A087980 -id:A087980 - cp's OEIS frontend

A363537 Rewrite A087980(n) = Product_{i=1..m} p(i)^e(i) instead as Sum_{i=1..m} 2^(i-1), where m = omega(A087980(n)) = A001221(A087980(n)).

0, 1, 2, 4, 3, 8, 5, 16, 9, 32, 6, 17, 64, 10, 33, 128, 18, 7, 65, 12, 256, 34, 11, 129, 20, 512, 66, 19, 257, 36, 1024, 13, 130, 24, 35, 513, 68, 2048, 21, 258, 40, 67, 1025, 132, 4096, 37, 514, 72, 14, 131, 2049, 25, 260, 48, 8192, 69, 1026, 136, 22, 259, 4097, 41, 516, 80, 16384, 133, 2050, 264, 38

Offset: 1

Michael De Vlieger, Jun 09 2023

nonn

Permutation of nonnegative numbers.

Rewriting nonnegative numbers n = Sum_{i=1..A000120(n)} 2^i instead as Product_{i=1..A000120(n)} p(i)^(e(i)+1) gives A362227.

			Table relating this sequence to A087980, where b(n) = A087980(n), f(n) = A067255(n), g(n) = A272011(n), and a(n)_2 the binary expansion of a(n):
   n   b(n)  f(b(n))  a(n)  g(a(n))   a(n)_2
   1     1   0         0
   2     2   1         1    0             1
   3     4   2         2    1            1.
   4     8   3         4    2           1..
   5    12   2,1       3    1,0          11
   6    16   4         8    3          1...
   7    24   3,1       5    2,0         1.1
   8    32   5        16    4         1....
   9    48   4,1       9    3,0        1..1
  10    64   6        32    5        1.....
  11    72   3,2       6    2,1         11.
  12    96   5,1      17    4,0       1...1
  13   128   7        64    6       1......
  14   144   4,2      10    3,1        1.1.
  15   192   6,1      33    5,0      1....1
  16   256   8       128    7      1.......
  17   288   5,2      18    4,1       1..1.
  18   360   3,2,1     7    2,1,0       111
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11157 (a(11157) = 2^64.)
Michael De Vlieger, Log log scatterplot of a(n), n = 1..1203278.
Michael De Vlieger, Plot S(n,k) in row a(n) of A272011 at (x,y) = (n,k), n = 1..2048.

Cf. A000079, A001221, A006939, A087980, A362227.

m = 15; f[n_] := Times @@ MapIndexed[Prime[First[#2]]^(#1 + 1) &, Length[#] - Position[#, 1][[All, 1]]] &[IntegerDigits[n, 2]]; SortBy[Select[Array[{#, f[#]} &, 2^(m + 1)], Last[#] <= 2^m &], Last][[All, 1]]

a(2^k) = 2^(k-1) for k > 0.

a(A006939(k)) = 2^k-1 for k > 0.

A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2310

Offset: 1

David W. Wilson

All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n, sorted.
A111059 is a subsequence. - Reinhard Zumkeller, Jul 05 2010
Choie et al. (2007) call these "Hardy-Ramanujan integers". - Jean-François Alcover, Aug 14 2014
The exponents k1, k2, ... can be read off Abramowitz & Stegun p. 831, column labeled "pi".
For all such sequences b for which it holds that b(n) = b(A046523(n)), the sequence which gives the indices of records in b is a subsequence of this sequence. For example, A002182 which gives the indices of records for A000005, A002110 which gives them for A001221 and A000079 which gives them for A001222. - Antti Karttunen, Jan 18 2019
The prime signature corresponding to a(n) is given in row n of A124832. - M. F. Hasler, Jul 17 2019

