A067255 -id:A067255 - cp's OEIS frontend

A001222 Number of prime divisors of n counted with multiplicity (also called big omega of n, bigomega(n) or Omega(n)).

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 6, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 5, 4, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 6, 1, 3, 3, 4, 1, 3, 1, 4, 3, 2, 1, 5, 1, 3, 2

Offset: 1

N. J. A. Sloane

Maximal number of terms in any factorization of n.
Number of prime powers (not including 1) that divide n.
Sum of exponents in prime-power factorization of n. - Daniel Forgues, Mar 29 2009
Sum_{d|n} 2^(-A001221(d) - a(n/d)) = Sum_{d|n} 2^(-a(d) - A001221(n/d)) = 1 (see Dressler and van de Lune link). - Michel Marcus, Dec 18 2012
Row sums in A067255. - Reinhard Zumkeller, Jun 11 2013
Conjecture: Let f(n) = (x+y)^a(n), and g(n) = x^a(n), and h(n) = (x+y)^A046660(n) * y^A001221(n) with x, y complex numbers and 0^0 = 1. Then f(n) = Sum_{d|n} g(d)*h(n/d). This is proved for x = 1-y (see Dressler and van de Lune link). - Werner Schulte, Feb 10 2018
Let r, s be some fixed integers. Then we have:
  (1) The sequence b(n) = Dirichlet convolution of r^bigomega(n) and s^bigomega(n) is multiplicative with b(p^e) = (r^(e+1)-s^(e+1))/(r-s) for prime p and e >= 0. The case r = s leads to b(p^e) = (e+1)*r^e.
  (2) The sequence c(n) = Dirichlet convolution of r^bigomega(n) and mu(n)*s^bigomega(n) is multiplicative with c(p^e) = (r-s)*r^(e-1) and c(1) = 1 for prime p and e > 0 where mu(n) = A008683(n). - Werner Schulte, Feb 20 2019
a(n) is also the length of the composition series for every solvable group of order n. - Miles Englezou, Apr 25 2024

			16=2^4, so a(16)=4; 18=2*3^2, so a(18)=3.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 119, #12, omega(n).
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 48-57.
M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 92.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 844.
Benoit Cloitre, A tauberian approach to RH, arXiv:1107.0812 [math.NT], 2011.
Robert E. Dressler and Jan van de Lune, Some remarks concerning the number theoretic functions omega and Omega, Proc. Amer. Math. Soc. 41 (1973), 403-406.
G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92. Also Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI (2000): 262-275.
Douglas E. Iannucci and Urban Larsson, Game values of arithmetic functions, arXiv:2101.07608 [math.NT], 2021. Section 1.1.1. pp. 4-5.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.10.
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Roundness
Wolfram Research, First 50 numbers factored
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n
Index entries for sequences computed from indices in prime factorization

Cf. A001221 (omega, primes counted without multiplicity), A008836 (Liouville's lambda, equal to (-1)^a(n)), A046660, A144494, A074946, A134334.
Bisections give A091304 and A073093. A086436 is essentially the same sequence. Cf. A022559 (partial sums), A066829 (parity), A092248 (parity of omega).
Sequences listing n such that a(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13),  A069275 (r = 14),  A069276 (r = 15),  A069277 (r = 16),  A069278 (r = 17),  A069279 (r = 18),  A069280 (r = 19),  A069281 (r = 20). - Jason Kimberley, Oct 02 2011
Cf. A079149 (primes adj. to integers with at most 2 prime factors, a(n)<=2).
Cf. A000120, A020639, A091202, A091222, A156552, A267116, A268387.
Cf. A027748 (without repetition).
Cf. A000010.

Concatenation([0],List([2..150],n->Length(Factors(n)))); # Muniru A Asiru, Feb 21 2019

import Math.NumberTheory.Primes.Factorisation (factorise)
a001222 = sum . snd . unzip . factorise
-- Reinhard Zumkeller, Nov 28 2015

using Nemo
function NumberOfPrimeFactors(n; distinct=true)
    distinct && return length(factor(ZZ(n)))
    sum(e for (p, e) in factor(ZZ(n)); init=0)
end
println([NumberOfPrimeFactors(n, distinct=false) for n in 1:60])  # Peter Luschny, Jan 02 2024

[n eq 1 select 0 else &+[p[2]: p in Factorization(n)]: n in [1..120]]; // Bruno Berselli, Nov 27 2013

with(numtheory): seq(bigomega(n), n=1..111);

Array[ Plus @@ Last /@ FactorInteger[ # ] &, 105]
PrimeOmega[Range[120]] (* Harvey P. Dale, Apr 25 2011 *)

PARI
```
vector(100,n,bigomega(n))
```

from sympy import primeomega
def a(n): return primeomega(n)
print([a(n) for n in range(1, 112)]) # Michael S. Branicky, Apr 30 2022

[sloane.A001222(n) for n in (1..120)] # Giuseppe Coppoletta, Jan 19 2015

[gp.bigomega(n) for n in range(1,131)] # G. C. Greubel, Jul 13 2024

(define (A001222 n) (let loop ((n n) (z 0)) (if (= 1 n) z (loop (/ n (A020639 n)) (+ 1 z)))))
;; Requires also A020639 for which an equally naive implementation can be found under that entry. - Antti Karttunen, Apr 12 2017

n = Product_(p_j^k_j) -> a(n) = Sum_(k_j).
Dirichlet g.f.: ppzeta(s)*zeta(s). Here ppzeta(s) = Sum_{p prime} Sum_{k>=1} 1/(p^k)^s. Note that ppzeta(s) = Sum_{p prime} 1/(p^s-1) and ppzeta(s) = Sum_{k>=1} primezeta(k*s). - Franklin T. Adams-Watters, Sep 11 2005
Totally additive with a(p) = 1.
a(n) = if n=1 then 0 else a(n/A020639(n)) + 1. - Reinhard Zumkeller, Feb 25 2008
a(n) = Sum_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011
a(n) = A022559(n) - A022559(n-1).
G.f.: Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)). - Ilya Gutkovskiy, Jan 25 2017
a(n) = A091222(A091202(n)) = A000120(A156552(n)). - Antti Karttunen, circa 2004 and Mar 06 2017
a(n) >= A267116(n) >= A268387(n). - Antti Karttunen, Apr 12 2017
Sum_{k=1..n} 2^(-A001221(gcd(n,k)) - a(n/gcd(n,k)))/phi(n/gcd(n,k)) = Sum_{k=1..n} 2^(-a(gcd(n,k)) - A001221(n/gcd(n,k)))/phi(n/gcd(n,k)) = 1, where phi = A000010. - Richard L. Ollerton, May 13 2021
a(n) = a(A046523(n)) = A007814(A108951(n)) = A061395(A122111(n)) = A056239(A181819(n)) = A048675(A293442(n)). - Antti Karttunen, Apr 30 2022

