A181816 -id:A181816 - cp's OEIS frontend

A181818 Products of superprimorials (A006939).

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 360, 384, 512, 576, 720, 768, 1024, 1152, 1440, 1536, 1728, 2048, 2304, 2880, 3072, 3456, 4096, 4320, 4608, 5760, 6144, 6912, 8192, 8640, 9216, 11520, 12288, 13824, 16384, 17280, 18432, 20736, 23040, 24576, 27648, 32768

Offset: 1

Matthew Vandermast, Nov 30 2010

nonn

Sorted list of positive integers with a factorization Product p(i)^e(i) such that (e(1) - e(2)) >= (e(2) - e(3)) >= ... >= (e(k-1) - e(k)) >= e(k), with k = A001221(n), and p(k) = A006530(n) = A000040(k), i.e., the prime factors p(1) .. p(k) must be consecutive primes from 2 onward. - Comment clarified by Antti Karttunen, Apr 28 2022
Subsequence of A025487. A025487(n) belongs to this sequence iff A181815(n) is a member of A025487.
If prime signatures are considered as partitions, these are the members of A025487 whose prime signature is conjugate to the prime signature of a member of A182863. - Matthew Vandermast, May 20 2012

			2, 12, and 360 are all superprimorials (i.e., members of A006939). Therefore, 2*2*12*360 = 17280 is included in the sequence.
From _Gus Wiseman_, Aug 12 2020 (Start):
The sequence of factorizations (which are unique) begins:
    1 = empty product
    2 = 2
    4 = 2*2
    8 = 2*2*2
   12 = 12
   16 = 2*2*2*2
   24 = 2*12
   32 = 2*2*2*2*2
   48 = 2*2*12
   64 = 2*2*2*2*2*2
   96 = 2*2*2*12
  128 = 2*2*2*2*2*2*2
  144 = 12*12
  192 = 2*2*2*2*12
  256 = 2*2*2*2*2*2*2*2
(End)

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

A181817 rearranged in numerical order. Also includes all members of A000079, A001021, A006939, A009968, A009992, A066120, A166475, A167448, A181813, A181814, A181816, A182763.
Subsequence of A025487, A055932, A087980, A130091, A181824.
A001013 is the version for factorials.
A336426 is the complement.
A336496 is the version for superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A317829 counts factorizations of superprimorials.
Cf. A022915, A076954, A304686, A325368, A336419, A336420, A336421, A353518 (characteristic function).

Select[Range[100],PrimePi[First/@If[#==1,{}, FactorInteger[#]]]==Range[ PrimeNu[#]]&&LessEqual@@Differences[ Append[Last/@FactorInteger[#],0]]&] (* Gus Wiseman, Aug 12 2020 *)

firstdiffs0forward(vec) = { my(v=vector(#vec)); for(n=1,#v,v[n] = vec[n]-if(#v==n,0,vec[1+n])); (v); };
A353518(n) = if(1==n,1,my(f=factor(n), len=#f~); if(primepi(f[len,1])!=len, return(0), my(diffs=firstdiffs0forward(f[,2])); for(i=1,#diffs-1,if(diffs[i+1]>diffs[i],return(0))); (1)));
isA181818(n) = A353518(n); \\ Antti Karttunen, Apr 28 2022

A181817 a(n) is the smallest integer that, when divided by any divisor of A025487(n), yields a member of A025487.

Original entry on oeis.org

1, 2, 4, 12, 8, 24, 16, 48, 360, 32, 144, 96, 720, 64, 288, 192, 1440, 128, 576, 4320, 384, 75600, 1728, 2880, 256, 1152, 8640, 768, 151200, 3456, 5760, 512, 2304, 17280, 1536, 302400, 6912, 129600, 11520, 1024, 51840, 4608, 907200, 20736, 34560, 3072, 604800, 13824, 259200, 23040, 2048

Offset: 1

Matthew Vandermast, Nov 30 2010

nonn

A permutation of A181818.

			For any divisor d of 6 (d = 1, 2, 3, 6), 12/d (12, 6, 4, 2) is always a member of A025487. 12 is the smallest number with this relationship to 6; therefore, since 6 = A025487(4), a(4) = 12.

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

Cf. A025487, A108951, A181816, A181818.

If A025487(n) = Product prime(i)^e(i), then a(n) = Product A002110(i)^e(i). I.e., a(n) = A108951(A025487(n)).
If A025487(n) = Product A002110(i)^e(i), then a(n) = Product A006939(i)^e(i).
a(n) = A025487(n) * A181816(n).

A181811 a(n) = smallest integer that, upon multiplying any divisor of n, produces a member of A025487.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 30, 1, 4, 6, 210, 2, 2310, 30, 12, 1, 30030, 4, 510510, 6, 60, 210, 9699690, 2, 36, 2310, 8, 30, 223092870, 12, 6469693230, 1, 420, 30030, 180, 4, 200560490130, 510510, 4620, 6, 7420738134810, 60, 304250263527210, 210, 24, 9699690

Offset: 1

Matthew Vandermast, Nov 30 2010

Each member of A025487 appears infinitely often, and exactly once among odd values of n. a(m) = a(n) iff A000265(m) = A000265(n).

			For any divisor d of 6 (d = 1, 2, 3, 6), 2d (2, 4, 6, 12) is always a member of A025487. 2 is the smallest integer with this relationship to 6; therefore, a(6)=2.

Amiram Eldar, Table of n, a(n) for n = 1..2370
Index to divisibility sequences

Cf. A064989, A108951, A181812, A181813, A181816.

from sympy import primerange, factorint
from operator import mul
from functools import reduce
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a(n):
    f = factorint(n)
    return 1 if n==1 else (reduce(mul, [P(i)**f[i] for i in f]))//n
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017

If n = Product p(i)^e(i), then a(n) = Product A002110(i-1)^e(i). Sequence is completely multiplicative.
a(n) = A108951(n)/n.
a(n) = A108951(A064989(n)). - Antti Karttunen, Dec 31 2023

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A181818 Products of superprimorials (A006939).

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A181817 a(n) is the smallest integer that, when divided by any divisor of A025487(n), yields a member of A025487.

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A181811 a(n) = smallest integer that, upon multiplying any divisor of n, produces a member of A025487.

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