A283477 -id:A283477 - cp's OEIS frontend

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).

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.

A106831 Define a triangle in which the entries are of the form +-1/(b!c!d!e!...), where the order of the factorials is important; read the triangle by rows and record and expand the denominators.

Original entry on oeis.org

2, 6, 4, 24, 12, 12, 8, 120, 48, 36, 24, 48, 24, 24, 16, 720, 240, 144, 96, 144, 72, 72, 48, 240, 96, 72, 48, 96, 48, 48, 32, 5040, 1440, 720, 480, 576, 288, 288, 192, 720, 288, 216, 144, 288, 144, 144, 96, 1440, 480, 288, 192, 288, 144, 144, 96, 480, 192, 144, 96, 192

Offset: 0

N. J. A. Sloane, May 22 2005

Row n has 2^n terms. Row 0 is +1/2!. An entry +-1/b!c!d!... has two children, a left child -+1/(a+1)!b!c!... and a right child +-1/2!b!c!d!...

Let S_n = sum of entries in row n of the triangle. Then for n > 0, n!S_{n-1} is the Bernoulli number B_n.

			Woon's "Bernoulli Tree" begins like this (see also the given Wikipedia-link). This sequence gives the values of the denominators:
                                     +1
                                    ────
                                     2!
                 -1                 /  \                  +1
                ──── ............../    \.............. ─────
                 3!                                      2!2!
        +1        .         -1                 -1         .         +1
       ────      / \       ────               ────       / \      ──────
        4! ...../   \..... 2!3!               3!2! ...../   \.... 2!2!2!
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    -1      +1         +1       -1         +1      -1          -1       +1
   ────    ────       ────     ──────     ────   ──────      ──────  ────────
    5!     2!4!       3!3!     2!2!3!     4!2!   2!3!2!      3!2!2!  2!2!2!2!
etc.

Reinhard Zumkeller, Rows n = 0..13 of table, flattened
Wikipedia, Bernoulli number: A binary tree representation
S. C. Woon, A tree for generating Bernoulli numbers, Math. Mag., 70 (1997), 51-56.
Index entries for sequences related to Bernoulli numbers.

Cf. A242179 (numerators), A050925, A050932, A000142.
Cf. A001013, A060054, A075180, A164555, A027642, A246660.
Cf. A323505 (mirror image), and also A005940, A283477, A322827 for other similar trees.

a106831 n k = a106831_tabf !! n !! n
a106831_row n = a106831_tabf !! n
a106831_tabf = map (map (\(, , left, right) -> left * right)) $
   iterate (concatMap (\(x, f, left, right) -> let f' = f * x in
   [(x + 1, f', f', right), (3, 2, 2, left * right)])) [(3, 2, 2, 1)]
-- Reinhard Zumkeller, May 05 2014

Contribution from Peter Luschny, Jun 12 2009: (Start)
The routine computes the triangle row by row and gives the numbers with their sign.
Thus A(1)=[2]; A(2)=[ -6,4]; A(3)=[24,-12,-12,8]; etc.
A := proc(n) local k, i, j, m, W, T; k := 2;
W := array(0..2^n); W[1] := [1,`if`(n=0,1,2)];
for i from 1 to n-1 do for m from k by 2 to 2*k-1 do
T := W[iquo(m,2)]; W[m] := [ -T[1],T[2]+1,seq(T[j],j=3..nops(T))];
W[m+1] := [T[1],2,seq(T[j],j=2..nops(T))]; od; k := 2*k; od;
seq(W[i][1]*mul(W[i][j]!,j=2..nops(W[i])),i=iquo(k,2)..k-1) end:
seq(print(A(i)),i=1..5); (End)

a [n_] := Module[{k, i, j, m, w, t}, k = 2; w = Array[0&, 2^n]; w[[1]] := {1, If[n == 0, 1, 2]}; For[i = 1, i <= n-1, i++, For[m = k, m <= 2*k-1 , m = m+2, t = w[[Quotient[m, 2]]]; w[[m]] = {-t[[1]], t[[2]]+1, Sequence @@ Table[t[[j]], {j, 3, Length[t]}]}; w[[m+1]] = {t[[1]], 2, Sequence @@ Table[t[[j]], {j, 2, Length[t]}]}]; k = 2*k]; Table[w[[i, 1]]*Product[w[[i, j]]!, {j, 2, Length[w[[i]]]}], {i, Quotient[k, 2], k-1}]]; Table[a[i] , {i, 1, 6}] // Flatten // Abs (* Jean-François Alcover, Dec 20 2013, translated from Maple *)

