A324342 -id:A324342 - cp's OEIS frontend

A324343 Lexicographically earliest positive sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A324342(i) = A324342(j), for all i, j >= 0.

1, 2, 2, 3, 2, 4, 5, 6, 2, 4, 7, 8, 9, 10, 11, 12, 2, 4, 7, 13, 7, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 2, 4, 7, 13, 25, 14, 18, 22, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 2, 4, 7, 13, 50, 51, 52, 53, 50, 54, 55, 56, 38, 39, 57, 58, 59, 60, 55, 56, 61, 62, 63, 64, 19, 65, 66, 67, 16, 68, 69, 70, 71, 72, 73, 74, 75, 63, 76

Offset: 0

Antti Karttunen, Feb 24 2019

nonn

Restricted growth sequence transform of the ordered pair [A278222(n), A324342(n)], or equally, of [A286622(n), A324342(n)].

For all i, j: a(i) = a(j) => A324344(i) = A324344(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A278222, A286622, A324342, A324344.

Cf. also A323889, A324345.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A002110(n) = prod(i=1,n,prime(i));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
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));
A324343aux(n) = [A278222(n), A324342(n)];
v324343 = rgs_transform(vector(1+up_to,n,A324343aux(n-1)));
A324343(n) = v324343[1+n];

A324344 Lexicographically earliest positive sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A324342(i) = A324342(j), for all i, j >= 0.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 5, 2, 3, 4, 6, 7, 8, 9, 10, 2, 3, 4, 11, 4, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2, 3, 4, 11, 22, 11, 15, 19, 14, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 2, 3, 4, 11, 44, 5, 23, 45, 44, 12, 16, 46, 32, 33, 33, 47, 48, 18, 16, 46, 26, 37, 49, 50, 16, 51, 52, 53, 13, 54, 55, 56, 57, 36, 58, 38, 59, 49, 60

Offset: 0

Antti Karttunen, Feb 24 2019

nonn

Restricted growth sequence transform of the ordered pair [A000120(n), A324342(n)].

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000120, A324342, A324343.

Cf. also A318310.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
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));
A324344aux(n) = [hammingweight(n), A324342(n)];
v324344 = rgs_transform(vector(1+up_to,n,A324344aux(n-1)));
A324344(n) = v324344[1+n];

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

Original entry on oeis.org

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

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

A324382 Minimal number of primorials that add to the n-th highly composite number: a(n) = A276150(A002182(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 6, 4, 6, 8, 2, 4, 6, 8, 12, 16, 20, 12, 14, 18, 12, 12, 12, 12, 12, 12, 12, 24, 8, 8, 8, 4, 16, 8, 16, 8, 16, 24, 16, 32, 6, 14, 30, 12, 18, 18, 24, 12, 18, 18, 24, 18, 36, 8, 14, 32, 28, 6, 24, 38, 12, 18, 36, 20, 24, 30, 40, 26, 10, 40, 20, 30, 18, 38, 26, 36, 36, 24, 24, 44, 50, 48, 14, 42

Offset: 1

Antti Karttunen, Feb 26 2019

nonn

Among the first 10000 highly composite numbers, only in two cases a(n) < A112779(n). This happens on A002182(12) = 240 and A002182(18) = 2520. Note that A112779(n) gives the number of primorials needed when A002182(n) is expressed as a product [not as a sum] of primorials.

			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) = 1+1 = 2. (Note that 240 = 30*2*2*2).
For n=18, A002182(18) = 2520 = "110000" in primorial base because 2520 = 1*A002110(5) + 1*A002110(4) = 2310+210, thus a(18) = 1+1 = 2. (Note that 2520 = 210*6*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) = 1+6+7 = 14. (Note that 45360 = 210*6*6*6).

Cf. A002110, A002182, A276150, A324381.

Cf. also A108602, A112778, A112779, A324342.

A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A324382(n) = A276150(A002182(n));

a(n) = A276150(A002182(n)).

a(n) >= A324381(n).

A324386 a(n) = A324383(A006068(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 4, 4, 4, 2, 2, 6, 6, 1, 2, 4, 8, 4, 4, 6, 12, 2, 8, 6, 10, 6, 22, 10, 8, 1, 4, 4, 6, 2, 8, 6, 8, 4, 6, 12, 14, 2, 16, 10, 16, 2, 8, 16, 4, 6, 14, 8, 24, 6, 30, 18, 20, 6, 26, 18, 26, 1, 6, 8, 8, 4, 12, 12, 6, 8, 12, 14, 18, 4, 20, 20, 20, 4, 16, 16, 8, 12, 28, 16, 10, 12, 22, 26, 14, 12, 34, 20, 22, 2, 12

Offset: 0

Antti Karttunen, Feb 27 2019

nonn

This is most likely equal to A276150(A086141(n)), apart from the different offset used in A086141.

The same comments about the parity of terms as in A324383 and A324387 apply also here, except here 1's occur at positions given by 2^k - 1.

Cf. A002110, A006068, A025487, A086141, A276150, A322827, A324342, A324382.

