A074206 -id:A074206 - cp's OEIS frontend

A174725 a(n) = (A074206(n) + A008683(n))/2.

1, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 2, 2, 4, 0, 4, 0, 4, 2, 2, 0, 10, 1, 2, 2, 4, 0, 6, 0, 8, 2, 2, 2, 13, 0, 2, 2, 10, 0, 6, 0, 4, 4, 2, 0, 24, 1, 4, 2, 4, 0, 10, 2, 10, 2, 2, 0, 22, 0, 2, 4, 16, 2, 6, 0, 4, 2, 6, 0, 38, 0, 2, 4, 4, 2

Offset: 1

Mats Granvik, Mar 28 2010

nonn

From Mats Granvik, May 25 2017: (Start)
A074206(n) = A002033(n-1) = a(n) + A174726(n).
A008683(n) = a(n) - A174726(n).
Let m = size of matrix a matrix T, and let T be defined as follows:
T(n,k) = if m = 1 then 1 else if mod(n, k) = 0 then if and(n = k, n = m) then 0 else 1 else if and(n = 1, k = m) then 1 else 0
a(n) is then the number of permutation matrices with a positive contribution in the determinant of matrix T. The determinant of T is equal to the Möbius function A008683, see Mathematica program below for how to compute the determinant.
A174726 is the number of permutation matrices with a negative contribution in the determinant of matrix T.
(End)
From Gus Wiseman, Jan 04 2021: (Start)
Also the number of ordered factorizations of n into an even number of factors > 1. The non-ordered case is A339846. For example, the a(n) factorizations for n = 12, 24, 30, 32, 36 are:
  (2*6)  (3*8)      (5*6)   (4*8)      (4*9)
  (3*4)  (4*6)      (6*5)   (8*4)      (6*6)
  (4*3)  (6*4)      (10*3)  (16*2)     (9*4)
  (6*2)  (8*3)      (15*2)  (2*16)     (12*3)
         (12*2)     (2*15)  (2*2*2*4)  (18*2)
         (2*12)     (3*10)  (2*2*4*2)  (2*18)
         (2*2*2*3)          (2*4*2*2)  (3*12)
         (2*2*3*2)          (4*2*2*2)  (2*2*3*3)
         (2*3*2*2)                     (2*3*2*3)
         (3*2*2*2)                     (2*3*3*2)
                                       (3*2*2*3)
                                       (3*2*3*2)
                                       (3*3*2*2)
(End)

Mats Granvik, Table of n, a(n) for n = 1..10000

Cf. A008683, A051731.
The odd version is A174726.
The unordered version is A339846.
A001055 counts factorizations, with strict case A045778.
A058696 counts partitions of even numbers, ranked by A300061.
A074206 counts ordered factorizations, with strict case A254578.
A251683 counts ordered factorizations by product and length.
Other cases of even length:
- A024430 counts set partitions of even length.
- A027187 counts partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A067661 counts strict partitions of even length.
- A332305 counts strict compositions of even length
Cf. A002033, A027193, A028260, A050320, A058695, A236913, A316439, A320655, A320656, A339890, A378216 (Dirichlet inverse).

(* From Mats Granvik, May 25 2017: (Start) *)
Clear[t, nn]; nn = 77; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k == 1, Sum[t[n, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; Monitor[Table[Sum[If[Mod[n, k] == 0, MoebiusMu[k]*t[n/k, 1], 0], {k, 1, 77}], {n, 1, nn}], n]
(* The Möbius function as a determinant *) Table[Det[Table[Table[If[m == 1, 1, If[Mod[n, k] == 0, If[And[n == k, n == m], 0, 1], If[And[n == 1, k == m], 1, 0]]], {k, 1, m}], {n, 1, m}]], {m, 1, 42}]
(* (End) *)
ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n],EvenQ@*Length]],{n,100}] (* Gus Wiseman, Jan 04 2021 *)

a(n) = (Mobius transform of a(n)) + (Mobius transform of A174726). - Mats Granvik, Apr 04 2010
From Mats Granvik, May 25 2017: (Start)
This sequence is the Moebius transform of A074206.
a(n) = (A074206(n) + A008683(n))/2.
(End)
G.f. A(x) satisfies: A(x) = x + Sum_{i>=2} Sum_{j>=2} A(x^(i*j)). - Ilya Gutkovskiy, May 11 2019

References to A002033(n-1) changed to A074206(n) by Antti Karttunen, Nov 23 2024

A163272 Numbers k such that k = A074206(k), the number of ordered factorizations of k.