			The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10001 (first 291 terms from Will Nicholes)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
Kevin Broughan, Equivalents of the Riemann Hypothesis, Vol. 1: Arithmetic Equivalents, Cambridge University Press, 2017. See section 8.2, "Hardy-Ramanujan Numbers".
YoungJu Choie, Nicolas Lichiardopol, Pieter Moree and Patrick Solé, On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, Vol. 19, No. 2 (2007), pp. 357-372. See section 5, p. 367.
Asaf Cohen Antonir and Asaf Shapira, An Elementary Proof of a Theorem of Hardy and Ramanujan (2022). arXiv:2207.09410 [math.NT]
Michael De Vlieger, Relations of A025487 to A002110, A002182, and A002201.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, pp. 9-10.
G. H. Hardy and S. Ramanujan, Asymptotic formulae for the distribution of integers of various types, Proc. London Math. Soc, Ser. 2, Vol. 16 (1917), pp. 112-132. Also published in the book Collected Papers of Srinivasa Ramanujan, Chelsea, 1962, pages 245-261.
Jeffery Kline, On the eigenstructure of sparse matrices related to the prime number theorem, Linear Algebra and its Applications (2020) Vol. 584, 409-430.
L. B. Richmond, Asymptotic results for partitions (I) and the distribution of certain integers, Journal of Number Theory, Vol. 8, No. 4 (1976), pp. 372-389. See page 388.

Subsequence of A055932, A191743, and A324583.
Cf. A025488, A051282, A036041, A051466, A061394, A124832, A161360, A166469, A181815, A181817, A283980, A306802, A322584, A322585 (characteristic function), A329897, A329898, A329899, A329900, A329904, A330683.
Cf. A085089, A101296 (left inverses).
Equals range of values taken by A046523.
Cf. A178799 (first differences), A247451 (squarefree kernel), A146288 (number of divisors).
Subsequences of this sequence include: A000079, A000142, A000400, A001013, A001813, A002110, A002182, A005179, A006939, A025527, A056836, A061742, A064350, A066120, A087980, A097212, A097213, A111059, A119840, A119845, A126098, A129912, A140999, A166338, A166470, A166472, A166473, A166475, A167448, A168262, A168263, A168264, A179215, A181555, A181804, A181806, A181809, A181818, A181822, A181823, A181824, A181825, A181826, A181827, A182763, A182862, A182863, A212170, A220264, A220423, A250269, A250270, A260633, A266047, A284456, A300357, A304938, A329894, A330687; also A037019 and A330681 (when sorted), possibly also A289132.
Rearrangements of this sequence include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822, A322827, A329886, A329887.
Cf. also array A124832 (row n = prime signature of a(n)) and A304886, A307056.

import Data.Set (singleton, fromList, deleteFindMin, union)
a025487 n = a025487_list !! (n-1)
a025487_list = 1 : h [b] (singleton b) bs where
   (_ : b : bs) = a002110_list
   h cs s xs'@(x:xs)
     | m <= x    = m : h (m:cs) (s' `union` fromList (map (* m) cs)) xs'
     | otherwise = x : h (x:cs) (s  `union` fromList (map (* x) (x:cs))) xs
     where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 06 2013

isA025487 := proc(n)
    local pset,omega ;
    pset := sort(convert(numtheory[factorset](n),list)) ;
    omega := nops(pset) ;
    if op(-1,pset) <> ithprime(omega) then
        return false;
    end if;
    for i from 1 to omega-1 do
        if padic[ordp](n,ithprime(i)) < padic[ordp](n,ithprime(i+1)) then
            return false;
        end if;
    end do:
    true ;
end proc:
A025487 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        1 ;
    else
        for a from procname(n-1)+1 do
            if isA025487(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A025487(n),n=1..100) ; # R. J. Mathar, May 25 2017

PrimeExponents[n_] := Last /@ FactorInteger[n]; lpe = {}; ln = {1}; Do[pe = Sort@PrimeExponents@n; If[ FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[ln, n]], {n, 2, 2350}]; ln (* Robert G. Wilson v, Aug 14 2004 *)
(* Second program: generate all terms m <= A002110(n): *)
f[n_] := {{1}}~Join~
  Block[{lim = Product[Prime@ i, {i, n}],
   ww = NestList[Append[#, 1] &, {1}, n - 1], dec},
   dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]];
   Map[Block[{w = #, k = 1},
      Sort@ Prepend[If[Length@ # == 0, #, #[[1]]],
        Product[Prime@ i, {i, Length@ w}] ] &@ Reap[
         Do[
          If[# < lim,
             Sow[#]; k = 1,
             If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w,
             If[k == 1,
               MapAt[# + 1 &, w, k],
               PadLeft[#, Length@ w, First@ #] &@
                 Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]],
           {i, Infinity}] ][[-1]]
] &, ww]]; Sort[Join @@ f@ 13] (* Michael De Vlieger, May 19 2018 *)

