A163272 -id:A163272 - cp's OEIS frontend

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

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

A122408 Numbers n such that A067824(n) = n.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 32, 40, 64, 128, 224, 256, 264, 512, 1024, 2048, 4096, 5632, 8192, 16384, 26624, 32768, 65536, 72192, 131072, 154752, 262144, 524288, 557056, 1048576, 2072576, 2097152, 2490368, 4194304, 5537792, 8388608, 10518528, 16777216, 33554432, 48234496

Offset: 1

Reinhard Zumkeller, Sep 03 2006

nonn

			m=40, the proper divisors of 40 are 1, 2, 4, 5, 8, 10 and 20:
A067824(40) = 1+Sum(A067824(d): 1<=d<40) = 1+(1+2+4+2+8+6+16) = 40, therefore 40 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..100

A000079 is a subsequence.

Cf. A002033, A067824, A074206, A163272.

A002033(a(n)) = A074206(a(n)) = a(n)/2.

A067824(a(n)) = a(n).

More terms from Amiram Eldar, Aug 31 2019

A270308 Numbers which are less than the number of their ordered factorizations.

Original entry on oeis.org

72, 96, 120, 144, 192, 216, 240, 288, 336, 360, 384, 432, 480, 504, 540, 576, 600, 648, 672, 720, 768, 840, 864, 960, 1008, 1080, 1152, 1200, 1260, 1296, 1344, 1440, 1512, 1536, 1584, 1620, 1680, 1728, 1800, 1872, 1920, 1944, 2016, 2112, 2160, 2240, 2304, 2400

Offset: 1

Paolo P. Lava, Mar 15 2016

Also numbers which are less than the number of their perfect factorizations or gozinta chains.

The least odd term of this sequence is 51288546684375. - Amiram Eldar, Apr 11 2020

			The ordered partitions of 72 are 76, of 96 are 112, of 120 are 132, etc.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Paolo P. Lava)

Cf. A002033, A074206, A163272.

P:=proc(q) local a,j,k,m; a:= array(1..q); for k from 1 to q do a[k]:=0 od; a[1]:=1;
for j from 2 to q do for m from 1 to j-1 do if j mod m=0 then a[j]:= a[j]+a[m]; fi; od;
if j

f[1] = 1; f[n_] := DivisorSum[n, f[#] &, # < n &]; Select[Range[2400], f[#] > # &] (* Amiram Eldar, Apr 11 2020 *)

Solution of the inequation n < A002033(n-1) = A074206(n).

A307867 Numbers m such that K(m)/m > K(j)/j for all j < m, where K(m) is the Kalmár function (A074206).

Original entry on oeis.org

1, 72, 96, 144, 240, 288, 480, 576, 720, 1152, 1440, 2880, 4320, 5760, 8640, 11520, 17280, 34560, 51840, 69120, 103680, 138240, 207360, 241920, 311040, 345600, 362880, 414720, 483840, 622080, 725760, 829440, 967680, 1244160, 1451520, 1935360, 2073600, 2419200

Offset: 1

Amiram Eldar, May 02 2019

nonn

Subsequence of A307866.

Amiram Eldar, Table of n, a(n) for n = 1..393

Cf. A004394, A163272, A307866.

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

A318251 Lesser of amicable numbers pair (m, n) such that n = H(m) and m = H(n) where H(n) = A074206(n) is the number of ordered factorizations of n.

Original entry on oeis.org

6144, 19329024, 939524096, 4026531840, 309237645312, 6146186280960, 52158082842624, 29273397577908224

Offset: 1

Amiram Eldar, Aug 22 2018

The larger numbers in each pair are in A318252.
Analogous to A002025 as A163272 is analogous to A000396.
If p and 4p+1 are primes then 2^(4p-1)*p is in this sequence, therefore if A023212 is infinite then also this sequence is.
The terms were calculated using an extended list of terms of A025487.

			6144 is in the sequence since A074206(6144) = 13312 and A074206(13312) = 6144.

Cf. A023212, A074206, A090866, A163272, A318252.

f(n) = if( n<2, n>0, my(A = divisors(n)); sum(k=1, #A-1, f(A[k])));
isok(n)={my(a=f(n)); a>n && f(a)==n;} \\ Michel Marcus, Sep 26 2018

A318252 Larger of amicable numbers pair (m, n) such that n = H(m) and m = H(n) where H(n) = A074206(n) is the number of ordered factorizations of n.

