A330743 -id:A330743

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A329902

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

+10
16

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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)

FORMULA

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).

MATHEMATICA

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 *)

CROSSREFS

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

Subsequence of A181815.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 22 2019

EXTENSIONS

More linking formulas added by Antti Karttunen, Jan 13 2020

STATUS

approved

A328521

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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

FORMULA

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

MATHEMATICA

(* 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 *)

PROG

(PARI) 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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

David A. Corneth, Jan 04 2020

STATUS

approved

A330748

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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)

FORMULA

a(n) = min{k: A112778(k)=n}.

A002182(a(n)) = A328521(n).

A329902(a(n)) = A330743(n).

PROG

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

(PARI)

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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