A106831off1(n) = if(!n,1, my(rl=1, m=1); while(n,if(!(n%2), rl++, m *= ((1+rl)!); rl=1); n >>= 1); (m));
A106831(n) = A106831off1(1+n); \\ Antti Karttunen, Jan 16 2019

A001511(n) = (1+valuation(n,2));
A106831r1(n) = if(!n,1,if(n%2, 2*A106831r1((n-1)/2), (1+A001511(n))*A106831r1(n/2))); \\ Implements the given recurrence.
A106831(n) = A106831r1(1+n); \\ Antti Karttunen, Jan 16 2019

From Antti Karttunen, Jan 16 2019: (Start)
If sequence is shifted one term to the right, then the following recurrence works:
a(0) = 1; and for n > 0, a(2n) = (1+A001511(2n))*a(n), a(2n+1) = 2*a(n).
(End)

More terms from Franklin T. Adams-Watters, Apr 28 2006

Example section reillustrated by Antti Karttunen, Jan 16 2019

A323505 Mirror image of (denominators of) Bernoulli tree, A106831.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 12, 24, 16, 24, 24, 48, 24, 36, 48, 120, 32, 48, 48, 96, 48, 72, 96, 240, 48, 72, 72, 144, 96, 144, 240, 720, 64, 96, 96, 192, 96, 144, 192, 480, 96, 144, 144, 288, 192, 288, 480, 1440, 96, 144, 144, 288, 144, 216, 288, 720, 192, 288, 288, 576, 480, 720, 1440, 5040, 128, 192, 192, 384, 192, 288, 384, 960, 192, 288, 288

Offset: 0

Antti Karttunen, Jan 16 2019

In contrast to A106831 which follows Woon's original indexing (and orientation), this variant starts with value a(0) = 1, with the rest of terms having an index incremented by one, thus allowing for a simple recurrence.

Sequence contains only terms of A001013 and each a(n) is a multiple of A246660(n).

			This sequence can be represented as a binary tree:
                                       1
                                       |
                    ...................2....................
                   4                                        6
         8......../ \........12                 12........./ \.......24
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    16       24         24       48         24       36         48      120
  32  48   48  96     48  72   96  240    48  72   72  144    96  144 240  720
etc.

Antti Karttunen, Table of n, a(n) for n = 0..16383
S. C. Woon, A tree for generating Bernoulli numbers, Math. Mag., 70 (1997), 51-56.
Index entries for sequences related to Bernoulli numbers.

Cf. A000079 (left edge), A000142 (right edge), A001013, A001511, A036987, A054429, A246660, A323506, A323508.

Cf. A106831 and also A005940, A283477, A322827 for other similar trees.

A001511(n) = (1+valuation(n,2));
A036987(n) = !bitand(n,1+n);
A323505(n) = if(!n,1,if(!(n%2), 2*A323505(n/2), (A001511(n+1)+1-A036987(n))*A323505((n-1)/2)));

A054429(n) = if(!n,n,((3<<#binary(n\2))-n-1)); \\ From A054429
A106831r1(n) = if(!n,1,if(n%2, 2*A106831r1((n-1)/2), (1+A001511(n))*A106831r1(n/2))); \\ Recurrence for A106831, when prepended with 1, thus shifted one term right
A323505(n) = A106831r1(A054429(n));

a(0) = 1; and for n > 0, if n is even, a(n) = 2*a(n/2), and if n is odd, a(n) = (A001511(n+1)+1-A036987(n)) * a((n-1)/2).
For n > 0, a(n) = b(A054429(n)), where b(n) = A106831(n-1).
a(n) = A246660(n) * A323506(n).
a(n) = A323508(A005940(1+n)).

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.

Original entry on oeis.org

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];

A324287 a(n) = A002487(A005187(n)).