More terms from David W. Wilson

A007814 Exponent of highest power of 2 dividing n, a.k.a. the binary carry sequence, the ruler sequence, or the 2-adic valuation of n.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0

Offset: 1

John Tromp, Dec 11 1996

This sequence is an exception to my usual rule that when every other term of a sequence is 0 then those 0's should be omitted. In this case we would get A001511. - N. J. A. Sloane
To construct the sequence: start with 0,1, concatenate to get 0,1,0,1. Add + 1 to last term gives 0,1,0,2. Concatenate those 4 terms to get 0,1,0,2,0,1,0,2. Add + 1 to last term etc. - Benoit Cloitre, Mar 06 2003
The sequence is invariant under the following two transformations: increment every element by one (1, 2, 1, 3, 1, 2, 1, 4, ...), put a zero in front and between adjacent elements (0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, ...). The intermediate result is A001511. - Ralf Hinze (ralf(AT)informatik.uni-bonn.de), Aug 26 2003
Fixed point of the morphism 0->01, 1->02, 2->03, 3->04, ..., n->0(n+1), ..., starting from a(1) = 0. - Philippe Deléham, Mar 15 2004
Fixed point of the morphism 0->010, 1->2, 2->3, ..., n->(n+1), .... - Joerg Arndt, Apr 29 2014
a(n) is also the number of times to repeat a step on an even number in the hailstone sequence referenced in the Collatz conjecture. - Alex T. Flood (whiteangelsgrace(AT)gmail.com), Sep 22 2006
Let F(n) be the n-th Fermat number (A000215). Then F(a(r-1)) divides F(n)+2^k for r = k mod 2^n and r != 1. - T. D. Noe, Jul 12 2007
The following relation holds: 2^A007814(n)*(2*A025480(n-1)+1) = A001477(n) = n. (See functions hd, tl and cons in [Paul Tarau 2009].)
a(n) is the number of 0's at the end of n when n is written in base 2.
a(n+1) is the number of 1's at the end of n when n is written in base 2. - M. F. Hasler, Aug 25 2012
Shows which bit to flip when creating the binary reflected Gray code (bits are numbered from the right, offset is 0). That is, A003188(n) XOR A003188(n+1) == 2^A007814(n). - Russ Cox, Dec 04 2010
The sequence is squarefree (in the sense of not containing any subsequence of the form XX) [Allouche and Shallit]. Of course it contains individual terms that are squares (such as 4). - Comment expanded by N. J. A. Sloane, Jan 28 2019
a(n) is the number of zero coefficients in the n-th Stern polynomial, A125184. - T. D. Noe, Mar 01 2011
Lemma: For n < m with r = a(n) = a(m) there exists n < k < m with a(k) > r. Proof: We have n=b2^r and m=c2^r with b < c both odd; choose an even i between them; now a(i2^r) > r and n < i2^r < m. QED. Corollary: Every finite run of consecutive integers has a unique maximum 2-adic valuation. - Jason Kimberley, Sep 09 2011
a(n-2) is the 2-adic valuation of A000166(n) for n >= 2. - Joerg Arndt, Sep 06 2014
a(n) = number of 1's in the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} p_j-th prime (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(24)=3; indeed, the partition having Heinz number 24 = 2*2*2*3  is [1,1,1,2]. - Emeric Deutsch, Jun 04 2015
a(n+1) is the difference between the two largest parts in the integer partition having viabin number n (0 is assumed to be a part). Example: a(20) = 2. Indeed, we have 19 = 10011_2, leading to the Ferrers board of the partition [3,1,1]. For the definition of viabin number see the comment in A290253. - Emeric Deutsch, Aug 24 2017
Apart from being squarefree, as noted above, the sequence has the property that every consecutive subsequence contains at least one number an odd number of times. - Jon Richfield, Dec 20 2018
a(n+1) is the 2-adic valuation of Sum_{e=0..n} u^e = (1 + u + u^2 + ... + u^n), for any u of the form 4k+1 (A016813). - Antti Karttunen, Aug 15 2020
{a(n)} represents the "first black hat" strategy for the game of countably infinitely many hats, with a probability of success of 1/3; cf. the Numberphile link below. - Frederic Ruget, Jun 14 2021
a(n) is the least nonnegative integer k for which there does not exist i+j=n and a(i)=a(j)=k (cf. A322523). - Rémy Sigrist and Jianing Song, Aug 23 2022

			2^3 divides 24, so a(24)=3.
From _Omar E. Pol_, Jun 12 2009: (Start)
Triangle begins:
  0;
  1,0;
  2,0,1,0;
  3,0,1,0,2,0,1,0;
  4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0;
  5,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0;
  6,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,5,0,1,0,2,...
(End)

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 27.
K. Atanassov, On the 37th and the 38th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 83-85.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

T. D. Noe, Table of n, a(n) for n = 1..10000
Joerg Arndt, Subset-lex: did we miss an order?, arXiv:1405.6503 [math.CO], 2014.
K. Atanassov, On Some of Smarandache's Problems
Alain Connes, Caterina Consani, and Henri Moscovici, Zeta zeros and prolate wave operators, arXiv:2310.18423 [math.NT], 2023.
Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, (2009) Discrete Math., 309 (2009), 6245-6254.
Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, arXiv:0901.1397 [math.CO], 2009.
M. Hassani, Equations and inequalities involving v_p(n!), J. Inequ. Pure Appl. Math. 6 (2005) vol. 2, #29.
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 61. Book's website
R. Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi, Section 3.2.3.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., 274 (2004), 147-160.
Francis Laclé, 2-adic parity explorations of the 3n+ 1 problem, hal-03201180v2 [cs.DM], 2021.
Shuo Li, Palindromic length sequence of the ruler sequence and of the period-doubling sequence, arXiv:2007.08317 [math.CO], 2020.
Nicolas Mallet, Trial for a proof of the Syracuse conjecture, arXiv preprint arXiv:1507.05039 [math.GM], 2015.
S. Mazzanti, Plain Bases for Classes of Primitive Recursive Functions, Mathematical Logic Quarterly, 48 (2002).
Matthew Andres Moreno, Luis Zaman, and Emily Dolson, Structured Downsampling for Fast, Memory-efficient Curation of Online Data Streams, arXiv:2409.06199 [cs.DS], 2024. See p. 5.
Sascha Mücke, Coding Nuggets Faster QUBO Brute-Force Solving, TU Dortmund Univ. (Germany 2023).
S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.
Numberphile, Hat Problems - Numberphile
Giovanni Pighizzini, Limited Automata: Properties, Complexity and Variants, International Conference on Descriptional Complexity of Formal Systems (DCFS 2019) Descriptional Complexity of Formal Systems, Lecture Notes in Computer Science (LNCS, Vol. 11612) Springer, Cham, 57-73.
Simon Plouffe, On the values of the functions ... [zeta and Gamma] ..., arXiv preprint arXiv:1310.7195 [math.NT], 2013.
A. Postnikov (MIT) and B. Sagan, What power of two divides a weighted Catalan number?, arXiv:math/0601339 [math.CO], 2006.
Lara Pudwell and Eric Rowland, Avoiding fractional powers over the natural numbers, arXiv:1510.02807 [math.CO] (2015). Also Electronic Journal of Combinatorics, Volume 25(2) (2018), #P2.27. See Section 2.
Ville Salo, Decidability and Universality of Quasiminimal Subshifts, arXiv preprint arXiv:1411.6644 [math.DS], 2014.
Vladimir Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2014.
F. Smarandache, Only Problems, Not Solutions!.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Paul Tarau, A Groupoid of Isomorphic Data Transformations. Calculemus 2009, 8th International Conference, MKM 2009, pp. 170-185, Springer, LNAI 5625.
P. M. B. Vitanyi, An optimal simulation of counter machines, SIAM J. Comput, 14:1(1985), 1-33.
Eric Weisstein's World of Mathematics, Binary, Binary Carry Sequence, and Double-Free Set.
Wikipedia, P-adic order.
Index entries for sequences that are fixed points of mappings
Index entries for sequences related to binary expansion of n

