A087980 -id:A087980 - cp's OEIS frontend

A133811 Numbers that are primally tight and have strictly ascending powers.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 162, 163, 167, 169, 173, 179, 181, 191

Offset: 1

Olivier Gérard, Sep 23 2007

nonn

All numbers of the form p_1^k1*p_2^k2*...*p_n^k_n, where k1 < k2 < ... < k_n and the p_i are n successive primes.
Subset of A073491, A133810.
Different from A082377 starting n=16.
Different from A000961 (prime powers) starting n=13.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A025487, A087980, A073491, A133808-A133813.

Cf. A124010, A027748, A049084.

a133811 n = a133811_list !! (n-1)
a133811_list = 1 : filter f [2..] where
   f x = (and $ zipWith (<) eps $ tail eps) &&
         (all (== 1) $ zipWith (-) (tail ips) ips)
     where ips = map a049084 $ a027748_row x
           eps = a124010_row x
-- Reinhard Zumkeller, Nov 07 2012

isok(n) = {my(f = factor(n)); my(nbf = #f~); my(lastp = 0); for (i=1, nbf, if (lastp && (f[i, 1] != nextprime(lastp+1)), return (0)); lastp = f[i, 1];); for (j=2, nbf, if (f[j,2] <= f[j-1,2], return (0));); return (1);} \\ Michel Marcus, Jun 04 2014

A363063 Positive integers k such that the largest power of p dividing k is larger than or equal to the largest power of q dividing k (i.e., A305720(k,p) >= A305720(k,q)) for all primes p and q with p < q.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, 512, 576, 720, 768, 864, 1024, 1152, 1440, 1536, 1728, 2048, 2304, 2880, 3072, 3456, 4096, 4320, 4608, 5760, 6144, 6912, 8192, 8640, 9216, 10368, 11520, 12288, 13824, 16384, 17280, 18432

Offset: 1

Pontus von Brömssen, May 16 2023

nonn

Includes all products of terms in A347284, but there are also other terms such as 4320.

Closed under multiplication. - Peter Munn, May 21 2023

			151200 = 2^5 * 3^3 * 5^2 * 7 is a term, because 2^5 >= 3^3 >= 5^2 >= 7.
72 = 2^3 * 3^2 is not a term, because 2^3 < 3^2.
40 = 2^3 * 3^0 * 5 is not a term, because 3^0 < 5.
From _Michael De Vlieger_, May 19 2023: (Start)
Sequence read as an irregular triangle delimited by appearance of 2^m:
     1
     2
     4
     8     12
    16     24
    32     48
    64     96
   128    144    192
   256    288    384
   512    576    720    768    864
  1024   1152   1440   1536   1728
  2048   2304   2880   3072   3456
  4096   4320   4608   5760   6144   6912
  8192   8640   9216  10368  11520  12288  13824
  ... (End)

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot p^e | a(n) at (x,y) = (n, pi(p)), n = 1..1024, showing multiplicity e with a color function such that e = 1 is black, e = 2 is red, e = 3 is orange, etc., 12X vertical exaggeration. On the bottom, a color code represents a(n) is empty product (black), prime (red), composite prime power (gold), neither squarefree nor prime power (blue).
Michael De Vlieger, Plot multiplicities e in a(n) = Product p^e at (x,y) = (e, -n) for n = 1..1024, 8X horizontal exaggeration.

Cf. A181818, A305720, A347284, A363098 (primitive terms).

Subsequence of: A087980, {1} U A363122.

Select[Range[20000], # == 1 || PrimePi[(f = FactorInteger[#])[[-1, 1]]] == Length[f] && Greater @@ (Power @@@ f) &] (* Amiram Eldar, May 16 2023 *)

from sympy import nextprime
primes = [2] # global list of first primes
def f(kmax,pi,ppmax):
    # Generate numbers up to kmax with nonincreasing prime-powers <= ppmax, starting at the (pi+1)-st prime.
    if len(primes) <= pi: primes.append(nextprime(primes[-1]))
    p0 = primes[pi]
    ppmax = min(ppmax,kmax)
    if ppmax < p0:
        yield 1
        return
    pp = 1
    while pp <= ppmax:
        for x in f(kmax//pp,pi+1,pp):
            yield pp*x
        pp *= p0
def A363063_list(kmax):
    return sorted(f(kmax,0,kmax))

A133810 Numbers that are primally tight and have weakly ascending powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 16, 17, 18, 19, 23, 25, 27, 29, 30, 31, 32, 35, 36, 37, 41, 43, 47, 49, 53, 54, 59, 61, 64, 67, 71, 73, 75, 77, 79, 81, 83, 89, 97, 101, 103, 105, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 143, 149, 150, 151, 157, 162

Offset: 1

Olivier Gérard, Sep 23 2007

nonn

All numbers of the form p_1^k1*p_2^k2*...*p_n^k_n, where k1 <= k2 <= ... <= k_n and the p_i are n successive primes.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A025487, A087980, A073491, A133808-A133813.

Cf. A124010, A027748, A049084.

a133810 n = a133810_list !! (n-1)
a133810_list = 1 : filter f [2..] where
   f x = (and $ zipWith (<=) eps $ tail eps) &&
         (all (== 1) $ zipWith (-) (tail ips) ips)
     where ips = map a049084 $ a027748_row x
           eps = a124010_row x
-- Reinhard Zumkeller, Nov 07 2012

A328524 T(n,k) is the k-th smallest least integer of prime signatures for partitions of n into distinct parts; triangle T(n,k), n>=0, 1<=k<=A000009(n), read by rows.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 72, 64, 96, 144, 360, 128, 192, 288, 432, 720, 256, 384, 576, 864, 1440, 2160, 512, 768, 1152, 1728, 2592, 2880, 4320, 10800, 1024, 1536, 2304, 3456, 5184, 5760, 8640, 12960, 21600, 75600, 2048, 3072, 4608, 6912, 10368, 11520

Offset: 0

Alois P. Heinz, Feb 18 2020

			Triangle T(n,k) begins:
     1;
     2;
     4;
     8,   12;
    16,   24;
    32,   48,   72;
    64,   96,  144,  360;
   128,  192,  288,  432,  720;
   256,  384,  576,  864, 1440, 2160;
   512,  768, 1152, 1728, 2592, 2880, 4320, 10800;
  1024, 1536, 2304, 3456, 5184, 5760, 8640, 12960, 21600, 75600;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Eric Weisstein's World of Mathematics, Prime Signature
Wikipedia, Partition (number theory)
Wikipedia, Prime signature
Index entries for sequences related to prime signature

Column k=1-3 give: A000079, A003945 for n>2, A116453 for n>4.
Row sums give A332626.
Last elements of rows give A332644.
Cf. A000009, A087443 (for all partitions), A087980 (as sorted sequence).

b:= proc(n, i, j) option remember; `if`(i*(i+1)/2 x*ithprime(j)^i,
       b(n-i, min(n-i, i-1), j+1))[], b(n, i-1, j)[]]))
    end:
T:= n-> sort(b(n$2, 1))[]:
seq(T(n), n=0..12);

b[n_, i_, j_] := b[n, i, j] = If[i(i+1)/2 < n, {}, If[n == 0, {1}, Join[# * Prime[j]^i& /@ b[n - i, Min[n - i, i - 1], j + 1], b[n, i - 1, j]]]];
T[n_] := Sort[b[n, n, 1]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 07 2020, after Maple *)

A133809 Numbers that are primally tight, have 2 as first prime and strictly ascending powers.

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 32, 54, 64, 108, 128, 162, 256, 324, 486, 512, 648, 972, 1024, 1458, 1944, 2048, 2250, 2916, 3888, 4096, 4374, 5832, 8192, 8748, 11250, 11664, 13122, 16384, 17496, 23328, 26244, 32768, 33750, 34992, 39366, 52488, 56250, 65536

Offset: 1

Olivier Gérard, Sep 23 2007

nonn

All numbers of the form 2^k1*p_2^k2*...*p_n^k_n, where k1 < k2 < ... < k_n and the p_i are the n first primes.

Subset of A073491, A133811 and A133808.

			36 = 2^2*3^2 with both exponents being equal is not in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A025487, A087980, A073491, A133808-A133813.