isA025487(n)=my(k=valuation(n,2),t);n>>=k;forprime(p=3,default(primelimit),t=valuation(n,p);if(t>k,return(0),k=t);if(k,n/=p^k,return(n==1))) \\ Charles R Greathouse IV, Jun 10 2011

factfollow(n)={local(fm, np, n2);
  fm=factor(n); np=matsize(fm)[1];
  if(np==0,return([2]));
  n2=n*nextprime(fm[np,1]+1);
  if(np==1||fm[np,2]Franklin T. Adams-Watters, Dec 01 2011 */

is(n) = {if(n==1, return(1)); my(f = factor(n));  f[#f~, 1] == prime(#f~) && vecsort(f[, 2],,4) == f[, 2]} \\ David A. Corneth, Feb 14 2019

upto(Nmax)=vecsort(concat(vector(logint(Nmax,2),n,select(t->t<=Nmax,if(n>1,[factorback(primes(#p),Vecrev(p)) || p<-partitions(n)],[1,2]))))) \\ M. F. Hasler, Jul 17 2019

\\ For fast generation of large number of terms, use this program:
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A025487list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista,t))); i++); vecsort(Vec(lista)); }; \\ Returns a list of terms up to the term 2^e.
v025487 = A025487list(101);
A025487(n) = v025487[n];
for(n=1,#v025487,print1(A025487(n), ", ")); \\ Antti Karttunen, Dec 24 2019

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
N = 2310
nmax = 2^floor(log(N,2))
sorted([j for j in (prod(sharp_primorial(t[0])^t[1] for k, t in enumerate(factor(n))) for n in (1..nmax)) if j <= N])
# Giuseppe Coppoletta, Jan 26 2015

What can be said about the asymptotic behavior of this sequence? - Franklin T. Adams-Watters, Jan 06 2010
Hardy & Ramanujan prove that there are exp((2 Pi + o(1))/sqrt(3) * sqrt(log x/log log x)) members of this sequence up to x. - Charles R Greathouse IV, Dec 05 2012
From Antti Karttunen, Jan 18 & Dec 24 2019: (Start)
A085089(a(n)) = n.
A101296(a(n)) = n [which is the first occurrence of n in A101296, and thus also a record.]
A001221(a(n)) = A061395(a(n)) = A061394(n).
A007814(a(n)) = A051903(a(n)) = A051282(n).
a(A101296(n)) = A046523(n).
a(A306802(n)) = A002182(n).
a(n) = A108951(A181815(n)) = A329900(A181817(n)).
If A181815(n) is odd, a(n) = A283980(a(A329904(n))), otherwise a(n) = 2*a(A329904(n)).
(End)
Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A002110(n)) = A161360. - Amiram Eldar, Oct 20 2020

Offset corrected by Matthew Vandermast, Oct 19 2008

Minor correction by Charles R Greathouse IV, Sep 03 2010

A181818 Products of superprimorials (A006939).

Original entry on oeis.org

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

A133808 Numbers that are primally tight, have 2 as first prime and weakly ascending powers.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 18, 30, 32, 36, 54, 64, 108, 128, 150, 162, 210, 216, 256, 324, 450, 486, 512, 648, 750, 900, 972, 1024, 1296, 1458, 1470, 1944, 2048, 2250, 2310, 2916, 3750, 3888, 4096, 4374, 4500, 5832, 6750, 7350, 7776, 8192, 8748, 10290, 11250

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, A133810.

			10 = 2*5 with missing prime factor 3 between 2 and 5 is not in the sequence.
12 = 2^2*3 with 2's exponent > 3's exponent is not in the sequence.

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

Cf. A025487, A087980, A073491, A133809-A133813.

Cf. A027748, A124010, A151800, A000040.

import Data.Set (singleton, deleteFindMin, insert)
a133808 n = a133808_list !! (n-1)
a133808_list = 1 : f (singleton (2, 2, 1)) where
   f s = y : f (insert (y * p, p, e + 1) $ insert (y * q^e, q, e) s')
             where q = a151800 p
                   ((y, p, e), s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 13 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

A304405 Number of partitions of n in which the sequence of the sum of the same summands is nondecreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 31, 37, 52, 61, 80, 97, 127, 147, 189, 220, 277, 325, 402, 469, 578, 665, 804, 933, 1121, 1282, 1537, 1754, 2081, 2374, 2793, 3179, 3739, 4232, 4923, 5587, 6477, 7305, 8445, 9519, 10949, 12323, 14110, 15825, 18099, 20229, 23005