Original entry on oeis.org

0, 1, 48, 1280, 2496, 28672, 29808, 454656, 2342912, 11534336, 57409536, 218103808, 34753216512, 73014444032, 583041810432, 1305670057984, 2624225017856, 404620279021568, 467515780104192, 1014849232437248, 4446425022726144, 5806013294837760, 46545625738641408

Offset: 1

Mats Granvik, Jul 24 2009

nonn

From Mauro Fiorentini, Jul 15 2018: (Start)
If p is an odd prime, 2^(2*p - 2)*p belongs to the sequence, so the sequence is infinite.
If n^2 + 6*n + 6 = 2*p*q is twice the product of two distinct odd primes, 2^n*p*q belongs to the sequence.
No number of the form 2^n*p^2, with p odd prime, belongs to the sequence. (End)
For every possible prime signature (see A025487) there can be at most one number having it in this sequence. - David A. Corneth, Jul 15 2018
2*10^14 < a(18) <= 404620279021568. Also terms: 467515780104192, 1014849232437248, 4446425022726144, 5806013294837760, and 46545625738641408. - Giovanni Resta, Jul 16 2018
These numbers are named "super-perfect numbers" (Miller), "gamma-perfect numbers" (Sandor & Crstici), "factor-perfect numbers" (Knopfmacher & Mays) and "balanced numbers" (Brown). - Amiram Eldar, Aug 22 2018
From David A. Corneth, Aug 23 2018: (Start)
Suppose one searches terms below u. We have A074206(m * t) > A074206(m) for m, t > 1 so if A074206(m) > u we needn't check any value A074206(m * t) where m * t < u.
All terms < 10^25 except 29809 are of the form 4^e * s where s is a squarefree odd number. (End)

J. Sandor and B. Crstici, Handbook of Number Theory II, Springer, 2004, pp. 54-55.

David A. Corneth, Table of n, a(n) for n = 1..49 (terms < 10^30)
Peter Brown, Number of Ordered Factorizations, 2004.
Martin Klazar and Florian Luca, On the maximal order of numbers in the "factorisatio numerorum" problem, Journal of Number Theory, Vol. 124, No. 2 (2007), pp. 470-490.
Arnold Knopfmacher and M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, Vol. 1, No. 4 (2005), pp. 563-581, DOI: 10.1142/S1793042105000315.
Michael D. Miller, A recursively defined divisor function, The Fibonacci Quarterly, Vol. 13 (1975), pp. 199-204.
Project Euler, Problem 548: Gozinta Chains.

Cf. A025487, A074206.

A074206 := proc(n) option remember; if n <= 1 then n; else add(procname(d), d=numtheory[divisors](n) minus {n}) ; end if; end proc: for n from 1 do if n = A074206(n) then printf("%d,\n",n) ; end if; end do: \\ R. J. Mathar, Aug 01 2009

term(n) = {my(f = A074206(n)); if(factor(n)[, 2] == factor(f)[, 2], f, 0)};
isok(n) = term(n) == n;  \\ David A. Corneth, Jul 15 2018

a(6)-a(7) from R. J. Mathar, Aug 01 2009
a(8)-a(9) from Nathaniel Johnston, Dec 04 2010
a(10)-a(12) from Mauro Fiorentini, Dec 07 2015
a(13)-a(17) from Giovanni Resta, Jul 16 2018, following a suggestion from David A. Corneth
a(18)-a(23) from Amiram Eldar, Aug 22 2018, following the same suggestion with an extended list of terms of A025487.

