A328524 -id:A328524 - cp's OEIS frontend

A087980 Numbers with strictly decreasing prime exponents.

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 72, 96, 128, 144, 192, 256, 288, 360, 384, 432, 512, 576, 720, 768, 864, 1024, 1152, 1440, 1536, 1728, 2048, 2160, 2304, 2592, 2880, 3072, 3456, 4096, 4320, 4608, 5184, 5760, 6144, 6912, 8192, 8640, 9216, 10368, 10800

Offset: 1

Rainer Rosenthal, Oct 27 2003

This representation provides a natural ordering between strictly decreasing sequences of natural numbers. Let f and g be such sequences with f(1) > f(2) > ... > f(m) and g(1) > g(2) > ... > g(n). Define f < g iff p^f < p^g, where p^f is short for Product(i=1..m) p_i^f(i) and p^g is defined likewise as Product(i=1..n) p_i^g(i).
Note that "strictly decreasing sequences of natural numbers" is another way to say "partitions into distinct parts".
Also products of primorial numbers p_1#^k_1 * p_2#^k_2 * ... * p_n#^k_n where all k_i > 0.
A124010(a(n),k+1) < A124010(a(n),k), 1 <= k < A001221(a(n)). - Reinhard Zumkeller, Apr 13 2015
Numbers whose prime indices cover an initial interval of positive integers with strictly decreasing multiplicities. Intersection of A055932 and A304686. First differs from A181818 in having 72. - Gus Wiseman, Oct 21 2022

			The sequence starts with a(1)=1, a(2)=2, a(3)=4 and a(4)=8. The next term is a(5)=12 = 2^2*3^1 = p_1^k_1 * p_2^k_2 with k_1=2 > k_2=1.

T. D. Noe, Table of n, a(n) for n = 1..1000

The weak (weakly decreasing) version is A025487.
The weak opposite (weakly increasing) version is A133808.
The opposite (strictly increasing) version is A133809.
For strictly decreasing prime signature we have A304686.
Cf. A000009, A000040, A002110, A027748, A055932, A124010, A133813, A317791, A328524, A357864.

import Data.List (isPrefixOf)
a087980 n = a087980_list !! (n-1)
a087980_list = 1 : filter f [2..] where
   f x = isPrefixOf ps a000040_list && all (< 0) (zipWith (-) (tail es) es)
         where ps = a027748_row x; es = a124010_row x
-- Reinhard Zumkeller, Apr 13 2015

selQ[k_] := Module[{n = k, e = IntegerExponent[k, 2], t}, n /= 2^e; For[p = 3, True, p = NextPrime[p], t = IntegerExponent[n, p]; If[t == 0, Return[n == 1]]; If[t >= e, Return[False]]; e = t; n /= p^e]];
Select[Range[12000], selQ] (* Jean-François Alcover, Mar 27 2020, after first PARI program *)

is(n)=my(e=valuation(n,2),t); n>>=e; forprime(p=3,, t=valuation(n,p); if(t==0, return(n==1)); if(t>=e, return(0)); e=t; n/=p^e) \\ Charles R Greathouse IV, Jun 25 2017

list(lim)=my(v=[],u=powers(2,logint(lim\=1,2)),w,p=2,t); forprime(q=3,, w=List(); for(i=1,#u, t=u[i]; for(e=1,valuation(u[i],p)-1, t*=q; if(t>lim, break); listput(w,t))); v=concat(v,Vec(u)); if(#w==0, break); u=w; p=q); Set(v) \\ Charles R Greathouse IV, Jun 25 2017

The numbers of the form Product(i=1..n) p_i^k_i where p_i = A000040(i) is the i-th prime and k_1 > k_2 > ... > k_n are positive natural numbers.

Compute x = 2^k_1 * 3^k_2 * 5^k_3 * 7^k_4 * 11^k_5 for k_1 > ... > k_5 allowing k_i = 0 for i > 1 and k_i = k_(i+1) in that case. Discard all x > 174636000 = 2^5*3^4*5^3*7^2*11 and enumerate those below. For more members take higher primes into account.

Edited by Franklin T. Adams-Watters, Apr 25 2006

Offset change to 1 by T. D. Noe, May 24 2010

A087443 Least integer of each prime signature ordered first by sum of exponents and then by least integer value.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 216, 240, 360, 840, 900, 1260, 4620, 30030, 128, 192, 288, 432, 480, 720, 1080, 1680, 1800, 2520, 6300, 9240, 13860, 60060, 510510, 256, 384, 576, 864, 960, 1296, 1440

Offset: 0

Ray Chandler, Sep 04 2003

A025487 in a different order.