Cf. A027748, A124010, A151800, A000040, A145108 (subsequence).

import Data.Set (singleton, deleteFindMin, insert)
a133809 n = a133809_list !! (n-1)
a133809_list = 1 : f (singleton (2, 2, 1)) where
   f s = y : f (insert (y*p, p, e+1) $ insert (y*q^(e+1), q, e+1) s')
             where q = a151800 p
                   ((y, p, e), s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 14 2015

isok(n) = {my(f = factor(n)); my(nbf = #f~); if (prod(i=1, nbf, prime(i)) ! = prod(i=1, nbf, f[i, 1]), return (0)); for (j=2, nbf, if (f[j,2] <= f[j-1,2], return (0));); return (1);} \\ Michel Marcus, Jun 04 2014

A133812 Numbers that are primally tight and have weakly descending powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 23, 24, 25, 27, 29, 30, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 59, 60, 61, 64, 67, 71, 72, 73, 77, 79, 81, 83, 89, 96, 97, 101, 103, 105, 107, 109, 113, 120, 121, 125, 127, 128, 131, 135, 137, 139, 143, 144

Offset: 1

Olivier Gérard, Sep 23 2007

nonn

All numbers of the form p_1^k1*p_2^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n and the p_i are n successive primes.

Differs from A073491 starting n=16.

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

Cf. A025487, A087980, A073491, A133808-A133813, A025487.

aQ[n_] := Module[{f=FactorInteger[n]}, p=f[[;;,1]]; e=f[[;;,2]]; PrimePi[p[[-1]]]-PrimePi[p[[1]]] == Length[p]-1 && AllTrue[Differences[e], #<=0 &]]; Join[{1}, Select[Range[2, 144], aQ]] (* Amiram Eldar, Jun 20 2019 *)

Cross-references from Charles R Greathouse IV, Dec 04 2009

A362227 a(n) = Product_{k=1..w(n)} p(k)^(S(n,k)-1), where set S(n,k) = row n of A272011 and w(n) = A000120(n) is the binary weight of n.

Original entry on oeis.org

1, 2, 4, 12, 8, 24, 72, 360, 16, 48, 144, 720, 432, 2160, 10800, 75600, 32, 96, 288, 1440, 864, 4320, 21600, 151200, 2592, 12960, 64800, 453600, 324000, 2268000, 15876000, 174636000, 64, 192, 576, 2880, 1728, 8640, 43200, 302400, 5184, 25920, 129600, 907200, 648000, 4536000, 31752000, 349272000, 15552

Offset: 0

Michael De Vlieger, Jun 08 2023

In other words, let S(n) contain place values of 1's in the binary expansion of n, ordered greatest to least, where S(n,1) = floor(log_2(n+1)) = A000523(n+1) and the remaining terms in S strictly decrease. This sequence reads S(n,k)+1 instead as a multiplicity of prime(k) so as to produce a number with strictly decreasing prime exponents.

			a(0) = 1 since 1 is the empty product.
a(1) = 2 since 1 = 2^0, s = {0}, hence a(1) = prime(1)^(0+1) = 2^1 = 2.
a(2) = 4 since 2 = 2^1, s = {1}, hence a(2) = 2^(1+1) = 4.
a(3) = 12 since 3 = 2^1+2^0, s = {1,0}, hence a(3) = 2^2*3^1 = 12, etc.
The table below relates first terms of this sequence greater than 1 to A272011 and A067255:
   n   A272011(n)  a(n)  A067255(a(n))
  ------------------------------------
   1   0             2   1
   2   1             4   2
   3   1,0          12   2,1
   4   2             8   3
   5   2,0          24   3,1
   6   2,1          72   3,2
   7   2,1,0       360   3,2,1
   8   3            16   4
   9   3,0          48   4,1
  10   3,1         144   4,2
  11   3,1,0       720   4,2,1
  12   3,2         432   4,3
  13   3,2,0      2160   4,3,1
  14   3,2,1     10800   4,3,2
  15   3,2,1,0   75600   4,3,2,1
  16   4            32   5
  ...
This sequence appears below, seen as an irregular triangle T(m,j) delimited by 2^m where j = 1..2^(m-1) for m > 0:
   1;
   2;
   4, 12;
   8, 24,  72, 360;
  16, 48, 144, 720, 432, 2160, 10800, 75600;
  ...
T(m,1) = 2^m.
T(m,2^(m-1)) = A006939(m) for m > 0.