A050324 Number of ordered factorizations indexed by prime signatures: A074206(A025487).

Original entry on oeis.org

1, 1, 2, 3, 4, 8, 8, 20, 13, 16, 26, 48, 44, 32, 76, 112, 132, 64, 208, 176, 256, 75, 252, 368, 128, 544, 604, 576, 308, 768, 976, 256, 1376, 1888, 1280, 1076, 2208, 818, 2496, 512, 2316, 3392, 1460, 2568, 5536, 2816, 3408, 6080, 3172, 6208, 1024, 7968

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

This sequence can help to find terms for A163272, as has been done by Giovanni Resta. A074206(n) is computed only from the prime signature of n. If A074206(k) has the same prime signature as k then A074206(k) is in A163272. - David A. Corneth, Jul 16 2018

The number of ordered prime factorizations of n is A074206(n), not really A002033(n) = A074206(n-1). This has induced confusion in A002033 so it might be worth mentioning the distinction to be made. - M. F. Hasler, Oct 12 2018

David A. Corneth, Table of n, a(n) for n = 1..12651 (first 300 terms by R. J. Mathar)

Cf. A025487, A034776, A074206, A163272.

A050324 := proc(n)
    A002033(A025487(n)-1) ;
end proc: # R. J. Mathar, May 25 2017
A050324 := A074206 @ A025487; # M. F. Hasler, Oct 12 2018

A050324(n)=A074206(A025487(n)) \\ M. F. Hasler, Oct 12 2018

Edited to accommodate change in A025487's offset by Matthew Vandermast, Nov 27 2009

A307866 K-champion numbers: numbers m such that K(m) > K(j) for all j < m, where K(m) is the Kalmár function (A074206).

Original entry on oeis.org

0, 1, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 192, 240, 288, 360, 432, 480, 576, 720, 864, 960, 1152, 1440, 1728, 1920, 2160, 2304, 2880, 3456, 4320, 5760, 6912, 8640, 11520, 17280, 23040, 25920, 30240, 34560, 46080, 51840, 60480, 69120, 86400, 103680, 120960

Offset: 1

Amiram Eldar, May 02 2019

nonn

The corresponding record values are 0, 1, 2, 3, 4, 8, 20, 26, 48, 76, 112, 132, 208, ... (see the link for more values).
Deléglise et al. (2008) proved that the number of powerful (A001694) terms in this sequence is finite. They ask whether a(391) = 485432135516160000 (the 112th powerful term) is the largest. - Amiram Eldar, Aug 20 2019
Is abs(omega(a(n)) - omega(a(n+1))) <= 1? (Cf. A001221.) - David A. Corneth, Apr 16 2020

David A. Corneth, Table of n, a(n) for n = 1..884 (terms 1..762 from Amiram Eldar, calculated by Deléglise et al.)
David A. Corneth, Table of n, a(n) for n = 1..1544 if abs(omega(a(n)) - omega(a(n + 1))) <= 1
M. Deléglise, M. O. Hernane, and J.-L. Nicolas, Grandes valeurs et nombres champions de la fonction arithmétique de Kalmár, Journal of Number Theory, Vol. 128, No. 6 (2008), pp. 1676-1716.
M. Deléglise, M. O. Hernane, and J.-L. Nicolas, Tables des 761 premiers champions de la fonction de Kalmar, alternative link.
Amiram Eldar, Table of n, a(n), A074206(a(n)), calculated by Deléglise et al.
T. M. A. Fink, Number of ordered factorizations and recursive divisors, arXiv:2307.16691 [math.NT], 2023.

