A324381 -id:A324381 - 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

A324382 Minimal number of primorials that add to the n-th highly composite number: a(n) = A276150(A002182(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 6, 4, 6, 8, 2, 4, 6, 8, 12, 16, 20, 12, 14, 18, 12, 12, 12, 12, 12, 12, 12, 24, 8, 8, 8, 4, 16, 8, 16, 8, 16, 24, 16, 32, 6, 14, 30, 12, 18, 18, 24, 12, 18, 18, 24, 18, 36, 8, 14, 32, 28, 6, 24, 38, 12, 18, 36, 20, 24, 30, 40, 26, 10, 40, 20, 30, 18, 38, 26, 36, 36, 24, 24, 44, 50, 48, 14, 42

Offset: 1

Antti Karttunen, Feb 26 2019

nonn

Among the first 10000 highly composite numbers, only in two cases a(n) < A112779(n). This happens on A002182(12) = 240 and A002182(18) = 2520. Note that A112779(n) gives the number of primorials needed when A002182(n) is expressed as a product [not as a sum] of primorials.

			For n=12, A002182(12) = 240, which is written as "11000" in primorial base (A049345) because 240 = 1*A002110(4) + 1*A002110(3) = 210+30, thus a(12) = 1+1 = 2. (Note that 240 = 30*2*2*2).
For n=18, A002182(18) = 2520 = "110000" in primorial base because 2520 = 1*A002110(5) + 1*A002110(4) = 2310+210, thus a(18) = 1+1 = 2. (Note that 2520 = 210*6*2).
For n=26, A002182(26) = 45360 = "1670000" in primorial base because 45360 = 1*A002110(6) + 6*A002110(5) + 7*A002110(4), thus a(26) = 1+6+7 = 14. (Note that 45360 = 210*6*6*6).

Cf. A002110, A002182, A276150, A324381.

Cf. also A108602, A112778, A112779, A324342.

A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A324382(n) = A276150(A002182(n));

a(n) = A276150(A002182(n)).

a(n) >= A324381(n).

A324581 a(n) = A276086(A002182(n)).

Original entry on oeis.org

2, 3, 9, 5, 25, 625, 35, 875, 49, 2401, 117649, 77, 184877, 456533, 14641, 1771561, 214358881, 143, 20449, 2924207, 418161601, 8550986578849, 174859124550883201, 3575694237941010577249, 23298085122481, 1599034490244763, 32698656291015158587, 30466726698629, 39841104144361, 52099905419549, 89093921102069, 152355876914189, 260537564663909

Offset: 1

Antti Karttunen, Mar 09 2019

nonn

Note that gcd(a(n), A002182(n)) = A324198(A002182(n)) = 1 for all n because each term of A002182 is a product of primorial numbers (A002110).

Antti Karttunen, Table of n, a(n) for n = 1..670
Michael De Vlieger, Small chart of first terms.
Michael De Vlieger, 2400 x 3600 chart of first terms.
Index entries for sequences related to primorial base
Index entries for sequences related to primorial numbers

Subsequence of A324576.

Cf. A002110, A002182, A276086, A324198, A324381, A324382, A324582.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 20], s = DivisorSigma[0, Range[10^5]], t}, t = Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]; Array[Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[(*a002182[[#]]*)t[[#]], b] &, Length@ t]] (* Michael De Vlieger, Mar 18 2019 *)

\\ A002182 assumed to be precomputed
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324581(n) = A276086(A002182(n));

a(n) = A276086(A002182(n)).
a(n) = A324582(n)/A002182(n).
A001221(a(n)) = A324381(n).
A001222(a(n)) = A324382(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).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A324382 Minimal number of primorials that add to the n-th highly composite number: a(n) = A276150(A002182(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A324581 a(n) = A276086(A002182(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula