A108951 -id:A108951 - cp's OEIS frontend

A238690 Let each integer m (1 <= m <= n) be factorized as m = prime_m(1)prime_m(2)...*prime_m(bigomega(m)), with the primes sorted in nonincreasing order. Then a(n) is the number of values of m such that each prime_m(i) <= prime_n(i).

1, 2, 3, 3, 4, 5, 5, 4, 6, 7, 6, 7, 7, 9, 9, 5, 8, 9, 9, 10, 12, 11, 10, 9, 10, 13, 10, 13, 11, 14, 12, 6, 15, 15, 14, 12, 13, 17, 18, 13, 14, 19, 15, 16, 16, 19, 16, 11, 15, 16, 21, 19, 17, 14, 18, 17, 24, 21, 18, 19, 19, 23, 22, 7, 22, 24, 20, 22, 27, 23, 21

Offset: 1

Matthew Vandermast, Apr 28 2014

nonn

Equivalently, a(n) equals the number of values of m such that each value of A238689 T(m,k) <= A238689 T(n,k). (Since the prime factorization of 1 is the empty factorization, we consider each prime_1(i) not to be greater than prime_n(i) for all positive integers n.)
Suppose we say that n "covers" m iff both m and n are factorized as described in the sequence definition and each prime_m(i) <= prime_n(i). At least three sequences (A037019, A108951 and A181821) have the property that a(m) divides a(n) iff n "covers" m.  These sequences are also divisibility sequences (i.e., sequences with the property that a(m) divides a(n) if m divides n), since any positive integer "covers" each of its divisors.
For any positive integers m and k, the following integer sequences (with n >= 0) are arithmetic progressions:
1. The sequence b(n) = a(m*(2^n)).
2. The sequence b(n) = a(m*(prime(n+k))) if prime(k) >= A006530(m).
Also, a(n) = the number of distinct prime signatures that occur among the divisors of any integer m such that A181819(m) = n and/or A238745(m) = n.
Number of skew partitions whose numerator has Heinz number n, where a skew partition is a pair y/v of integer partitions such that the diagram of v fits inside the diagram of y. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Feb 24 2018

			The prime factorizations of integers 1 through 9, with prime factors sorted from largest to smallest:
1 - the empty factorization (no prime factors)
2 = 2
3 = 3
4 = 2*2
5 = 5
6 = 3*2
7 = 7
8 = 2*2*2
9 = 3*3
To find a(9), we consider 9 = 3*3. There are 6 positive integers (1, 2, 3, 4, 6 and 9) which satisfy the following criteria:
1) The largest prime factor, if one exists, is not greater than 3;
2) The second-largest prime factor, if one exists, is not greater than 3;
3) The total number of prime factors (counting repeated factors) does not exceed 2.
Therefore, a(9) = 6.
From _Gus Wiseman_, Feb 24 2018: (Start)
Heinz numbers of the a(15) = 9 partitions contained within the partition (32) are 1, 2, 3, 4, 5, 6, 9, 10, 15. The a(15) = 9 skew partitions are (32)/(), (32)/(1), (32)/(11), (32)/(2), (32)/(21), (32)/(22), (32)/(3), (32)/(31), (32)/(32).
Corresponding diagrams are:
  o o o   . o o   . o o   . . o   . . o   . . o   . . .   . . .   . . .
  o o     o o     . o     o o     . o     . .     o o     . o     . .    (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Rearrangement of A115728, A115729 and A238746. A116473(n) is the number of times n appears in the sequence.

Cf. A000041, A000085, A000720, A056239, A063834, A112798, A122111, A153452, A215366, A238689, A259478, A259480, A296150, A296188, A296561, A297388, A299925, A299926, A299966, A299967.

undptns[y_]:=Select[Tuples[Range[0,#]&/@y],OrderedQ[#,GreaterEqual]&];
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[undptns[Reverse[primeMS[n]]]],{n,100}] (* Gus Wiseman, Feb 24 2018 *)

a(n) = A085082(A108951(n)) = A085082(A181821(n)).
a(n) = a(A122111(n)).
a(prime(n)) = a(2^n) = n+1.
a((prime(n))^m) = a((prime(m))^n) = binomial(n+m, n).
a(A002110(n)) = A000108(n+1).
A000005(n) <= a(n) <= n.

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

A336835 Number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2

Offset: 1

Antti Karttunen, Aug 07 2020

nonn

It holds that a(n) <= A336836(n) for all n, because sigma(n) <= A003961(n) for all n (see A286385 for a proof).
The first 3 occurs at n = 19399380, the first 4 at n = 195534950863140268380. See A336389.
If x and y are relatively prime (i.e., gcd(x,y) = 1), then a(x*y) >= max(a(x),a(y)). Compare to a similar comment in A336915.

			For n = 120, sigma(120) = 360 >= 2*120, thus 120 is not deficient, and we get the next number by applying the prime shift, A003961(120) = 945, and sigma(945) = 1920 >= 945*2, so neither 945 is deficient, so we prime shift once again, and A003961(945) = 9625, which is deficient, as sigma(9625) = 14976 < 2*9625. Thus after two iteration steps we encounter a deficient number, and therefore a(120) = 2.

Cf. A003961, A005100, A005940, A023196, A046523, A047802, A071364, A108951, A286385, A294934, A336834, A337474, A337475.
Cf. A336389 (position of the first occurrence of a term >= n).
Differs from A294936 for the first time at n=120.
Cf. also A246271, A252459, A336836 and A336915 for similar iterations.

Array[-1 + Length@ NestWhileList[If[# == 1, 1, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]] &, #, DivisorSigma[1, #] >= 2 # &] &, 120] (* Michael De Vlieger, Aug 27 2020 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A336835(n) = { my(i=0); while(sigma(n) >= (n+n), i++; n = A003961(n)); (i); };

If A294934(n) = 1, a(n) = 0, otherwise a(n) = 1 + a(A003961(n)).
From Antti Karttunen, Aug 21-Sep 01 2020: (Start)
For all n >= 1,
a(A046523(n)) >= a(n).
a(A071364(n)) >= a(n).
a(A108951(n)) = A337474(n).
a(A025487(n)) = A337475(n).
(End)

A181822 a(n) = member of A025487 whose prime signature is conjugate to the prime signature of A025487(n).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 210, 60, 8, 2310, 36, 420, 24, 30030, 180, 4620, 120, 510510, 1260, 72, 60060, 16, 900, 840, 9699690, 13860, 360, 1021020, 48, 6300, 9240, 223092870, 180180, 2520, 19399380, 240, 69300, 216, 120120, 6469693230, 1800, 3063060, 144, 44100, 27720, 446185740, 1680, 900900, 1080, 2042040, 200560490130, 12600, 58198140, 32, 720

Offset: 1

Matthew Vandermast, Dec 07 2010

A permutation of the members of A025487.

If integers m and n have conjugate prime signatures, then A001222(m) = A001222(n), A071625(m) = A071625(n), A085082(m) = A085082(n), and A181796(m) = A181796(n).

			A025487(5) = 8 = 2^3 has a prime signature of (3). The partition that is conjugate to (3) is (1,1,1), and the member of A025487 with that prime signature is 30 = 2*3*5 (or 2^1*3^1*5^1).  Therefore, a(5) = 30.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Conjugate Partition

Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821.

A181825 lists members of A025487 with self-conjugate prime signatures. See also A181823-A181824, A181826-A181827.

f[n_] := Block[{ww, dec}, dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]]; ww = NestList[Append[#, 1] &, {1}, # - 1] &[-2 + Length@ NestWhileList[NextPrime@ # &, 1, Times @@ {##} <= n &, All] ]; {{{0}}}~Join~Map[Block[{w = #, k = 1}, Sort@ Apply[Join, {{ConstantArray[1, Length@ w]}, If[Length@ # == 0, #, #[[1]]] }] &@ Reap[Do[If[# <= n, Sow[w]; k = 1, If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w, If[k == 1, MapAt[# + 1 &, w, k], PadLeft[#, Length@ w, First@ #] &@ Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]], {i, Infinity}] ][[-1]] ] &, ww]]; Sort[Map[{Times @@ MapIndexed[Prime[First@ #2]^#1 &, #], Times @@ MapIndexed[Prime[First@ #2]^#1 &, Table[LengthWhile[#1, # >= j &], {j, #2}]] & @@ {#, Max[#]}} &, Join @@ f[2310]]][[All, -1]] (* Michael De Vlieger, Oct 16 2018 *)

partitionConj(v)=vector(v[1],i,sum(j=1,#v,v[j]>=i))
primeSignature(n)=vecsort(factor(n)[,2]~,,4)
f(n)=if(n==1, return(1)); my(e=partitionConj(primeSignature(n))~); factorback(concat(Mat(primes(#e)~),e))
A025487=[2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768];
concat(1, apply(f, A025487)) \\ Charles R Greathouse IV, Jun 02 2016

If A025487(n) = Product p(i)^e(i), then a(n) = Product A002110(e(i)). I.e., a(n) = A108951(A181819(A025487(n))). a(n) also equals A108951(A181820(n)).

A124859 Multiplicative with p^e -> primorial(e), p prime and e > 0.

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 30, 6, 4, 2, 12, 2, 4, 4, 210, 2, 12, 2, 12, 4, 4, 2, 60, 6, 4, 30, 12, 2, 8, 2, 2310, 4, 4, 4, 36, 2, 4, 4, 60, 2, 8, 2, 12, 12, 4, 2, 420, 6, 12, 4, 12, 2, 60, 4, 60, 4, 4, 2, 24, 2, 4, 12, 30030, 4, 8, 2, 12, 4, 8, 2, 180, 2, 4, 12, 12, 4, 8, 2, 420, 210, 4, 2, 24, 4, 4

Offset: 1

Reinhard Zumkeller, Nov 10 2006

			From _Michael De Vlieger_, Mar 06 2017: (Start)
a(2) = 2 since 2 = 2^1, thus primorial p_1# = 2.
a(4) = 6 since 4 = 2^2, thus primorial p_2# = 2*3 = 6.
a(6) = 4 because 6 is squarefree with omega(6)=2, thus 2^2 = 4.
a(8) = 30 since 8 = 2^3, thus primorial p_3# = 2*3*5 = 30.
a(10) = 4 since 10 is squarefree with omega(10)=2, thus 2^2 = 4.
a(12) = 12 since 12 = 2^1 * 3^2, thus primorials p_1# * p_2# = 2*6 = 12.
(End)

Indranil Ghosh, Table of n, a(n) for n = 1..5000 (first 1000 terms from R. Zumkeller)
Eric Weisstein's World of Mathematics, Prime Factorization
Eric Weisstein's World of Mathematics, Primorial
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A000079, A001221, A001222, A002110, A005117, A028234, A046523, A067029, A072048, A108951, A156552, A181819, A181821, A238745, A278159, A278222.

A124859 := proc(n)
    local a,pf;
    a := 1;
    for pf in ifactors(n)[2] do
        a := a*A002110(pf[2]) ;
    end do:
    a ;
end proc:
seq(A124859(n),n=1..80) ; # R. J. Mathar, Oct 06 2017

Table[Which[n == 1, 1, SquareFreeQ@ n, 2^PrimeNu@ n, True, Times @@ Map[Times @@ Prime@ Range@ # &, #[[All, -1]]]] &@ FactorInteger@ n, {n, 86}] (* Michael De Vlieger, Mar 06 2017 *)

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = prod(j=1, f[k,2], prime(j)); f[k,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 16 2015

from sympy.ntheory.factor_ import core
from sympy import factorint, primorial, primefactors
from operator import mul
def omega(n): return 0 if n==1 else len(primefactors(n))
def a(n):
    f=factorint(n)
    return n if n<3 else 2**omega(n) if core(n) == n else reduce(mul, [primorial(f[i]) for i in f]) # Indranil Ghosh, May 13 2017

(define (A124859 n) (cond ((= 1 n) 1) (else (* (A002110 (A067029 n)) (A124859 (A028234 n)))))) ;; Antti Karttunen, Mar 06 2017

a(A000040(x)^n) = A002110(n); a(A002110(n)) = A000079(n);
a(A005117(n)) = 2^A001221(A005117(n)) = A072048(n);
A001221(a(n)) = A051903(n); A001222(a(n)) = A001222(n).
From Antti Karttunen, Mar 06 2017: (Start)
a(1) = 1, for n > 1, a(n) = A002110(A067029(n)) * a(A028234(n)).
a(n) = A278159(A156552(n)).
a(A278159(n)) = A278222(n).
a(a(n)) = A046523(n). [after Matthew Vandermast's May 19 2012 formula for the latter sequence]
A181819(a(n)) = A238745(n). [after Matthew Vandermast's formula for the latter sequence]
(End)
a(n) = A108951(A181819(n)). [Primorial inflation of the prime shadow of n] - Antti Karttunen, Sep 15 2023

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.

A329044 a(n) = A064989(A324886(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 6, 15, 25, 11, 81, 13, 49, 15625, 36, 17, 225, 19, 625, 117649, 121, 23, 135, 60025, 169, 21, 2401, 29, 21875, 31, 10, 1771561, 289, 697540921, 50625, 37, 361, 4826809, 35, 41, 77, 43, 14641, 84035, 529, 47, 375, 161212051, 3603000625, 24137569, 28561, 53, 441, 2474329, 5764801, 47045881, 841, 59, 42875, 61, 961

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Cf. A034386, A064989, A108951, A276086, A324886, A329045.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A329044(n) = A064989(A324886(n));

a(n) = A064989(A324886(n)).

a(A000040(n)) = A000040(n).

A373158 Fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

0, 2, 6, 4, 30, 8, 210, 6, 12, 32, 2310, 10, 30030, 212, 36, 8, 510510, 14, 9699690, 34, 216, 2312, 223092870, 12, 60, 30032, 18, 214, 6469693230, 38, 200560490130, 10, 2316, 510512, 240, 16, 7420738134810, 9699692, 30036, 36, 304250263527210, 218, 13082761331670030, 2314, 42, 223092872, 614889782588491410, 14

Offset: 1

Antti Karttunen, May 27 2024

nonn

Completely additive with a(p^e) = e * A002110(A000720(p)).

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences related to primorial numbers

Cf. also A000040, A000720, A002110, A003961, A034386, A060389, A108951, A276085, A276154, A373984 [= A108951(n)-a(n)], A373985 [= gcd(a(n), A108951(n))], A373986, A373987.

A373158(n) = { my(f=factor(n)); sum(i=1, #f~, f[i, 2]*prod(i=1,primepi(f[i, 1]),prime(i))); }; \\ corrected Jun 25 2024

From Antti Karttunen, Jun 25 2024, Oct 28 2024: (Start)
a(n) = A276085(A003961(n)).
For n >= 1, a(A000040(n)) = A002110(n), a(A002110(n)) = A060389(n).
(End)

Data [first incorrect term was at a(8)] and the faulty PARI-program corrected by Antti Karttunen, Jun 25 2024

A277120 Number of branching factorizations of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 11, 1, 3, 3, 15, 1, 11, 1, 11, 3, 3, 1, 45, 2, 3, 5, 11, 1, 19, 1, 51, 3, 3, 3, 62, 1, 3, 3, 45, 1, 19, 1, 11, 11, 3, 1, 195, 2, 11, 3, 11, 1, 45, 3, 45, 3, 3, 1, 113, 1, 3, 11, 188, 3, 19, 1, 11, 3, 19, 1, 345, 1, 3, 11, 11, 3

Offset: 1

Michel Marcus, Oct 01 2016

nonn

Per the formula, a(n) = 1 at prime n. As the sequence extends, additional values become more frequent than 1. These values can be characterized, for example, a(n) = 19 is seen at n corresponding to A007304, a(n) = 3 is seen at n corresponding to A006881, a(n) = 113 is seen at n corresponding to A085987. - Bill McEachen, Dec 28 2023
From Antti Karttunen, Jan 02 2024: (Start)
The value of a(n) depends only on the prime signature of n. In other words, for all i, j >= 1, it holds that A101296(i) = A101296(j) => a(i) = a(j). Moreover, it seems that the converse proposition also holds, that for all i, j >= 1, a(i) = a(j) => A101296(i) = A101296(j), i.e., for each distinct prime signature there exists a distinct value of a(n). This has been empirically checked up to the first 21001 prime signatures in A025487 (see A366884), and can be proved if one can show that the latter sequence (equally: A366377) is injective. If this conjecture holds, it would imply an unlimited number of statements like those given in the previous comment (see the formula section of A101296).
Questions: Are there any terms of the form 10k+4 or 10k+6? What is the asymptotic density of terms of the form 10k+5 (those ending with digit "5")? Compare to the data shown in A366884.
For squarefree n > 1, a(n) is never even, and apparently, never a multiple of five. See comments in A052886.
(End)