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 3, 5, 4, 1, 4, 7, 5, 7, 7, 5, 5, 1, 5, 9, 7, 10, 11, 8, 7, 9, 9, 7, 13, 8, 3, 10, 6, 1, 6, 11, 9, 13, 15, 11, 10, 13, 14, 11, 21, 13, 5, 17, 9, 11, 11, 9, 19, 12, 5, 18, 19, 11, 3, 13, 7, 18, 15, 4, 7, 1, 7, 13, 11, 16, 19, 14, 13, 17, 19, 15, 29, 18, 7, 24, 13, 16, 17, 14, 30, 19, 8, 29, 31, 18, 5, 22, 12, 31, 26, 7

Offset: 0

Antti Karttunen, Feb 20 2019

nonn

The motivation for this kind of sequence was a question: what kind of simply defined non-injective functions f exist such that this sequence can be defined as their function, e.g., as a(n) = g(f(n)), where g is a nontrivial integer-valued function? The same question can also be asked about A324288, A324337 and A324338. Note that A005187, A283477 and A006068 used in their definitions are all injections. Of course, A324377(n) = A000265(A005187(n)) fills the bill as A002487(n) = A002487(A000265(n)), but are there any less obvious solutions? - Antti Karttunen, Feb 28 2019

Antti Karttunen, Table of n, a(n) for n = 0..16385
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 Stern's sequences

Cf. A000265, A002487, A005187, A006068, A283477, A324286, A324288, A324293, A324337, A324338, A324377.

A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); }; \\ Modified from the one given in A002487, sign not actually needed here.
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A324287(n) = A002487(A005187(n));

from functools import reduce
def A324287(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin((n<<1)-n.bit_count())[-1:2:-1],(1,0))) if n else 0 # Chai Wah Wu, May 05 2023

a(n) = A002487(A005187(n)).
a(n) = A324286(A283477(n)).
a(n) = A002487(A324377(n)).

A342456 A276086 applied to the primorial inflation of Doudna-tree, where A276086(n) is the prime product form of primorial base expansion of n.

Original entry on oeis.org

2, 3, 5, 9, 7, 25, 35, 15, 11, 49, 117649, 625, 717409, 1225, 55, 225, 13, 121, 1771561, 2401, 36226650889, 184877, 1127357, 875, 902613283, 514675673281, 3780549773, 1500625, 83852850675321384784127, 3025, 62004635, 21, 17, 169, 4826809, 14641, 8254129, 143, 2924207, 77, 8223741426987700773289, 59797108943, 546826709

Offset: 0

Antti Karttunen and Michael De Vlieger, Mar 14 2021

nonn

This sequence (which could be viewed as a binary tree, like the underlying A005940 and A329886) is similar to A324289, but unlike its underlying tree A283477 that generates only numbers that are products of distinct primorial numbers (i.e., terms of A129912), here the underlying tree A329886 generates all possible products of primorial numbers, i.e., terms of A025487, but in different order.

Cf. A005940, A025487, A108951, A129912, A276086, A283980, A324886, A342457 [= 2*A246277(a(n))], A342461 [= A001221(a(n))], A342462 [= A001222(a(n))], A342463 [= A342001(a(n))], A342464 [= A051903(a(n))].

Cf. A324289 (a subset of these terms, in different order).

Block[{a, f, r = MixedRadix[Reverse@ Prime@ Range@ 24]}, f[n_] :=
Times @@ MapIndexed[Prime[First[#2]]^#1 &, Reverse@ IntegerDigits[n, r]]; 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[f@ a[#] &, 43, 0]] (* Michael De Vlieger, Mar 17 2021 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
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)));
A342456(n) = A276086(A329886(n));

a(n) = A276086(A329886(n)) = A324886(A005940(1+n)).
For all n >= 0, gcd(a(n), A329886(n)) = 1.
For all n >= 1, A055396(a(n))-1 = A061395(A329886(n)) = A290251(n) = 1+A080791(n).
For all n >= 0, a(2^n) = A000040(2+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).

A283985 Sums of distinct terms of A143293: a(n) = Sum_{k>=0} A030308(n,k)*A143293(k).