Cf. A001221, A001694, A002093, A033833, A074206, A163272, A330686 (after primorial deflation).

a[0] = 0; a[1] = 1; a[n_] := a[n] = Total[a /@ Most[Divisors[n]]]; s = {}; am=-1; Do[a1 = a[n]; If[a1>am, am=a1; AppendTo[s, n]], {n, 0, 10000}]; s

For n >= 1, a(1+n) = A108951(A330686(n)). - Antti Karttunen, Dec 31 2019

A378647 Dirichlet convolution of A074206 and A103977, where A074206 is the number of ordered factorizations of n, and A103977 is the Zumkeller deficiency of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 13, 14, 15, 16, 17, 22, 19, 22, 21, 22, 23, 40, 25, 26, 27, 28, 29, 42, 31, 32, 33, 34, 35, 64, 37, 38, 39, 52, 41, 54, 43, 44, 45, 46, 47, 96, 49, 50, 51, 52, 53, 70, 55, 64, 57, 58, 59, 126, 61, 62, 63, 64, 65, 78, 67, 68, 69, 74, 71, 176, 73, 74, 75, 76, 77, 90, 79, 120, 81

Offset: 1

Antti Karttunen, Dec 03 2024

nonn

Möbius transform of A378648, which is the Dirichlet convolution of A067824 and A103977.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000027, A008683, A067824, A074206, A103977, A263837, A378648 (inverse Möbius transform), A378649 (Möbius transform), A378650 [= a(n)-n], A378655 (Dirichlet inverse).

A378647(n) = sumdiv(n,d,moebius(d)*A378648(n/d));

A378647(n) = sumdiv(n,d,A074206(d)*A103977(n/d));

a(n) = Sum_{d|n} A074206(d)*A103977(n/d).
a(n) = Sum_{d|n} A008683(d)*A378648(n/d).
a(n) = Sum_{d|n} A067824(d)*A378644(n/d).
a(n) = A378650(n)+n, with a(n) = n if and only if n is a non-abundant number (A263837).

A361662 Least number k >= 1 such that A074206(k) is divisible by n.

Original entry on oeis.org

1, 4, 6, 8, 24, 48, 96, 12, 216, 24, 60, 48, 30, 96, 210, 32, 288, 216, 72, 24, 216, 60, 240, 48, 210, 36, 6480, 96, 15552, 4320, 7560, 64, 120, 288, 2520, 216, 5040, 72, 960, 768, 2520, 216, 576, 60, 83160, 240, 7680, 48, 18480, 13860, 7776, 144, 1152, 6480

Offset: 1

Pontus von Brömssen, Mar 20 2023

nonn

a(n) exists for all n. (This is problem 5 of the first round of the British Mathematical Olympiad 2022/2023.)

All terms are in A025487.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
British Mathematical Olympiad, Round 1, Problem 5, 2022/2023.
Index to sequences related to Olympiads.

Cf. A025487, A074206, A361663, A361664, A361666.

f(n)={if(!n, 0, my(sig=factor(n)[, 2], m=vecsum(sig)); sum(k=0, m, prod(i=1, #sig, binomial(sig[i]+k-1, k-1))*sum(r=k, m, binomial(r, k)*(-1)^(r-k))))}; \\ A074206
a(n) = my(k=1); while (f(k) % n, k++); k; \\ Michel Marcus, Mar 23 2023

a(n) = A025487(A361663(n)).

A173382 Partial sums of A074206.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 9, 10, 14, 16, 19, 20, 28, 29, 32, 35, 43, 44, 52, 53, 61, 64, 67, 68, 88, 90, 93, 97, 105, 106, 119, 120, 136, 139, 142, 145, 171, 172, 175, 178, 198, 199, 212, 213, 221, 229, 232, 233, 281, 283, 291, 294, 302, 303, 323, 326, 346, 349, 352, 353, 397, 398, 401, 409, 441, 444, 457

Offset: 0

Jonathan Vos Post, Feb 17 2010

Partial sums of number of ordered factorizations of n.

