A324341 -id:A324341 - cp's OEIS frontend

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

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

A324342 If 2n = 2^e1 + ... + 2^ek [e1 .. ek distinct], then a(n) is the minimal number of primorials (A002110) that add to A002110(e1) * ... * A002110(ek).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 6, 6, 1, 2, 6, 2, 10, 10, 8, 16, 1, 2, 6, 12, 6, 12, 24, 20, 18, 20, 28, 28, 26, 6, 18, 24, 1, 2, 6, 12, 14, 12, 20, 6, 18, 18, 22, 26, 38, 20, 16, 16, 24, 32, 42, 44, 34, 50, 68, 70, 36, 54, 60, 54, 70, 56, 60, 82, 1, 2, 6, 12, 12, 6, 18, 36, 12, 24, 28, 34, 34, 50, 50, 72, 22, 26, 28, 34, 38, 54, 40, 52, 28, 38, 56

Offset: 0

Antti Karttunen, Feb 23 2019

nonn

When A283477(n) is written in primorial base (A049345), then a(n) is the sum of digits (with unlimited digit values), thus also the minimal number of primorials (A002110) that add to A283477(n).
Number of prime factors in A324289(n), counted with multiplicity.
Each subsequence starting at each n = 2^k is converging towards A283477: 1, 2, 6, 12, 30, 60, 180, 360, 210, 420, etc. See also comments in A324289.

Cf. A001222, A002110, A049345, A276086, A276150, A283477, A324289, A324341.

A002110(n) = prod(i=1,n,prime(i));
A030308(n,k) = bittest(n,k);
A283477(n) = prod(i=0,#binary(n),if(0==A030308(n,i),1,A030308(n,i)*A002110(1+i)));
A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A324342(n) = A276150(A283477(n));

a(n) = A276150(A283477(n)).
a(n) = A001222(A324289(n)) = A001222(A276086(A283477(n))).
a(n) >= A324341(n).
a(2^n) = 1 for all n >= 0.

A324289 a(n) = A276086(A283477(n)).

Original entry on oeis.org

2, 3, 5, 25, 7, 49, 117649, 184877, 11, 121, 1771561, 143, 36226650889, 59797108943, 546826709, 299019449675770681, 13, 169, 4826809, 23298085122481, 8254129, 68130645548641, 17750592470222918406076697669, 406193515012381653451063, 8223741426987700773289, 1553319630709265128413587, 1977089672816762887718980502697827

Offset: 0

Antti Karttunen, Feb 23 2019

nonn

All primes are present, and furthermore, each subsequence starting at each n = 2^k is converging towards p^A283477(0), p^A283477(1), p^A283477(2), p^A283477(3), ..., where p = A000040(2+k). For example, for a(2^4) = a(16), the prime is A000040(2+4) = 13, and its powers 13^1, 13^2, 13^6 and 13^12 occur in successive positions from a(16) to a(19). See also comments in A324342.

Cf. A000040, A001222, A276086, A283477, A324341, A324342.

A002110(n) = prod(i=1,n,prime(i));
A030308(n,k) = bittest(n,k);
A283477(n) = prod(i=0,#binary(n),if(0==A030308(n,i),1,A030308(n,i)*A002110(1+i)));
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324289(n) = A276086(A283477(n));

a(n) = A276086(A283477(n)).
For n >= 0, a(2^n) = A000040(2+n).
A001221(a(n)) = A324341(n).
A001222(a(n)) = A324342(n).

A342461 Number of nonzero digits when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 15 2021

nonn

Cf. A001221, A005940, A108951, A267263, A329886, A342456, A342457, A342462.

Cf. also A324341, A329040.

A342461(n) = omega(A342456(n)); \\ Other code as in A342456.

a(n) = A001221(A342456(n)) = A001221(A342457(n)).
a(n) = A267263(A329886(n)) = A329040(A005940(1+n)).
a(n) <= A342462(n).
For n >= 0, a(2^n) = 1.

A324381 Number of nonzero digits when the n-th highly composite number is written in primorial base: a(n) = A267263(A002182(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 3, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 4

Offset: 1

Antti Karttunen, Feb 26 2019

nonn

			For n=12, A002182(12) = 240, which is written as "11000" in primorial base (A049345) because 240 = 1*A002110(4) + 1*A002110(3) = 210+30, thus a(12) = 2, as there are two nonzero digits.
For n=18, A002182(18) = 2520 = "110000" in primorial base because 2520 = 1*A002110(5) + 1*A002110(4) = 2310+210, thus a(18) = 2.
For n=26, A002182(26) = 45360 = "1670000" in primorial base because 45360 = 1*A002110(6) + 6*A002110(5) + 7*A002110(4), thus a(26) = 3, as there are three nonzero digits.

Cf. A002100, A002182, A049345, A267263, A324382.

Cf. also A324341.

A267263(n) = { my(s=0); forprime(p=2, , if(n%p, s++, if(n==0, return(s))); n\=p); }; \\ From A267263
A324381(n) = A267263(A002182(n));

a(n) = A267263(A002182(n)).

a(n) <= A324382(n).

cp's OEIS Frontend

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

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A324342 If 2n = 2^e1 + ... + 2^ek [e1 .. ek distinct], then a(n) is the minimal number of primorials (A002110) that add to A002110(e1) * ... * A002110(ek).

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A324289 a(n) = A276086(A283477(n)).

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A342461 Number of nonzero digits when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

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A324381 Number of nonzero digits when the n-th highly composite number is written in primorial base: a(n) = A267263(A002182(n)).

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