			In this scheme, the following factorizations of 12 are counted as distinct: 12, 2 x 6, 2 x (2 x 3), 2 x (3 x 2), 3 x 4, 3 x (2 x 2), 4 x 3, (2 x 2) x 3, 6 x 2, (2 x 3) x 2, (3 x 2) x 2, thus a(12) = 11. - _Antti Karttunen_, Nov 02 2016, based on the illustration given at page 14 of Knopfmacher & Mays paper.
The following factorizations of 30 are counted as distinct: 30, 2 x 15, 15 x 2, 3 x 10, 10 x 3, 5 x 6, 6 x 5, 2 x (3 x 5), 2 x (5 x 3), 3 x (2 x 5), 3 x (5 x 2), 5 x (2 x 3), 5 x (3 x 2), (2 x 3) x 5, (2 x 5) x 3, (3 x 2) x 5, (3 x 5) x 2, (5 x 2) x 3, (5 x 3) x 2, thus a(30) = 19. - _Antti Karttunen_, Jan 02 2024

Daniel Mondot, Table of n, a(n) for n = 1..9999
H.-K. Hwang, Théorèmes limites pour les structures combinatoires et les fonctions arithmétiques, PhD thesis, in Ecole Polytechnique, 1994. See page 198.
A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See page 14.
Index entries for sequences computed from exponents in factorization of n

Cf. A007317, A052886, A074206, A101296, A277130, A366377 [= a(A108951(n))], A366884 [= a(A025487(n))].

After n=1 differs from A104725 for the next time at n=32, where a(32) = 51, while A104725(32) = 52.

#include 
#define MAX 10000
/* Number of branching factorizations of n. */
unsigned long n, m, a, b, p, x, nbr[MAX];
int main(void)
{
  for (x=n=1; nDaniel Mondot, Oct 01 2016 */

v[n_] := v[n] = If[n == 1, 0, 1 + Sum[If[d == 1 || d^2 > n, 0, If[d^2 == n, 1, 2]*v[d]*v[n/d]], {d, Divisors[n]}]]; Table[v[n], {n, 1, 100}] (* Vaclav Kotesovec, Jan 13 2024, after Antti Karttunen *)

A277120(n) = if(1==n, 0, 1+sumdiv(n, d, if((1==d)||(d*d)>n,0,if((d*d)==n,1,2)*A277120(d)*A277120(n/d)))); \\ Antti Karttunen, Nov 02 2016, after Daniel Mondot's C-program above.

seq(n)={my(v=vector(n)); for(n=2, n, v[n] = 1 + sumdiv(n, d, v[d]*v[n/d])); v} \\ Andrew Howroyd, Nov 17 2018

a(1) = 0; for n > 1, a(n) = 1 + Sum_{d|n, 1 < d < n} a(d)*a(n/d). - Antti Karttunen, Nov 02 2016, after Daniel Mondot's C program, simplified Dec 30 2023.

For all n >= 1, a(prime^n) = A007317(n), and a(product of n distinct primes) = A052886(n). - Antti Karttunen, Dec 31 2023

More terms from Daniel Mondot, Oct 01 2016

cp's OEIS Frontend

A238690 Let each integer m (1 <= m <= n) be factorized as m = prime_m(1)*prime_m(2)*...*prime_m(bigomega(m)), with the primes sorted in nonincreasing order. Then a(n) is the number of values of m such that each prime_m(i) <= prime_n(i).

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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).

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A336835 Number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=n.

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A181822 a(n) = member of A025487 whose prime signature is conjugate to the prime signature of A025487(n).

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A124859 Multiplicative with p^e -> primorial(e), p prime and e > 0.

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

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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.

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A238690 Let each integer m (1 <= m <= n) be factorized as m = prime_m(1)prime_m(2)...*prime_m(bigomega(m)), with the primes sorted in nonincreasing order. Then a(n) is the number of values of m such that each prime_m(i) <= prime_n(i).