a(n) is also the number of ways to tile a strip of length at most log(n) with tiles having lengths {log(k) : k>=2}. - David Bevan, Jan 07 2025

			a(96) = 0 + 1 + 1 + 1 + 2 + 1 + 3 + 1 + 4 + 2 + 3 + 1 + 8 + 1 + 3 + 3 + 8 + 1 + 8 + 1 + 8 + 3 + 3 + 1 + 20 + 2 + 3 + 4 + 8 + 1 + 13 + 1 + 16 + 3 + 3 + 3 + 26 + 1 + 3 + 3 + 20 + 1 + 13 + 1 + 8 + 8 + 3 + 1 + 48 + 2 + 8 + 3 + 8 + 1 + 20 + 3 + 20 + 3 + 3 + 1 + 44 + 1 + 3 + 8 + 32 + 3 + 13 + 1 + 8 + 3 + 13 + 1 + 76 + 1 + 3 + 8 + 8 + 3 + 13 + 1 + 48 + 8 + 3 + 1 + 44 + 3 + 3 + 3 + 20 + 1 + 44 + 3 + 8 + 3 + 3 + 3 + 112.

Shikao Ikehara, On Kalmar's Problem in “Factorisatio Numerorum”, Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, Vol. 21 (1939) pp. 208-219.
Shikao Ikehara, On Kalmar's Problem in “Factorisatio Numerorum” II, Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, Vol. 23 (1941) pp. 767-774.
Kalmár, Laszlo. "Über die mittlere Anzahl der Produktdarstellungen der Zahlen.(Erste Mitteilung)'." Acta Litt. ac Scient. Szeged 5 (1931): 95-107.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000
David Bevan and Julien Condé, Introducing irrational enumeration: analytic combinatorics for objects of irrational size, arXiv:2412.14682 [math.CO], 2024. See p. 11.
Ann Clifton, Eva Czabarka, Kevin Liu, Sarah Loeb, Utku Okur, Laszlo Szekely, and Kristina Wicke, Universal rooted phylogenetic tree shapes and universal tanglegrams, arXiv:2308.06580 [math.CO], 2023.
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A074206, A002033, A001055, A050324, A000670.

A025523 is an essentially identical sequence.

Clear[a]; a[0] = 0; a[1] = 1; a[n_] := a[n] = 1 + Sum[a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 0, 100}] (* Vaclav Kotesovec, Jan 31 2019 *)
Clear[a]; a[0] = 0; a[1] = 1; a[n_] := a[n] = Total[a /@ Most[Divisors[n]]]; Join[{0}, Accumulate[a /@ Range[100]]] (* Vaclav Kotesovec, Jan 31 2019, after Jean-François Alcover, faster *)

a(n) = Sum_{i=0..n} A074206(i).

a(n) ~ -n^r / (r*Zeta'(r)), where r = A107311 = 1.728647238998183618135103... is the root of the equation Zeta(r) = 2. - Vaclav Kotesovec, Jan 31 2019

Terms corrected by N. J. A. Sloane, May 04 2016

A174888 Triangle read by rows. Row sums = Mobius function A008683, row counts of nonzero elements = A074206.

Original entry on oeis.org

1, 0, -1, 0, -1, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, -1, 1, -1, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0

Offset: 1

Mats Granvik, Apr 01 2010

This table cannot be completed as a triangle beyond row 95 because the ordered factorization value for 96 is 112, which is greater than the row length 96. Values in the first column are either zero or one, and are chosen so that the number of nonzero elements in each row equals A074206 while rows sums equals A008683. The row indices for nonzero elements in the 1st column are A174891.

			Table begins:
n\k|....1....2....3....4....5....6....7....8....9...10
---|--------------------------------------------------
1..|....1
2..|....0...-1
3..|....0...-1....0
4..|....1...-1....0....0
5..|....0...-1....0....0....0
6..|....1...-1....1....0....0....0
7..|....0...-1....0....0....0....0....0
8..|....1...-1....1...-1....0....0....0....0
9..|....1...-1....0....0....0....0....0....0....0
10.|....1...-1....1....0....0....0....0....0....0....0
Notice that the 30th row begins with a zero:
0,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1

Cf. A008683, A074206, A174889, A174891, A174892.