Cf. also A324383, A324387 (permutations of this sequence) and A324380, A324390.

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
A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
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));
A324383(n) = A276150(A322827(n));
A324386(n) = A324383(A006068(n));

a(n) = A324383(A006068(n)) = A276150(A322827(A006068(n))).

a(A000225(n)) = 1 for all n.

A324387 Minimal number of primorials (A002110) that add to the n-th number that is a product of primorials: a(n) = A276150(A025487(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 4, 1, 2, 2, 4, 2, 4, 4, 4, 4, 6, 8, 6, 8, 1, 2, 2, 6, 6, 6, 10, 2, 4, 4, 6, 8, 6, 10, 4, 8, 6, 8, 12, 6, 10, 6, 8, 12, 10, 8, 12, 12, 10, 16, 12, 20, 1, 2, 6, 8, 10, 6, 10, 8, 10, 16, 14, 20, 2, 4, 12, 10, 10, 14, 10, 16, 12, 20, 6, 6, 10, 8, 10, 12, 20, 4, 8, 14, 14, 20, 14, 10, 16, 14, 24, 6, 12, 12

Offset: 1

Antti Karttunen, Feb 27 2019

nonn

A098719 gives the positions of ones in this sequence. See also comments in A324383.

Antti Karttunen, Table of n, a(n) for n = 1..10001 (based on the b-file of A025487 as computed by Will Nicholes and Franklin T. Adams-Watters)
Index entries for sequences related to primorial base
Index entries for sequences related to primorial numbers

Cf. A002110, A025487, A098719 (positions of ones), A276150, A324342.

Cf. A324382 for a subsequence, and A324383, A324386 for permutations of this sequence.

A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A324387(n) = A276150(A025487(n));

a(n) = A276150(A025487(n)).

A342462 Sum of 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, 2, 1, 2, 2, 2, 1, 2, 6, 4, 6, 4, 2, 4, 1, 2, 6, 4, 10, 6, 6, 4, 8, 12, 10, 8, 22, 4, 8, 2, 1, 2, 6, 4, 6, 2, 6, 2, 18, 10, 8, 6, 18, 12, 16, 4, 26, 16, 24, 8, 20, 14, 4, 6, 26, 16, 14, 8, 30, 6, 8, 4, 1, 2, 6, 4, 14, 12, 12, 8, 18, 12, 24, 4, 8, 12, 14, 4, 24, 20, 28, 20, 26, 16, 16, 12, 32, 26, 24, 14, 28, 16

Offset: 0

Antti Karttunen, Mar 15 2021

nonn

From David A. Corneth's Feb 27 2019 comment in A276150 follows that the only odd terms in this sequence are 1's occurring at 0 and at two's powers.

Subsequences starting at each n = 2^k are slowly converging towards A329886: 1, 2, 6, 4, 30, 12, 36, 8, 210, 60, 180, 24, etc.. Compare also to the behaviors of A324342 and A342463.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to primorial base

Cf. A001222, A005940, A108951, A276150, A329886, A342456, A342457, A342461, A342463, A342464.

Cf. also A324342, A324383, A324387, A324888.

A342462(n) = bigomega(A342456(n)); \\ Other code as in A342456.

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)};
A329886(n) = if(n<2,1+n,if(!(n%2),A283980(A329886(n/2)),2*A329886(n\2)));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A342462(n) = A276150(A329886(n));

a(n) = A001222(A342456(n)) = A001222(A342457(n)).
a(n) = A276150(A329886(n)) = A324888(A005940(1+n)).
a(n) >= A342461(n).
For n >= 0, a(2^n) = 1.

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).

A324341 If 2n = 2^e1 + ... + 2^ek [e1 .. ek distinct], then a(n) is the number of nonzero digits when A002110(e1) * ... * A002110(ek) is written in primorial base.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 23 2019

nonn

Number of nonzero digits when A283477(n) is represented in primorial base, A049345.

Number of distinct prime factors in A324289(n).

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to primorial base

Cf. A001221, A002110, A049345, A267263, A276086, A283477, A324289, 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)));
A267263(n) = { my(s=0); forprime(p=2, , if(n%p, s++, if(n==0, return(s))); n\=p); }; \\ From A267263
A324341(n) = A267263(A283477(n));

a(n) = A267263(A283477(n)).
a(n) = A001221(A324289(n)) = A001221(A276086(A283477(n))).
a(n) <= A324342(n).

cp's OEIS Frontend

A324343 Lexicographically earliest positive sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A324342(i) = A324342(j), for all i, j >= 0.

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A324344 Lexicographically earliest positive sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A324342(i) = A324342(j), for all i, j >= 0.

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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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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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A324382 Minimal number of primorials that add to the n-th highly composite number: a(n) = A276150(A002182(n)).

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A324386 a(n) = A324383(A006068(n)).

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A324387 Minimal number of primorials (A002110) that add to the n-th number that is a product of primorials: a(n) = A276150(A025487(n)).

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A342462 Sum of digits when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

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