Cf. A011371 (partial sums), A094267 (first differences), A001511 (bisection), A346070 (mod 4).
Bisection of A050605 and |A088705|. Pairwise sums are A050603 and A136480. Difference of A285406 and A281264.
This is Guy Steele's sequence GS(1, 4) (see A135416). Cf. A053398(1,n). Column/row 1 of table A050602.
Cf. A007949 (3-adic), A235127 (4-adic), A112765 (5-adic), A122841 (6-adic), A214411 (7-adic), A244413 (8-adic), A122840 (10-adic).
Cf. A000079, A002487, A006519, A051064, A055457, A063787, A215366, A220466.
Cf. A086463 (Dgf at s=2).

a007814 n = if m == 0 then 1 + a007814 n' else 0
            where (n', m) = divMod n 2
-- Reinhard Zumkeller, Jul 05 2013, May 14 2011, Apr 08 2011

a007814 n | odd n = 0 | otherwise = 1 + a007814 (n `div` 2)
--  Walt Rorie-Baety, Mar 22 2013

[Valuation(n, 2): n in [1..120]]; // Bruno Berselli, Aug 05 2013

ord := proc(n) local i,j; if n=0 then return 0; fi; i:=0; j:=n; while j mod 2 <> 1 do i:=i+1; j:=j/2; od: i; end proc: seq(ord(n), n=1..111);
A007814 := n -> padic[ordp](n,2): seq(A007814(n), n=1..111); # Peter Luschny, Nov 26 2010

Table[IntegerExponent[n, 2], {n, 64}] (* Eric W. Weisstein *)
IntegerExponent[Range[64], 2] (* Eric W. Weisstein, Feb 01 2024 *)
p=2; Array[ If[ Mod[ #, p ]==0, Select[ FactorInteger[ # ], Function[ q, q[ [ 1 ] ]==p ], 1 ][ [ 1, 2 ] ], 0 ]&, 96 ]
DigitCount[BitXor[x, x - 1], 2, 1] - 1; a different version based on the same concept: Floor[Log[2, BitXor[x, x - 1]]] (* Jaume Simon Gispert (jaume(AT)nuem.com), Aug 29 2004 *)
Nest[Join[ #, ReplacePart[ #, Length[ # ] -> Last[ # ] + 1]] &, {0, 1}, 5] (* N. J. Gunther, May 23 2009 *)
Nest[ Flatten[# /. a_Integer -> {0, a + 1}] &, {0}, 7] (* Robert G. Wilson v, Jan 17 2011 *)

PARI
```
A007814(n)=valuation(n,2);
```

import math
def a(n): return int(math.log(n - (n & n - 1), 2)) # Indranil Ghosh, Apr 18 2017

def A007814(n): return (~n & n-1).bit_length() # Chai Wah Wu, Jul 01 2022

sapply(1:100,function(x) sum(gmp::factorize(x)==2)) # Christian N. K. Anderson, Jun 20 2013

(define (A007814 n) (let loop ((n n) (e 0)) (if (odd? n) e (loop (/ n 2) (+ 1 e))))) ;; Antti Karttunen, Oct 06 2017