Michael De Vlieger, Table of n, a(n) for n = 0..16384

Cf. A000120, A006939, A067255, A087980, A272011.

Array[Times @@ MapIndexed[Prime[First[#2]]^(#1 + 1) &, Length[#] - Position[#, 1][[All, 1]] ] &[IntegerDigits[#, 2]] &, 48, 0]

This sequence, sorted, is A087980.
a(2^k) = 2^(k+1).
a(2^k-1) = A006939(k-1).

A145108 Multiples of 4 that are primally tight and have strictly ascending powers.

Original entry on oeis.org

4, 8, 16, 32, 64, 108, 128, 256, 324, 512, 648, 972, 1024, 1944, 2048, 2916, 3888, 4096, 5832, 8192, 8748, 11664, 16384, 17496, 23328, 26244, 32768, 34992, 52488, 65536, 67500, 69984, 78732, 104976, 131072, 139968, 157464, 209952, 236196, 262144

Offset: 1

Reikku Kulon, Oct 02 2008

nonn

All numbers of the form 2^k0*p_1^k1*p_2^k2*...*p_n^k_n, where 2 <= k0 < k1 < k2 < ... < k_n and the p_i are n successive primes.

Subset of A073491, A133808, A133809, A133810 and A133811.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A025487, A087980, A073491, A133808, A133809, A133810, A133811, A133812, A133813.

Cf. A008586.

a145108 n = a145108_list !! (n-1)
a145108_list = filter ((== 0) . (`mod` 4)) a133809_list
-- Reinhard Zumkeller, Apr 14 2015

A161812 Numbers with decreasing but not strictly decreasing prime signature.

Original entry on oeis.org

6, 30, 36, 60, 120, 180, 210, 216, 240, 420, 480, 840, 900, 960, 1080, 1260, 1296, 1680, 1800, 1920, 2310, 2520, 3360, 3600, 3840, 4620, 5040, 5400, 6300, 6480, 6720, 7200, 7560, 7680, 7776, 9240, 10080, 12600, 13440, 13860, 14400, 15120, 15360, 18480

Offset: 1

Peter Luschny, Jun 20 2009

All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n, and the '=' occurs at least once, sorted.

It appears that all highly composite numbers (A002182) greater than 720 are in this sequence. - Walter Roscello, Dec 24 2013

			a(1) = 2*3, a(2) = 2*3*5, a(3) = 2^2*3^2, a(4) = 2^2*3*5.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..279 from Walter Roscello)

Cf. A025487, A087980, A002182.

okQ[n_] := Module[{pp, ee}, {pp, ee} = Transpose[FactorInteger[n]]; Length[pp] == PrimePi[pp // Last] && GreaterEqual @@ ee && Sort[ee] != Union[ee]]; Select[Range[20000], okQ] (* Jean-François Alcover, Jun 17 2019 *)

Terms up to n=279 added by Walter Roscello, Dec 24 2013

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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A133811 Numbers that are primally tight and have strictly ascending powers.

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A363063 Positive integers k such that the largest power of p dividing k is larger than or equal to the largest power of q dividing k (i.e., A305720(k,p) >= A305720(k,q)) for all primes p and q with p < q.

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A133810 Numbers that are primally tight and have weakly ascending powers.

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A328524 T(n,k) is the k-th smallest least integer of prime signatures for partitions of n into distinct parts; triangle T(n,k), n>=0, 1<=k<=A000009(n), read by rows.

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A133809 Numbers that are primally tight, have 2 as first prime and strictly ascending powers.

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A133812 Numbers that are primally tight and have weakly descending powers.

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A362227 a(n) = Product_{k=1..w(n)} p(k)^(S(n,k)-1), where set S(n,k) = row n of A272011 and w(n) = A000120(n) is the binary weight of n.

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A145108 Multiples of 4 that are primally tight and have strictly ascending powers.

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