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Number of integer partitions of n with weakly decreasing run-sums, complement A357878. - Gus Wiseman, Oct 22 2022

			n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 2+1                  | 1, 2
  | 1+1+1                | 3
4 | 4                    | 4
  | 3+1                  | 1, 3
  | 2+2                  | 4
  | 2+1+1                | 2, 2
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 4+1                  | 1, 4
  | 3+2                  | 2, 3
  | 3+1+1                | 2, 3
  | 2+2+1                | 1, 4
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 5+1                  | 1, 5
  | 4+2                  | 2, 4
  | 4+1+1                | 2, 4
  | 3+3                  | 6
  | 3+2+1                | 1, 2, 3
  | 3+1+1+1              | 3, 3
  | 2+2+2                | 6
  | 2+2+1+1              | 2, 4
  | 1+1+1+1+1+1          | 6

The strict opposite version is A304430, ranked by A357864.
The strict version is A304428, ranked by A357862.
The opposite version is A304406, ranked by A357861.
Number of rows in A354584 summing to n that are strictly increasing.
These partitions are ranked by A357875.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A087980, A098859, A100883, A239312, A275870, A353832, A353864.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@Total/@Split[#]&]],{n,0,30}] (* Gus Wiseman, Oct 22 2022 *)

A304428 Number of partitions of n in which the sequence of the sum of the same summands is increasing.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 11, 14, 20, 26, 33, 41, 50, 64, 81, 97, 120, 150, 176, 210, 255, 303, 362, 426, 503, 595, 703, 816, 953, 1113, 1283, 1482, 1721, 1988, 2299, 2650, 3031, 3464, 3965, 4492, 5115, 5820, 6592, 7467, 8484, 9568, 10822, 12185, 13724, 15445, 17381, 19475, 21855

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Number of integer partitions of n with strictly decreasing run-sums. - Gus Wiseman, Oct 21 2022

			n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 2+1                  | 1, 2
  | 1+1+1                | 3
4 | 4                    | 4
  | 3+1                  | 1, 3
  | 2+2                  | 4
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 4+1                  | 1, 4
  | 3+2                  | 2, 3
  | 3+1+1                | 2, 3
  | 2+2+1                | 1, 4
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 5+1                  | 1, 5
  | 4+2                  | 2, 4
  | 4+1+1                | 2, 4
  | 3+3                  | 6
  | 3+2+1                | 1, 2, 3
  | 2+2+2                | 6
  | 2+2+1+1              | 2, 4
  | 1+1+1+1+1+1          | 6

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 71 terms from Seiichi Manyama)

The weak version is A304405, ranked by A357875.
The weak opposite version is A304406, ranked by A357861.
The opposite version is A304430, ranked by A357864.
Number of rows in A354584 summing to n that are strictly increasing.
These partitions are ranked by A357862, complement A357863.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A087980, A098859, A100471, A239312, A275870, A353832, A353864.

Table[Length[Select[IntegerPartitions[n],Greater@@Total/@Split[#]&]],{n,0,30}] (* Gus Wiseman, Oct 21 2022 *)

a(n) <= A304405(n).

A304430 Number of partitions of n in which the sequence of the sum of the same summands is decreasing.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 5, 6, 8, 10, 10, 13, 15, 18, 19, 22, 26, 33, 33, 38, 41, 50, 53, 60, 68, 77, 84, 94, 100, 116, 122, 136, 148, 172, 182, 206, 219, 246, 258, 281, 301, 341, 365, 397, 429, 466, 489, 528, 572, 623, 660, 728, 773, 849, 895, 968, 1019, 1120, 1188, 1288

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Number of integer partitions of n with strictly increasing run-sums. - Gus Wiseman, Oct 22 2022

			n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 1+1+1                | 3
4 | 4                    | 4
  | 2+2                  | 4
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 2+1+1+1              | 3, 2
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 3+3                  | 6
  | 2+2+2                | 6
  | 2+1+1+1+1            | 4, 2
  | 1+1+1+1+1+1          | 6

The weak opposite version is A304405, ranked by A357875.
The weak version is A304406, ranked by A357861.
The opposite version is A304428, ranked by A357862.
Number of rows in A354584 summing to n that are strictly decreasing.
These partitions are ranked by A357864.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A087980, A098859, A100471, A100881, A239312, A275870, A353832, A353833, A353864.