a(n) = A001511(n) - 1.
a(2*n) = A050603(2*n) = A001511(n).
a(n) = A091090(n-1) + A036987(n-1) - 1.
a(n) = 0 if n is odd, otherwise 1 + a(n/2). - Reinhard Zumkeller, Aug 11 2001
Sum_{k=1..n} a(k) = n - A000120(n). - Benoit Cloitre, Oct 19 2002
G.f.: A(x) = Sum_{k>=1} x^(2^k)/(1-x^(2^k)). - Ralf Stephan, Apr 10 2002
G.f. A(x) satisfies A(x) = A(x^2) + x^2/(1-x^2). A(x) = B(x^2) = B(x) - x/(1-x), where B(x) is the g.f. for A001151. - Franklin T. Adams-Watters, Feb 09 2006
Totally additive with a(p) = 1 if p = 2, 0 otherwise.
Dirichlet g.f.: zeta(s)/(2^s-1). - Ralf Stephan, Jun 17 2007
Define 0 <= k <= 2^n - 1; binary: k = b(0) + 2*b(1) + 4*b(2) + ... + 2^(n-1)*b(n-1); where b(x) are 0 or 1 for 0 <= x <= n - 1; define c(x) = 1 - b(x) for 0 <= x <= n - 1; Then: a(k) = c(0) + c(0)*c(1) + c(0)*c(1)*c(2) + ... + c(0)*c(1)...c(n-1); a(k+1) = b(0) + b(0)*b(1) + b(0)*b(1)*b(2) + ... + b(0)*b(1)...b(n-1). - Arie Werksma (werksma(AT)tiscali.nl), May 10 2008
a(n) = floor(A002487(n - 1) / A002487(n)). - Reikku Kulon, Oct 05 2008
Sum_{k=1..n} (-1)^A000120(n-k)*a(k) = (-1)^(A000120(n)-1)*(A000120(n) - A000035(n)). - Vladimir Shevelev, Mar 17 2009
a(A001147(n) + A057077(n-1)) = a(2*n). - Vladimir Shevelev, Mar 21 2009
For n>=1, a(A004760(n+1)) = a(n). - Vladimir Shevelev, Apr 15 2009
2^(a(n)) = A006519(n). - Philippe Deléham, Apr 22 2009
a(n) = A063787(n) - A000120(n). - Gary W. Adamson, Jun 04 2009
a(C(n,k)) = A000120(k) + A000120(n-k) - A000120(n). - Vladimir Shevelev, Jul 19 2009
a(n!) = n - A000120(n). - Vladimir Shevelev, Jul 20 2009
v_{2}(n) = Sum_{r>=1} (r / 2^(r+1)) Sum_{k=0..2^(r+1)-1} e^(2(k*Pi*i(n+2^r))/(2^(r+1))). - A. Neves, Sep 28 2010, corrected Oct 04 2010
a(n) mod 2 = A096268(n-1). - Robert G. Wilson v, Jan 18 2012
a(A005408(n)) = 1; a(A016825(n)) = 3; A017113(a(n)) = 5; A051062(a(n)) = 7; a(n) = (A037227(n)-1)/2. - Reinhard Zumkeller, Jun 30 2012
a((2*n-1)*2^p) = p, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 04 2013
a(n) = A067255(n,1). - Reinhard Zumkeller, Jun 11 2013
a(n) = log_2(n - (n AND n-1)). - Gary Detlefs, Jun 13 2014
a(n) = 1 + A000120(n-1) - A000120(n), where A000120 is the Hamming weight function. - Stanislav Sykora, Jul 14 2014
A053398(n,k) = a(A003986(n-1,k-1)+1); a(n) = A053398(n,1) = A053398(n,n) = A053398(2*n-1,n) = Min_{k=1..n} A053398(n,k). - Reinhard Zumkeller, Aug 04 2014
a((2*x-1)*2^n) = a((2*y-1)*2^n) for positive n, x and y. - Juri-Stepan Gerasimov, Aug 04 2016
a(n) = A285406(n) - A281264(n). - Ralf Steiner, Apr 18 2017
a(n) = A000005(n)/(A000005(2*n) - A000005(n)) - 1. - conjectured by Velin Yanev, Jun 30 2017, proved by Nicholas Stearns, Sep 11 2017
Equivalently to above formula, a(n) = A183063(n) / A001227(n), i.e., a(n) is the number of even divisors of n divided by number of odd divisors of n. - Franklin T. Adams-Watters, Oct 31 2018
a(n)*(n mod 4) = 2*floor(((n+1) mod 4)/3). - Gary Detlefs, Feb 16 2019
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - Amiram Eldar, Jul 11 2020
a(n) = 2*Sum_{j=1..floor(log_2(n))} frac(binomial(n, 2^j)*2^(j-1)/n). - Dario T. de Castro, Jul 08 2022
a(n) = A070939(n) - A070939(A030101(n)). - Andrew T. Porter, Dec 16 2022
a(n) = floor((gcd(n, 2^n)^(n+1) mod (2^(n+1)-1)^2)/(2^(n+1)-1)) (see Lemma 3.4 from Mazzanti's 2002 article). - Lorenzo Sauras Altuzarra, Mar 10 2024
a(n) = 1 - A088705(n). - Chai Wah Wu, Sep 18 2024

Formula index adapted to the offset of A025480 by R. J. Mathar, Jul 20 2010

Edited by Ralf Stephan, Feb 08 2014

A061395 Let p be the largest prime factor of n; if p is the k-th prime then set a(n) = k; a(1) = 0 by convention.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 3, 1, 7, 2, 8, 3, 4, 5, 9, 2, 3, 6, 2, 4, 10, 3, 11, 1, 5, 7, 4, 2, 12, 8, 6, 3, 13, 4, 14, 5, 3, 9, 15, 2, 4, 3, 7, 6, 16, 2, 5, 4, 8, 10, 17, 3, 18, 11, 4, 1, 6, 5, 19, 7, 9, 4, 20, 2, 21, 12, 3, 8, 5, 6, 22, 3, 2, 13, 23, 4, 7, 14, 10, 5, 24, 3, 6, 9, 11, 15

Offset: 1

Henry Bottomley, Apr 30 2001

Records occur at the primes. - Robert G. Wilson v, Dec 30 2007
For n > 1: length of n-th row in A067255. - Reinhard Zumkeller, Jun 11 2013
a(n) = the largest part of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(20) = 3; indeed, the partition having Heinz number 20 = 2*2*5 is [1,1,3]. - Emeric Deutsch, Jun 04 2015

			a(20) = 3 since the largest prime factor of 20 is 5, which is the 3rd prime.

Álvar Ibeas, Table of n, a(n) for n = 1..100000 (first 1000 terms from Harry J. Smith)
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A006530, A055396, A061394, A133674, A243055.

Cf. A029837, A087207.

a061395 = a049084 . a006530  -- Reinhard Zumkeller, Jun 11 2013

with(numtheory):
a:= n-> pi(max(1, factorset(n)[])):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 03 2013

Insert[Table[PrimePi[FactorInteger[n][[ -1]][[1]]], {n, 2, 120}], 0, 1] (* Stefan Steinerberger, Apr 11 2006 *)
f[n_] := PrimePi[ FactorInteger@n][[ -1, 1]]; Array[f, 94] (* Robert G. Wilson v, Dec 30 2007 *)

a(n) = if (n==1, 0, primepi(vecmax(factor(n)[,1]))); \\ Michel Marcus, Nov 14 2022

from sympy import primepi, primefactors
def a(n): return 0 if n==1 else primepi(primefactors(n)[-1])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017

A000040(a(n)) = A006530(n); a(n) = A049084(A006530(n)). - Reinhard Zumkeller, May 22 2003
A243055(n) = a(n) - A055396(n). - Antti Karttunen, Mar 07 2017
a(n) = A000720(A006530(n)). - Alois P. Heinz, Mar 05 2020
a(n) = A029837(A087207(n)+1). - Flávio V. Fernandes, Apr 24 2025

Definition reworded by N. J. A. Sloane, Jul 01 2008

A124010 Triangle in which first row is 0, n-th row (n>1) lists the exponents of distinct prime factors ("ordered prime signature") in the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 6, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1

Offset: 1

Franklin T. Adams-Watters, Nov 01 2006

A001222(n) = Sum(T(n,k), 1 <= k <= A001221(n)); A005361(n) = Product(T(n,k), 1 <= k <= A001221(n)), n>1; A051903(n) = Max(T(n,k): 1 <= k <= A001221(n)); A051904(n) = Min(T(n,k), 1 <= k <= A001221(n)); A067029(n) = T(n,1); A071178(n) = T(n,A001221(n)); A064372(n)=Sum(A064372(T(n,k)), 1 <= k <= A001221(n)). - Reinhard Zumkeller, Aug 27 2011
Any finite sequence of natural numbers appears as consecutive terms. - Paul Tek, Apr 27 2013
For n > 1: n-th row = n-th row of A067255 without zeros. - Reinhard Zumkeller, Jun 11 2013
Most often the prime signature is given as a sorted representative of the multiset of the nonzero exponents, either in increasing order, which yields A118914, or, most commonly, in decreasing order, which yields A212171. - M. F. Hasler, Oct 12 2018

			Initial values of exponents are:
1, [0]
2, [1]
3, [1]
4, [2]
5, [1]
6, [1, 1]
7, [1]
8, [3]
9, [2]
10, [1, 1]
11, [1]
12, [2, 1]
13, [1]
14, [1, 1]
15, [1, 1]
16, [4]
17, [1]
18, [1, 2]
19, [1]
20, [2, 1]
...