Original entry on oeis.org

0, 1, 3, 4, 9, 10, 12, 13, 39, 40, 42, 43, 48, 49, 51, 52, 249, 250, 252, 253, 258, 259, 261, 262, 288, 289, 291, 292, 297, 298, 300, 301, 2559, 2560, 2562, 2563, 2568, 2569, 2571, 2572, 2598, 2599, 2601, 2602, 2607, 2608, 2610, 2611, 2808, 2809, 2811, 2812, 2817, 2818, 2820, 2821, 2847, 2848, 2850, 2851, 2856, 2857, 2859, 2860, 32589

Offset: 0

Antti Karttunen, Mar 19 2017

nonn

Indexing starts from zero, with a(0) = 0.

Cf. A002110, A030308, A143293, A276085, A276156, A283477, A283984.

A143293(n) = { if(n==0, return(1)); my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); s; }; \\ This function from Charles R Greathouse IV, Feb 05 2014
A030308(n,k) = bittest(n,k);
A283985(n) = sum(i=0,(#binary(n)-1),A030308(n,i)*A143293(i));

from sympy import primorial, primepi, prime, primerange, factorint
from operator import mul
from functools import reduce
def a002110(n): return 1 if n<1 else primorial(n)
def a276085(n):
    f=factorint(n)
    return sum([f[i]*a002110(primepi(i) - 1) for i in f])
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 a276085(a108951(a019565(n)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 22 2017

(define (A283985 n) (A276085 (A283477 n)))

a(n) = Sum_{k>=0} A030308(n,k)*A143293(k).
a(n) = A276085(A283477(n)).
Other identities. For all n >= 0:
a(2^n) = A143293(n).

A283997 a(n) = n XOR A005187(floor(n/2)), where XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 2, 3, 15, 14, 2, 3, 6, 7, 5, 4, 31, 30, 2, 3, 6, 7, 5, 4, 14, 15, 13, 12, 5, 4, 4, 5, 63, 62, 2, 3, 6, 7, 5, 4, 14, 15, 13, 12, 5, 4, 4, 5, 30, 31, 29, 28, 5, 4, 4, 5, 13, 12, 12, 13, 4, 5, 7, 6, 127, 126, 2, 3, 6, 7, 5, 4, 14, 15, 13, 12, 5, 4, 4, 5, 30, 31, 29, 28, 5, 4, 4, 5, 13, 12, 12, 13, 4, 5, 7, 6, 62, 63, 61, 60, 5, 4, 4, 5, 13, 12, 12

Offset: 0

Antti Karttunen, Mar 19 2017

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

Cf. A003986, A005187, A006068, A283996, A283998, A283999.

Cf. also A279357, A283477.

Table[BitXor[n, 2 # - DigitCount[2 #, 2, 1] &@ Floor[n/2]], {n, 0, 106}] (* Michael De Vlieger, Mar 20 2017 *)

b(n) = if(n<1, 0, b(n\2) + n%2);
A(n) = 2*n - b(2*n);
for(n=0, 110, print1(bitxor(n, A(floor(n/2))),", ")) \\ Indranil Ghosh, Mar 25 2017

def A(n): return 2*n - bin(2*n)[2:].count("1")
print([n^A(n//2) for n in range(111)]) # Indranil Ghosh, Mar 25 2017

(define (A283997 n) (A003987bi n (A005187 (floor->exact (/ n 2))))) ;; Where A003987bi implements bitwise-XOR (A003987).

a(n) = n XOR A005187(floor(n/2)), where XOR is bitwise-xor (A003987).
a(n) = A283996(n) - A283998(n).
a(n) = A005187(n) - 2*A283998(n).
a(n) = A006068(n) XOR A283999(floor(n/2)).

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];

cp's OEIS Frontend

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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A106831 Define a triangle in which the entries are of the form +-1/(b!c!d!e!...), where the order of the factorials is important; read the triangle by rows and record and expand the denominators.

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A323505 Mirror image of (denominators of) Bernoulli tree, A106831.

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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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A324287 a(n) = A002487(A005187(n)).

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A342456 A276086 applied to the primorial inflation of Doudna-tree, where A276086(n) is the prime product form of primorial base expansion of n.

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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.

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