Table[Length[Select[IntegerPartitions[n],Less@@Total/@Split[#]&]],{n,0,30}] (* Gus Wiseman, Oct 22 2022 *)

a(n) <= A304406(n).

A304406 Number of partitions of n in which the sequence of the sum of the same summands is nonincreasing.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 6, 5, 9, 8, 11, 11, 20, 16, 20, 21, 32, 30, 41, 38, 50, 48, 62, 64, 89, 81, 97, 100, 123, 123, 151, 154, 187, 183, 221, 221, 279, 272, 312, 316, 377, 376, 446, 460, 531, 547, 628, 641, 754, 746, 841, 856, 990, 1007, 1145, 1167, 1325, 1346, 1519, 1567, 1776

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Number of integer partitions of n with weakly increasing run-sums. - Gus Wiseman, Oct 21 2022

			n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 1+1+1                | 3
4 | 4                    | 4
  | 2+2                  | 4
  | 2+1+1                | 2, 2
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 2+1+1+1              | 3, 2
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 3+3                  | 6
  | 3+1+1+1              | 3, 3
  | 2+2+2                | 6
  | 2+1+1+1+1            | 4, 2
  | 1+1+1+1+1+1          | 6

Cf. A100882.
These partitions are ranked by A357861.
The complement is A357865, ranked by A357850.
The opposite version is A304405, ranked by A357875.
The strict version is A304430, ranked by A357864.
The strict opposite version is A304428, ranked by A357862.
Number of rows in A354584 summing to n that are weakly decreasing.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A087980, A098859, A239312, A275870, A353832, A353833, A353864.

Table[Length[Select[IntegerPartitions[n],LessEqual@@Total/@Split[#]&]],{n,0,30}] (* Gus Wiseman, Oct 21 2022 *)

A212165 Numbers k such that the maximum exponent in its prime factorization is not less than the number of positive exponents (A051903(k) >= A001221(k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 100, 101, 103, 104

Offset: 1

Matthew Vandermast, May 22 2012

Union of A212164 and A212166.  Includes numerous subsequences that are subsequences of neither A212164 nor A212166.
Includes all factorials except A000142(3) = 6.
Observation: all terms in DATA section are also the first 65 numbers n whose difference between the arithmetic derivative of n and the sum of the divisors of n is nonnegative. - Omar E. Pol, Dec 19 2012

			10 = 2^1*5^1 has 2 distinct prime factors, hence, 2 positive exponents in its prime factorization (although 1s are often left implicit).  2 is larger than the maximal exponent in 10's prime factorization, which is 1. Therefore, 10 does not belong to the sequence. But 20 = 2^2*5^1 and 40 = 2^3*5^1 belong, since the largest exponents in their prime factorizations are 2 and 3 respectively.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages).

Complement of A212168.
See also A212167.
Subsequences (none of which are subsequences of A212164 or A212166) include A000079, A001021, A066120, A087980, A130091, A141586, A166475, A181818, A181823, A181824, A182763, A212169. Also includes all terms in A181813 and A181814.
Cf. A001221, A051903, A188654, A225230.

import Data.List (findIndices)
a212165 n = a212165_list !! (n-1)
a212165_list = map (+ 1) $ findIndices (<= 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] >= Length[f]]; Select[Range[1000], okQ] (* T. D. Noe, May 24 2012 *)

is(k) = {my(e = factor(k)[, 2]); !(#e) || vecmax(e) >= #e;} \\ Amiram Eldar, Sep 08 2024

A225230(a(n)) <= 0. - Reinhard Zumkeller, May 03 2013

cp's OEIS Frontend

A363537 Rewrite A087980(n) = Product_{i=1..m} p(i)^e(i) instead as Sum_{i=1..m} 2^(i-1), where m = omega(A087980(n)) = A001221(A087980(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

PARI

PARI

Sage

Formula

Extensions

A181818 Products of superprimorials (A006939).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A133808 Numbers that are primally tight, have 2 as first prime and weakly ascending powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

A304405 Number of partitions of n in which the sequence of the sum of the same summands is nondecreasing.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A304428 Number of partitions of n in which the sequence of the sum of the same summands is increasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A304430 Number of partitions of n in which the sequence of the sum of the same summands is decreasing.

Views

Author

Keywords

Comments

Examples