Reinhard Zumkeller, Rows n = 1..10000 of triangle, flattened
Index entries for sequences computed from exponents in factorization of n

Cf. A027748, A001221 (row lengths, n>1), A001222 (row sums), A027746, A020639, A064372, A067029 (first column).

Sorted rows: A118914, A212171.

a124010 n k = a124010_tabf !! (n-1) !! (k-1)
a124010_row 1 = [0]
a124010_row n = f n a000040_list where
   f 1 _      = []
   f u (p:ps) = h u 0 where
     h v e | m == 0 = h v' (e + 1)
           | m /= 0 = if e > 0 then e : f v ps else f v ps
           where (v',m) = divMod v p
a124010_tabf = map a124010_row [1..]
-- Reinhard Zumkeller, Jun 12 2013, Aug 27 2011

expts:=proc(n) local t1,t2,t3,t4,i; if n=1 then RETURN([0]); fi; if isprime(n) then RETURN([1]); fi; t1:=ifactor(n); if nops(factorset(n))=1 then RETURN([op(2,t1)]); fi; t2:=nops(t1); t3:=[]; for i from 1 to t2 do t4:=op(i,t1); if nops(t4) = 1 then t3:=[op(t3),1]; else t3:=[op(t3),op(2,t4)]; fi; od; RETURN(t3); end; # N. J. A. Sloane, Dec 20 2007
PrimeSignature := proc(n) local F, e, k; F := ifactors(n)[2]; [seq(e, e = seq(F[k][2], k = 1..nops(F)))] end:
ListTools:-Flatten([[0], seq(PrimeSignature(n), n = 1..73)]); # Peter Luschny, Jun 15 2025

row[1] = {0}; row[n_] := FactorInteger[n][[All, 2]] // Flatten; Table[row[n], {n, 1, 80}] // Flatten (* Jean-François Alcover, Aug 19 2013 *)

print1(0); for(n=2,50, f=factor(n)[,2]; for(i=1,#f,print1(", "f[i]))) \\ Charles R Greathouse IV, Nov 07 2014

A124010_row(n)=if(n,factor(n)[,2]~,[0]) \\ M. F. Hasler, Oct 12 2018

from sympy import factorint
def a(n):
    f=factorint(n)
    return [0] if n==1 else [f[i] for i in f]
for n in range(1, 21): print(a(n)) # Indranil Ghosh, May 16 2017

n = Product_k A027748(n,k)^a(n,k).

Name edited by M. F. Hasler, Apr 08 2022

A071178 Exponent of the largest prime factor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Benoit Cloitre, Jun 10 2002

a(n) = the multiplicity of the largest part in the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(18) = 2; indeed, the partition having Heinz number 18 = 2*3*3 is [1,2,2]. - Emeric Deutsch, Jun 04 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A006530, A053585, A067029, A067255, A070003, A124010, A215366.

a071178 = last . a124010_row -- Reinhard Zumkeller, Aug 27 2011

with(numtheory): with(padic):
a:= n-> `if`(n=1, 0, ordp(n, max(factorset(n)[]))):
seq(a(n), n=1..120);  # Alois P. Heinz, Jun 04 2015

a[n_] := FactorInteger[n] // Last // Last; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 12 2015 *)
Join[{0},Table[FactorInteger[n][[-1,2]],{n,2,120}]] (* Harvey P. Dale, Aug 02 2025 *)

a(n) = if(n == 1, 0, my(e = factor(n)[, 2]); e[#e]); \\ Amiram Eldar, Oct 02 2024

from sympy import factorint
def A071178(n): return max(factorint(n).items())[1] if n>1 else 0 # Chai Wah Wu, Oct 10 2023

a(n) = A124010(n, A001221(n)); A053585(n) = A006530(n)^a(n). - Reinhard Zumkeller, Aug 27 2011
a(n) = A067255(n, A001222(n)). - Reinhard Zumkeller, Jun 11 2013
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 (since the asymptotic density of A070003 is 0). - Amiram Eldar, Oct 02 2024

A133457 Irregular triangle read by rows: row n gives exponents in expression for n as a sum of powers of 2.

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 2, 1, 2, 0, 1, 2, 3, 0, 3, 1, 3, 0, 1, 3, 2, 3, 0, 2, 3, 1, 2, 3, 0, 1, 2, 3, 4, 0, 4, 1, 4, 0, 1, 4, 2, 4, 0, 2, 4, 1, 2, 4, 0, 1, 2, 4, 3, 4, 0, 3, 4, 1, 3, 4, 0, 1, 3, 4, 2, 3, 4, 0, 2, 3, 4, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 0, 5, 1, 5, 0, 1, 5, 2, 5, 0, 2, 5, 1, 2, 5, 0, 1, 2, 5, 3, 5, 0, 3, 5

Offset: 1

Masahiko Shin, Nov 27 2007

This sequence contains every increasing finite sequence. For example, the finite sequence {0,2,3,5} arises from n = 45.
Essentially A030308(n,k)*k, then entries removed where A030308(n,k)=0. - R. J. Mathar, Nov 30 2007
In the corresponding irregular triangle {a(n)+1}, the m-th row gives all positive integer roots m_i of polynomial {m,k}. - see link [Shevelev]; see also A264613. - Vladimir Shevelev, Dec 13 2015

			1 = 2^0.
2 = 2^1.
3 = 2^0 + 2^1.
4 = 2^2.
5 = 2^0 + 2^2.
etc. and reading the exponents gives the rows of the triangle.

Reinhard Zumkeller, Rows n = 1..1024 of triangle, flattened
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. (See Section 3, Theorem 21 and Section 8, Theorem 50)

Cf. A003071, A030308, A048793, A067255, A264613.

Cf. A073642 (row sums), A272011 (rows reversed).

a133457 n k = a133457_tabf !! (n-1) !! n
a133457_row n = a133457_tabf !! (n-1)
a133457_tabf = map (fst . unzip . filter ((> 0) . snd) . zip [0..]) $
                   tail a030308_tabf
-- Reinhard Zumkeller, Oct 28 2013, Feb 06 2013

A133457 := proc(n) local a,bdigs,i ; a := [] ; bdigs := convert(n,base,2) ; for i from 1 to nops(bdigs) do if op(i,bdigs) <> 0 then a := [op(a),i-1] ; fi ; od: a ; end: seq(op(A133457(n)),n=1..80) ; # R. J. Mathar, Nov 30 2007

Array[Join @@ Position[#, 1] - 1 &@ Reverse@ IntegerDigits[#, 2] &, 41] // Flatten (* Michael De Vlieger, Oct 08 2017 *)

a(n) = A048793(n) - 1.

More terms from R. J. Mathar, Nov 30 2007

A115627 Irregular triangle read by rows: T(n,k) = multiplicity of prime(k) as a divisor of n!.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 1, 4, 2, 1, 4, 2, 1, 1, 7, 2, 1, 1, 7, 4, 1, 1, 8, 4, 2, 1, 8, 4, 2, 1, 1, 10, 5, 2, 1, 1, 10, 5, 2, 1, 1, 1, 11, 5, 2, 2, 1, 1, 11, 6, 3, 2, 1, 1, 15, 6, 3, 2, 1, 1, 15, 6, 3, 2, 1, 1, 1, 16, 8, 3, 2, 1, 1, 1, 16, 8, 3, 2, 1, 1, 1, 1

Offset: 2

Franklin T. Adams-Watters, Jan 26 2006

The factorization of n! is n! = 2^T(n,1)*3^T(n,2)*...*p_(pi(n))^T(n,pi(n)) where p_k = k-th prime, pi(n) = A000720(n).
Nonzero terms of A085604; T(n,k) = A085604(n,k), k = 1..A000720(n). - Reinhard Zumkeller, Nov 01 2013
For n=2, 3, 4 and 5, all terms of the n-th row are odd. Are there other such rows? - Michel Marcus, Nov 11 2018
From Gus Wiseman, May 15 2019: (Start)
Differences between successive rows are A067255, so row n is the sum of the first n row-vectors of A067255 (padded with zeros on the right so that all n row-vectors have length A000720(n)). For example, the first 10 rows of A067255 are
  {}
  1
  0 1
  2 0
  0 0 1
  1 1 0
  0 0 0 1
  3 0 0 0
  0 2 0 0
  1 0 1 0
with column sums (8,4,2,1), which is row 10.
(End)
For all prime p > 7, 3*p > 2*nextprime(p), so for any n > 21 there will always be a prime p dividing n! with exponent 2 and there are no further rows with all entries odd. - Charlie Neder, Jun 03 2019

			From _Gus Wiseman_, May 09 2019: (Start)
Triangle begins:
   1
   1  1
   3  1
   3  1  1
   4  2  1
   4  2  1  1
   7  2  1  1
   7  4  1  1
   8  4  2  1
   8  4  2  1  1
  10  5  2  1  1
  10  5  2  1  1  1
  11  5  2  2  1  1
  11  6  3  2  1  1
  15  6  3  2  1  1
  15  6  3  2  1  1  1
  16  8  3  2  1  1  1
  16  8  3  2  1  1  1  1
  18  8  4  2  1  1  1  1
(End)
m such that 5^m||101!: floor(log(101)/log(5)) = 2 terms. floor(101/5) = 20. floor(20/5) = 4. So m = u_1 + u_2 = 20 + 4 = 24. - _David A. Corneth_, Jun 22 2014

T. D. Noe, Rows n = 2..300, flattened
H. T. Davis, Tables of the Mathematical Functions, Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX. [Annotated scan of pages 204-208 of Volume 2.] See Table 2 on page 206.
Wenguang Zhai, On the prime power factorization of n!, Journal of Number Theory, Volume 129, Issue 8, August 2009, pages 1820-1836.
Index entries for sequences related to factorial numbers

Row lengths are A000720.
Row-sums are A022559.
Row-products are A135291.
Row maxima are A011371.
Columns include A011371, A054861, A027868, A054896, A090617, A064458, A090620.
Cf. A090622, A090623, A000142, A115628.
Cf. A085604, A141809.
Cf. A034876, A067255, A071626, A076934, A322583, A325272, A325273, A325276, A325508, A325509.

a115627 n k = a115627_tabf !! (n-2) !! (k-1)
a115627_row = map a100995 . a141809_row . a000142
a115627_tabf = map a115627_row [2..]
-- Reinhard Zumkeller, Nov 01 2013

A115627 := proc(n,k) local d,p; p := ithprime(k) ; n-add(d,d=convert(n,base,p)) ; %/(p-1) ; end proc: # R. J. Mathar, Oct 29 2010

Flatten[Table[Transpose[FactorInteger[n!]][[2]], {n, 2, 20}]] (* T. D. Noe, Apr 10 2012 *)
T[n_, k_] := Module[{p, jm}, p = Prime[k]; jm = Floor[Log[p, n]]; Sum[Floor[n/p^j], {j, 1, jm}]]; Table[Table[T[n, k], {k, 1, PrimePi[n]}], {n, 2, 20}] // Flatten (* Jean-François Alcover, Feb 23 2015 *)

a(n)=my(i=2);while(n-primepi(i)>1,n-=primepi(i);i++);p=prime(n-1);sum(j=1,log(i)\log(p),i\=p) \\ David A. Corneth, Jun 21 2014

T(n,k) = Sum_{i=1..inf} floor(n/(p_k)^i). (Although stated as an infinite sum, only finitely many terms are nonzero.)

T(n,k) = Sum_{i=1..floor(log(n)/log(p_k))} floor(u_i) where u_0 = n and u_(i+1) = floor((u_i)/p_k). - David A. Corneth, Jun 22 2014

A252738 Row products of irregular table A005940: a(0) = 1; a(1) = 2; for n > 1: 2^(2^(n-2)) * a(n-1) * A003961(a(n-1)); also row products of A163511, A253563, A253565, and A332977.

Original entry on oeis.org

1, 2, 12, 2160, 2449440000, 8488905214204800000000000, 3025568387202006082882734693673523654400000000000000000000000000

Offset: 0

Antti Karttunen, Dec 21 2014

nonn

			From _Michael De Vlieger_, Jul 21 2023: (Start)
a(0) = 1 = product of {1},
a(1) = 2^1 = product of {2},
a(2) = 2^2 * 3^1 = product of {3, 2^2},
a(3) = 2^4 * 3^3 * 5^1 = product of {5, 2^1*3^1, 3^2, 2^3},
a(4) = 2^8 * 3^7 * 5^4 * 7^1 = product of
  {7, 2^1*5^1, 3^1*5^1, 2^2*3^1, 5^2, 2^1*3^2, 3^3, 2^4},
...
Table of e(n,k) where a(n) = Product_{k=1..n+1} prime(k)^e(n,k):
prime(k)|    2    3    5   7  11  13  17  19 23 29 31 ...
   n\k  |    1    2    3   4   5   6   7   8  9 10 11 ...
   ----------------------------------------------------
    0   |    1
    1   |    2    1
    2   |    4    3    1
    3   |    8    7    4   1
    4   |   16   15   11   5   1
    5   |   32   31   26  16   6   1
    6   |   64   63   57  42  22   7   1
    7   |  128  127  120  99  64  29   8   1
    8   |  256  255  247 219 163  93  37   9  1
    9   |  512  511  502 466 382 256 130  46 10  1
   10   | 1024 1023 1013 968 848 638 386 176 56 11  1
  ... (End)

Alois P. Heinz, Table of n, a(n) for n = 0..9

These are row products of irregular tables A005940, A163511, A253563 and A253565, which all are shaped like a binary tree.
Partial products of A252740.
Cf. A252737 (row sums), A252739 (divided by n), A252741 (divided by n!).
Cf. A000079, A000225, A000290, A003961, A163511, A191555, A266639, A267096, A332977.

