A330743 -id:A330743 - cp's OEIS frontend

A329902 Primorial deflation of the n-th highly composite number: the unique integer k such that A108951(k) = A002182(n).

1, 2, 4, 3, 6, 12, 9, 24, 10, 20, 15, 40, 30, 60, 28, 21, 56, 42, 84, 63, 168, 126, 336, 140, 66, 189, 280, 132, 99, 264, 198, 528, 220, 396, 297, 440, 792, 156, 117, 312, 234, 624, 260, 468, 351, 520, 936, 390, 1040, 1872, 780, 585, 306, 1560, 340, 612, 459, 680, 1224, 510, 1360, 2448, 1020, 765, 342, 2040, 1530, 684, 513

Offset: 1

Antti Karttunen, Dec 22 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (computed from the b-file of A002182)
Michael De Vlieger, List of a(n) for n = 1..779674 (gzipped text file, prepared from Flammenkamp's data)
Michael De Vlieger, List of a(n) for n = 1..500000 (noncompressed text file, an abridged version of above)

Cf. A002182, A002183, A056239, A108951, A112778, A112779, A306802, A319626, A324381, A324382, A324581, A324582, A324888, A329040, A329605, A329900, A330743, A330748.

Subsequence of A181815.

Map[Times @@ Prime@(TakeWhile[Reap[FixedPointList[Block[{k = 1}, While[Mod[#, Prime@ k] == 0, k++]; Sow[k - 1]; #/Product[Prime@ i, {i, k - 1}]] &, #]][[-1, 1]], # > 0 &]) &, Take[Import["https://oeis.org/b002182.txt", "Data"][[All, -1]], 69] ] (* Michael De Vlieger, Jan 13 2020, imports b-file at A002182 *)

a(n) = A329900(A002182(n)) = A319626(A002182(n)).
a(n) = A181815(A306802(n)).
A108951(a(n)) = A002182(n). [Highly composite numbers (undeflated)]
A056239(a(n)) = A112778(n). [Number of prime factors, counted with multiplicity]
A001222(a(n)) = A112779(n). [Largest exponent in the prime factorization]
A329605(a(n)) = A002183(n). [Number of divisors]
A329040(a(n)) = A324381(n).
A324888(a(n)) = A324382(n).
a(A330748(n)) = A330743(n).

More linking formulas added by Antti Karttunen, Jan 13 2020

A328521 Smallest highly composite number that has n prime factors counted with multiplicity.

Original entry on oeis.org

1, 2, 4, 12, 24, 48, 240, 720, 5040, 10080, 20160, 221760, 665280, 8648640, 17297280, 294053760, 2205403200, 27935107200, 293318625600, 1927522396800, 8995104518400, 26985313555200, 782574093100800, 24259796886124800, 48519593772249600, 1795224969573235200, 8976124847866176000, 368021118762513216000

Offset: 0

David A. Corneth, Jan 04 2020

nonn

a(n-1) differs from A133411(n) for n in A354880.

Question: Is this sequence strictly growing? If sequence A330748 is monotonic, so is this also, and vice versa. Note that the primorial deflation sequence, A330743, is not monotonic. - Antti Karttunen, Jan 14 2020

Amiram Eldar, Table of n, a(n) for n = 0..394

Cf. A001222 (bigomega), A002182 (highly composite numbers), A108951, A112778 (bigomega of HCN's), A330743 (primorial deflation), A330748 (indices in A002182).
Cf. also A133411.
Cf. A354880.

(* First load the function f at A025487, then: *)
Block[{s = Union@ Flatten@ f@ 17, t}, t = DivisorSigma[0, s]; s = Map[s[[FirstPosition[t, #][[1]] ]] &, Union@ FoldList[Max, t]]; t = PrimeOmega[s]; Array[s[[FirstPosition[t, #][[1]] ]] &, Max@ t + 1, 0]] (* Michael De Vlieger, Jan 12 2020 *)

a(n)=for(k=1,oo,bigomega(A2182[k])==n&&return(A2182[k])) \\ Global variable A2182 must hold a vector of values of A002182. - M. F. Hasler, Jan 08 2020

a(n) = A002182(A330748(n)) = A002182(min{k: A112778(k)=n}). - M. F. Hasler, Jan 08 2020

a(n) = A108951(A330743(n)), where A330743(n) is the first term k of A329902 for which A056239(k) = n. - Antti Karttunen, Jan 13 2020

A330748 Index of the smallest element in A002182 that has exactly n prime factors counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 12, 14, 19, 21, 23, 32, 37, 47, 50, 62, 70, 80, 91, 99, 105, 109, 124, 140, 143, 159, 166, 182, 198, 217, 221, 240, 253, 276, 297, 304, 327, 352, 357, 381, 398, 424, 449, 475, 485, 512, 540, 570, 584, 617, 642, 676, 704, 738, 765, 770, 805, 841, 877, 913, 937, 949, 985, 1021, 1058, 1096, 1134, 1169

Offset: 0

Antti Karttunen, suggested by M. F. Hasler, Jan 10 2020

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..4257 (computed using A. Flammenkamp's 779674-term HCN dataset; terms 0..329 from _Antti Karttunen_)

Cf. A001222, A002182, A112778, A328521, A329902, A330743.

A330748(n) = { for(k=1,#v112778,if(v112778[k]==n,return(k))); -(1/0); };

v329902 = readvec("a329902.txt"); \\ File for the first 779674 terms of A329902
A056239(n) = if(1==n,0,my(f=factor(n)); sum(i=1, #f~, f[i,2] * primepi(f[i,1])));
A330748list() = { my(m=Map(), lista=List([]), t); for(i=1, #v329902, t = A056239(v329902[i]); if(!mapisdefined(m,t), mapput(m,t,i))); for(n=0,oo,if(mapisdefined(m,n,&t), listput(lista,t), return(Vec(lista)))); };
v330748 = A330748list();
A330748(n) = v330748[1+n];
for(n=0,#v330748-1,write("b330748.txt", n, " ", A330748(n))); \\ Antti Karttunen, Jan 13 2020

a(n) = min{k: A112778(k)=n}.
A002182(a(n)) = A328521(n).
A329902(a(n)) = A330743(n).

cp's OEIS Frontend

A329902 Primorial deflation of the n-th highly composite number: the unique integer k such that A108951(k) = A002182(n).

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A328521 Smallest highly composite number that has n prime factors counted with multiplicity.

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A330748 Index of the smallest element in A002182 that has exactly n prime factors counted with multiplicity.

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