Original entry on oeis.org

13312, 81551360, 1946157056, 128580583424, 12695923326976, 33590071001088, 2182874178519040, 59672695062659072

Offset: 1

Amiram Eldar, Aug 22 2018

The lesser numbers in each pair are in A318251.
Analogous to A002046 as A163272 is analogous to A000396.
If p and 4p+1 are primes then 2^(4p-2)*(4p+1) is in this sequence.

			13312 is in the sequence since A074206(13312) = 6144 and A074206(6144) = 13312.

Cf. A023212, A074206, A090866, A163272, A318251.

f(n) = if( n<2, n>0, my(A = divisors(n)); sum(k=1, #A-1, f(A[k])));
isok(n)={my(a=f(n)); aMichel Marcus, Sep 26 2018

A334256 Numbers k such that H(k) = 2*k, where H(k) is the number of ordered factorizations of k (A074206).

Original entry on oeis.org

3072, 1310720, 469762048, 48378511622144, 14636698788954112, 1115414963960152064, 1254378597012249509888, 358899852698093036240896, 28472620903563746322679857152

Offset: 1

David A. Corneth and Amiram Eldar, Apr 20 2020

If p is an odd prime then 2^(4*p - 2) * p is a term, hence this sequence is infinite.
Since A074206(k) depends only on the prime signature (A124010) of k, then each term is of the form A050324(k)/2 = A074206(A025487(k))/2.
Besides terms of the form 2^(4*p - 2) * p at least 79 terms not of this form are known. For example, 1115414963960152064 = 2^46 * 11^2 * 131 is a term not of this form. To ease the search, can we narrow the possible prime signatures of terms?

			3072 is a term since A074206(3072) = 6144 = 2 * 3072.

David A. Corneth, List of found terms in form of oddpart * 2^(multiplicity of 2)

Subsequence of A270308.
Cf. A074206, A122408, A163272.
Cf. A025487, A050324, A124010.

h[1] = 1; h[n_] := h[n] = DivisorSum[n, h[#] &, # < n &]; Select[Range[1.5*10^6], h[#] == 2*# &]

PARI
```
is(n) = A074206(n) == n<<1
```

A327774 Composite numbers m such that tau_k(m) = m for some k, where tau_k is the k-th Piltz divisor function (A077592).

Original entry on oeis.org

18, 36, 75, 100, 200, 224, 225, 441, 560, 1183, 1344, 1920, 3025, 8281, 26011, 34606, 64009, 72030, 76895, 115351, 197173, 280041, 494209, 538265, 1168561, 1947271, 2927521, 3575881, 3613153, 3780295, 4492125, 7295401, 10665331, 11580409, 12511291, 13476375, 15381133

Offset: 1

Amiram Eldar, Sep 25 2019

nonn

The prime numbers are excluded from this sequence since tau_p(p) = p for all primes p.

The corresponding values of k are 3, 3, 5, 4, 4, 4, 5, 6, 4, 13, 4, 4, 10, 13, 37, 11, 22, 7, 13, 61, 73, 17, 37, 13, 46, 157, 58, 61, 193, 29, 9, 73, 277, 82, 37, 9, 313, ...

			18 is in the sequence since tau_3(18) = A007425(18) = 18.

Cf. A097989.

Cf. A007425, A007426, A034695, A061200, A077592, A111217, A111218, A111219, A111220, A111221, A111306, A163272.

fun[e_, k_] := Times @@ (Binomial[# + k - 1, k - 1] & /@ e); tau[n_, k_] := fun[ FactorInteger[n][[;; , 2]], k]; aQ[n_] := CompositeQ[n] && Module[{k = 2}, While[(t = tau[n, k]) < n, k++]; t == n]; Select[Range[10^5], aQ]

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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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A122408 Numbers n such that A067824(n) = n.

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A270308 Numbers which are less than the number of their ordered factorizations.

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A307867 Numbers m such that K(m)/m > K(j)/j for all j < m, where K(m) is the Kalmár function (A074206).

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A318251 Lesser of amicable numbers pair (m, n) such that n = H(m) and m = H(n) where H(n) = A074206(n) is the number of ordered factorizations of n.

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A318252 Larger of amicable numbers pair (m, n) such that n = H(m) and m = H(n) where H(n) = A074206(n) is the number of ordered factorizations of n.

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A334256 Numbers k such that H(k) = 2*k, where H(k) is the number of ordered factorizations of k (A074206).

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