Table[Times @@ Array[Prime[# + 1]^Sum[Binomial[n, # + j], {j, 0, n}] &, n + 1, 0], {n, 0, 5}] (* Michael De Vlieger, Jul 21 2023 *)

allocatemem(234567890);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A252738print(up_to_n) = { my(s, i=0, n=0); for(n=0, up_to_n, if(0 == n, s = 1, if(1 == n, s = 2; lev = vector(1); lev[1] = 2, oldlev = lev; lev = vector(2*length(oldlev)); s = 1; for(i = 0, (2^(n-1))-1, lev[i+1] = if((i%2),A003961(oldlev[(i\2)+1]),2*oldlev[(i\2)+1]); s *= lev[i+1]))); write("b252738.txt", n, " ", s)); }; \\ Counts them empirically.
A252738print(7);

(definec (A252738rec n) (if (<= n 1) (+ 1 n) (* (A000079 (A000079 (- n 2))) (A252738rec (- n 1)) (A003961 (A252738rec (- n 1)))))) ;; Implements the given recurrence; uses the memoizing definec-macro.
(define (A252738 n) (if (zero? n) 1 (mul A163511 (A000079 (- n 1)) (A000225 n))))
(define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))
;; Another alternative, implementing the new recurrence:
(definec (A252738 n) (if (<= n 1) (+ 1 n) (* (A267096 (- n 2)) (A000290 (A252738 (- n 1)))))) ;; Antti Karttunen, Feb 06 2016

a(0) = 1; a(1) = 2; for n > 1: a(n) = 2^(2^(n-2)) * a(n-1) * A003961(a(n-1)).
a(0) = 1; for n>=1: a(n) = Product_{k=A000079(n-1) .. A000225(n)} A163511(k) = Product_{k=2^(n-1) .. (2^n)-1} A163511(k).
a(0) = 1; a(1) = 2; for n > 1: a(n) = A267096(n-2) * a(n-1)^2. [Compare to the formulas of A191555] - Antti Karttunen, Feb 06 2016
From Michael De Vlieger, Jul 21 2023: (Start)
a(n) = Product_{k=1..n+1} prime(k)^e(n,k), where e(n,k) = k-th term in row n of A055248.
A067255(a(n)) = row n of A055248. (End)

Typos in the second formula corrected by Antti Karttunen, Feb 06 2016

A241918 Table of partitions where the ordering is based on the modified partial sums of the exponents of primes in the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5

Offset: 1

Antti Karttunen, May 03 2014, based on Marc LeBrun's Jan 11 2006 message on SeqFan mailing list

a(1) = 0 by convention (stands for an empty partition).

For n >= 2, A203623(n-1)+2 gives the index to the beginning of row n and for n>=1, A203623(n)+1 is the index to the end of row n.

			Table begins:
Row     Partition
[ 1]    0;         (stands for empty partition)
[ 2]    1;         (as 2 = 2^1)
[ 3]    1,1;       (as 3 = 2^0 * 3^1)
[ 4]    2;         (as 4 = 2^2)
[ 5]    1,1,1;     (as 5 = 2^0 * 3^0 * 5^1)
[ 6]    2,2;       (as 6 = 2^1 * 3^1)
[ 7]    1,1,1,1;   (as 7 = 2^0 * 3^0 * 5^0 * 7^1)
[ 8]    3;         (as 8 = 2^3)
[ 9]    1,2;       (as 9 = 2^0 * 3^2)
[10]    2,2,2;     (as 10 = 2^1 * 3^0 * 5^1)
[11]    1,1,1,1,1;
[12]    3,3;
[13]    1,1,1,1,1,1;
[14]    2,2,2,2;
[15]    1,2,2;     (as 15 = 2^0 * 3^1 * 5^1)
[16]    4;
[17]    1,1,1,1,1,1,1;
[18]    2,3;       (as 18 = 2^1 * 3^2)
etc.
If n is 2^k (k>=1), then the partition is a singleton {k}, otherwise, add one to the exponent of 2 (= A007814(n)), and subtract one from the exponent of the greatest prime dividing n (= A071178(n)), leaving the intermediate exponents as they are, and then take partial sums of all, thus resulting for e.g. 15 = 2^0 * 3^1 * 5^1 the modified sequence of exponents {0+1, 1, 1-1} -> {1,1,0}, whose partial sums {1,1+1,1+1+0} -> {1,2,2} give the corresponding partition at row 15.

Antti Karttunen, Table of n, a(n) for n = 1..10081; rows 1..521 flattened.
Marc LeBrun's original "crazy order" mapping for partitions (Copy of Marc's Jan 11 2006 message in OEIS Wiki)

For n>=2, the length of row n is given by A061395(n).
Cf. also A067255, A203623, A241914.
Other tables of partitions: A112798 (also based on prime factorization), A227739, A242628 (encoded in the binary representation of n), and A036036-A036037, A080576-A080577, A193073 for various lexicographical orderings.
Permutation A241909 maps between order of partitions employed here, and the order employed in A112798.
Permutation A122111 is induced when partitions in this list are conjugated.
A241912 gives the row numbers for which the corresponding rows in A112798 and here are the conjugate partitions of each other.

Table[If[n == 1, {0}, Function[s, Function[t, Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, s]]]]@ ConstantArray[0, Transpose[s][[1, -1]]]][FactorInteger[n] /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]], {n, 31}] // Flatten (* Michael De Vlieger, May 12 2017 *)

If A241914(n)=0 and A241914(n+1)=0, a(n) = A067255(n); otherwise, if A241914(n)=0 and A241914(n+1)>0, a(n) = A067255(n)+1; otherwise, if A241914(n)>0 and A241914(n+1)=0, a(n) = a(n-1) + A067255(n) - 1, otherwise, when A241914(n)>0 and A241914(n+1)>0, a(n) = a(n-1) + A067255(n).