First column = A174889.

Beginning from the second column: a(n)=if k<=A074206 then ((-1)^(k+1)) else 0.

A361664 a(n) = A074206(A361662(n))/n.

Original entry on oeis.org

1, 1, 1, 1, 4, 8, 16, 1, 28, 2, 4, 4, 1, 8, 5, 1, 32, 14, 4, 1, 12, 2, 16, 2, 3, 1, 1104, 4, 2944, 848, 804, 1, 4, 16, 164, 7, 544, 2, 64, 32, 140, 6, 32, 1, 6812, 8, 768, 1, 752, 286, 528, 4, 64, 552, 260, 2, 3904, 1472, 32, 424, 16, 402, 4, 1, 220, 2, 372, 8

Offset: 1

Pontus von Brömssen, Mar 20 2023

nonn

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

Cf. A074206, A361662, A361667, A361668.

a(n) = A361667(n) if n is not in A361668. The equality holds also for some n in A361668; for example, a(256) = A361667(256) = 1.

A160086 a(n) = A104725(n) - A074206(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 7, 0, 3, 0, 3, 0, 0, 0, 25, 0, 0, 1, 3, 0, 6, 0, 36, 0, 0, 0, 36, 0, 0, 0, 25, 0, 6, 0, 3, 3, 0, 0, 152, 0, 3, 0, 3, 0, 25, 0, 25, 0, 0, 0, 69, 0, 0, 3, 171, 0, 6, 0, 3, 0, 6, 0, 279, 0, 0, 3, 3, 0, 6, 0, 152, 7, 0, 0, 69, 0, 0, 0, 25, 0, 69, 0, 3, 0, 0, 0

Offset: 0

Augustine O. Munagi, May 01 2009

nonn

a(n) is also the excess of the number of labeled factorizations of n over the number of ordered factorizations (see the Munagi link for definition of labeled factorization)

			a(8)=1 because A074206(8)=4 and A104725(8)=5, so a(8)=5-4. The only labeled factorization of 8 which is not an ordered factorization is (2_1.2_3)(2_2). a(9)=0 because A074206(9)=2=A104725(9). The labeled factorizations of 9, namely (9_1) and (3_1)(3_2), are also ordered factorizations.

Antti Karttunen, Table of n, a(n) for n = 0..10000 (computed from the b-files of A074206 and A104725)
A. O. Munagi, Labeled factorization of integers, Electron. J. Combin. 16 (2009), no. 1, Research Paper 50, 17pp.

Cf. A074206, A104725.

a:=proc(n::integer) local u, r, i, j, k; if n<2 then return 0; end if; u:=map(x->x[2], ifactors(n)[2]); r:=add(u[i], i=1..nops(u)); add(add((-1)^i*binomial(k, i)*product(binomial(u[j]+k-i-1, u[j]), j=1..nops(u)), i=0..k-1)*(bell(k-1)-1), k=1..r); end proc: seq(a(n),n=0..99);

a(n) = Sum(ordfac(n,k)*(Bell(k-1)-1),k=1..Omega(n)), where ordfac(n,k)=number of ordered factorizations of n into k factors.

cp's OEIS Frontend

A174725 a(n) = (A074206(n) + A008683(n))/2.

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A163272 Numbers k such that k = A074206(k), the number of ordered factorizations of k.

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A050324 Number of ordered factorizations indexed by prime signatures: A074206(A025487).

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A307866 K-champion numbers: numbers m such that K(m) > K(j) for all j < m, where K(m) is the Kalmár function (A074206).

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A378647 Dirichlet convolution of A074206 and A103977, where A074206 is the number of ordered factorizations of n, and A103977 is the Zumkeller deficiency of n.

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A361662 Least number k >= 1 such that A074206(k) is divisible by n.

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A173382 Partial sums of A074206.

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Formula

Extensions

A174888 Triangle read by rows. Row sums = Mobius function A008683, row counts of nonzero elements = A074206.