = A227183(n).
A322585(a(n)) = 1.

cp's OEIS Frontend

A276150 Sum of digits when n is written in primorial base (A049345); minimal number of primorials (A002110) that add to n.

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A052330 Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k; then the numbers b_k*S_k are the next 2^k terms.

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A072411 LCM of exponents in prime factorization of n, a(1) = 1.

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A248663 Binary encoding of the prime factors of the squarefree part of n.

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A129912 Numbers that are products of distinct primorial numbers (see A002110).

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A329900 Primorial deflation of n: starting from x = n, repeatedly divide x by the largest primorial A002110(k) that divides it, until x is an odd number. Then a(n) = Product prime(k_i), for primorial indices k_1 >= k_2 >= ..., encountered in the process.

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