A347284 a(n) = Product_{j=1..A089576(n)} p_j^e_j with e_j = floor(e_(j-1)*log(p_(j-1))/log(p_j)) where the first factor is 2^n.

Original entry on oeis.org

1, 2, 12, 24, 720, 151200, 302400, 1814400, 4191264000, 8382528000, 251727315840000, 503454631680000, 3020727790080000, 1542111744113740800000, 3084223488227481600000, 92526704646824448000000, 555160227880946688000000, 1110320455761893376000000, 10769764221549079560253440000000

Offset: 0

Michael De Vlieger and David James Sycamore, Aug 26 2021

nonn

a(n) is the product of the largest prime power divisors p_j^e_j such that p_j^e_j < p_(j-1)^e_(j-1), beginning with p_1^e_1 = 2^n and proceeding with the next prime p until e_j = 0.
{a(n)} is a subset of A025487 which is a subset of A055932. All terms are products of primorials. No primes p_j for 1 <= j <= L have e = 0 with the exception of a(0) = 2^0. Let L = A001221(a(n)).
The largest primorial divisor P(L) = A2110(L).
For n > 0, all terms are even.
The greatest prime divisor p_L has multiplicity e_L = 1.
All multiplicities e are distinct; for 1 <= j <= L, the multiplicity e_j >= L - j + 1.
a(k) | a(n) for 0 <= k <= n.
The numbers q = a(n+1)/a(n) are primorials.
Finite intersection of A002182 and a(n) = {1, 2, 12, 360, 75600}.
Chernoff number A006939(L) | a(n). Quotient K = a(n) | A006939(L) is in A025487.
The prime shape of terms resembles a simplified map of the US state of Idaho.

			a(0) = 2^0 = 1;
a(1) = 2^1 = 2, since 3^1 > 2^1;
a(2) = 2^2 * 3^1, since 3^1 < 2^2 but 3^2 > 2^2, and since 5^1 > 3^1;
a(3) = 2^3 * 3^1, since 3^1 < 2^3 but 3^2 > 2^3, and 5^1 > 3^1;
a(4) = 2^4 * 3^2 * 5^1, since 3^2 < 2^4 yet 3^3 > 2^4, 5^1 < 3^2 yet 5^2 > 3^2, and 7^1 > 5^1; etc.
Prime shapes of a(n) for 2 <= n <= 5:
                                                     5  o
                                    4  o             4  x
                     3  o           3  x             3  x x
      2  x           2  x           2  x x           2  x x x
a(2)  1  X X   a(3)  1  X X   a(4)  1  X X X   a(5)  1  X X X X
         2 3            2 3            2 3 5            2 3 5 7
This demonstrates that a(n) is in A025487, that A002110(A001221(a(n))) is the greatest primorial divisor of a(n) as a consequence (prime divisors represented by capital X's), and Chernoff A006939(A001221(a(n))) | n, prime divisors represented by x's of any case. a(n) = A006939(A001221(a(n))) * k, k in A025487, represented by o's.
Because each multiplicity e is necessarily distinct, we may compactify a(n) using Sum_{k=1..omega(a(n))} 2^(e-1).
Prime shapes of a(12):
      12  o
      11  o
      10  o
       9  o
       8  o
       7  o o
       6  x o
       5  x x
       4  x x x
       3  x x x x
       2  x x x x x
a(12)  1  X X X X X X
          2 3 5 7 ...
a(12) = A006939(6) * 2^6 * 3^2
      = 5244319080000 * 64 * 9
      = 3020727790080000.
                                                 O
                                       O         x
                            O          x         x
                  O         x          x o       x x
         O        x         x o        x x o     x x x
  O      x o      x x       x x o      x x x o   x x x x
a(1)*6 = a(2)*2 = a(3)*30 = a(4)*210 = a(5)*2 =  a(6), etc., hence a(n) can be generated by a list of indices of primorials {1, 2, 1, 3, 4, 1, 1, 5, ...} and thereby be efficiently compactified.

Michael De Vlieger, Table of n, a(n) for n = 0..144
Michael De Vlieger, Bitmap resulting from binary compactification of a(n), 0 <= n <= 4096.
Michael De Vlieger, Animation of prime shapes of a(n) for 2 <= n <= 37, illustrating a(n) as a product of a particular sequence of primorials.

Cf. A000079, A001221, A002110, A002182, A006939, A067255, A089576, A347285, A347288, A347354.

Subsequence of A025487, A055932, A363063.

Array[Times @@ NestWhile[Append[#1, #2^Floor@ Log[#2, #1[[-1]]]] & @@ {#, Prime[Length@ # + 1]} &, {2^#}, Last[#] > 1 &] &, 18, 0] (* or *)
Block[{nn = 2^5, a = {}, b, e, i, m, p}, Array[Set[e[#], 0] &, Floor[2^# If[# <= 4, 1/2, -1 + 2^(7/(3 #))]] &[Ceiling@ Log2@ nn]]; Do[e[1]++; b = {2^e[1]}; Do[If[Last[b] == 1, Break[], i = e[j]; p = Prime[j]; While[p^i < b[[j - 1]], i++]; AppendTo[b, p^(i - 1)]; If[i > e[j], e[j]++]], {j, 2, k}]; AppendTo[a, Times @@ b], {k, nn}]; Prepend[a, 1]]
(* Generate up to 4096 terms from the bitmap image *)
With[{r = ImageData@ Import["https://oeis.org/A347284/a347284.png"]}, {1}~Join~Table[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, Reverse@ Position[r[[i]], 0.][[All, 1]]], {i, 20}]]
(* Generate up to 10000 terms using b-file at A347354 (numbers are large as n increases, limit nn is set to 120): *)
Block[{nn = 120, s, m}, s = Import["https://oeis.org/A347354/b347354.txt", "Data"][[1 ;; nn, -1]]; m = Prime@ Range@ Max[s]; {1}~Join~FoldList[Times, Map[Times @@ m[[1 ;; #]] &, s]]] (* Michael De Vlieger, Sep 25 2021 *)

a(n) = Product_{j=1..k} p_j^T(n,j) where T = A347285 and k = A089576(n).
Row n of A347285 yields row a(n) of A067255.
a(n) = product of row n of A347288.

Definition edited by Peter Munn, May 19 2023

cp's OEIS Frontend

A001222 Number of prime divisors of n counted with multiplicity (also called big omega of n, bigomega(n) or Omega(n)).

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A007814 Exponent of highest power of 2 dividing n, a.k.a. the binary carry sequence, the ruler sequence, or the 2-adic valuation of n.

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A061395 Let p be the largest prime factor of n; if p is the k-th prime then set a(n) = k; a(1) = 0 by convention.

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A124010 Triangle in which first row is 0, n-th row (n>1) lists the exponents of distinct prime factors ("ordered prime signature") in the prime factorization of n.

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A071178 Exponent of the largest prime factor of n.