			1;
2;
4,6;
8,12,30;
16,24,36,60,210;
32,48,72,120,180,420,2310;
64,96,144,216,240,360,840,900,1260,4620,30030;
128,192,288,432,480,720,1080,1680,1800,2520,6300,9240,13860,60060,510510;

Alois P. Heinz, Rows n = 0..26, flattened

Cf. A025487, A036035, A059901, A063008, A077569, A074140 (row sums), A328524.

b:= proc(n, i, l)
      `if`(n=0, [mul(ithprime(t)^l[t], t=1..nops(l))],
      `if`(i=1, b(0, 0, [l[], 1$n]), [b(n, i-1, l)[],
      `if`(i>n, [], b(n-i, i, [l[], i]))[]]))
    end:
T:= n-> sort(b(n$2, []))[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jun 13 2012

b[n_, i_, l_] := b[n, i, l] = If[n == 0, Join[{Product[Prime[t]^l[[t]], {t, 1, Length[l]}]}], If[i == 1, b[0, 0, Join[l, Table[1, {n}]]], Join[b[n, i - 1, l], If[i > n, {}, b[n - i, i, Append[l, i]]]]]];
T[n_] := Sort[b[n, n, {}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 06 2017, after Alois P. Heinz *)

A332626 Sum of least integers of prime signatures over all partitions of n into distinct parts.

Original entry on oeis.org

1, 2, 4, 20, 40, 152, 664, 1760, 5680, 24752, 138064, 356480, 1568320, 5886752, 32781664, 266420000, 726928960, 3135277952, 16299729664, 81402739520, 640678081600, 7084434124352, 18897678264064, 92846198695040, 464088929482240, 3347512310365952

Offset: 0

Alois P. Heinz, Feb 17 2020

nonn

			a(5) = 2^5 + 2^4*3^1 + 2^3*3^2 = 32 + 48 + 72 = 152.

Alois P. Heinz, Table of n, a(n) for n = 0..712
Eric Weisstein's World of Mathematics, Prime Signature
Wikipedia, Partition (number theory)
Wikipedia, Prime signature
Index entries for sequences related to prime signature

Row sums of A328524.

Cf. A000009, A025487, A074140.

b:= proc(n, i, j) option remember; `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..30);

b[n_, i_, j_] := b[n, i, j] = If[i(i+1)/2 < n, 0, If[n == 0, 1, b[n, i - 1, j] + Prime[j]^i b[n - i, Min[n - i, i - 1], j + 1]]];
a[n_] := b[n, n, 1];
a /@ Range[0, 30] (* Jean-François Alcover, May 04 2020, after Maple *)

a(n) = Sum_{k=1..A000009(n)} A328524(n,k).

A332644 Largest of the least integers of prime signatures over all partitions of n into distinct parts.

Original entry on oeis.org

1, 2, 4, 12, 24, 72, 360, 720, 2160, 10800, 75600, 151200, 453600, 2268000, 15876000, 174636000, 349272000, 1047816000, 5239080000, 36673560000, 403409160000, 5244319080000, 10488638160000, 31465914480000, 157329572400000, 1101307006800000, 12114377074800000

Offset: 0

Alois P. Heinz, Feb 18 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..712
Eric Weisstein's World of Mathematics, Prime Signature
Wikipedia, Partition (number theory)
Wikipedia, Prime signature
Index entries for sequences related to prime signature

Subsequence of A025487, A055932, A087980.

Cf. A000009, A000040, A000217, A002110, A002260, A003056, A001221, A001222, A007814, A037126, A046523, A123578, A328524.

b:= proc(n, i, j) option remember;
      `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*
      ithprime(n-(t-> t*(t+1)/2)(floor((sqrt(8*n-7)-1)/2))))
    end:
seq(a(n), n=0..30);

b[n_, i_, j_] := b[n, i, j] = If[i(i+1)/2 < n, 0, If[n == 0, 1, Max[b[n, i - 1, j], Prime[j]^i b[n - i, Min[n - i, i - 1], j + 1]]]];
a[n_] := b[n, n, 1];
a /@ Range[0, 30] (* Jean-François Alcover, May 07 2020, after 1st Maple program *)

a(n) = A328524(n,A000009(n)).
A001221(a(n)) = A003056(n).
A001222(a(n)) = n.
A046523(a(n)) = a(n).
a(n)/a(n-1) = A037126(n) = A000040(n-A000217(A003056(n))) for n > 0.
a(n) in { A025487 }.
a(n) in { A055932 }.
a(n) in { A087980 }.
A007814(a(n)) = A123578(n).

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A087980 Numbers with strictly decreasing prime exponents.

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A087443 Least integer of each prime signature ordered first by sum of exponents and then by least integer value.

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Maple

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A332626 Sum of least integers of prime signatures over all partitions of n into distinct parts.

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Programs

Maple

Mathematica

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A332644 Largest of the least integers of prime signatures over all partitions of n into distinct parts.

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Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula