A124832 -id:A124832 - cp's OEIS frontend

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

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

A085089 Number of distinct prime signatures arising up to n.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 02 2003

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A025487.

Cf. A025488(n) = a(2^n); A124832 (table of the distinct prime signatures in the order they occur).

A085089(n)=#Set(apply(t->vecsort(factor(t)[,2]), [1..n])) \\ Not very efficient for large n > 10^5, but very quick up to the point where stack overflow occurs. - M. F. Hasler, Jul 16 2019

More terms from Ray Chandler, Aug 17 2003

A025488 Number of distinct prime signatures of the positive integers up to 2^n.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 14, 18, 25, 32, 40, 51, 63, 80, 98, 119, 145, 173, 207, 248, 292, 346, 404, 473, 552, 639, 742, 855, 984, 1129, 1289, 1477, 1681, 1912, 2170, 2452, 2771, 3121, 3514, 3951, 4426, 4955, 5536, 6182, 6898, 7674, 8535, 9470, 10500, 11633, 12869

Offset: 0

David W. Wilson

nonn

The distinct prime signatures, in the order in which they occur, are listed in A124832. - M. F. Hasler, Jul 16 2019

The subsequence a(n) = A085089(2^n) is strictly increasing since it counts at least the additional prime signature (n) which did not occur for the previously considered numbers. All other partitions of n are prime signatures of numbers larger than 2^n and therefore counted only as part of later terms. - M. F. Hasler, Jul 17 2019

			From _M. F. Hasler_, Jul 16 2019: (Start)
For n = 0, the only integer k to be considered is 1, so the only prime signature is the empty one, (), whence a(0) = 1.
For n = 1, the integers k to be considered are {1, 2}; the prime signatures are {(), (1)}, whence a(1) = 2.
For n = 2, the integers k to be considered are {1, 2, 3, 4}; the distinct prime signatures are {(), (1), (2)}, whence a(2) = 3.
For n = 3, the integers k to be considered are {1, 2, 3, 4, 5, 6, 7, 8}; the distinct prime signatures are {(), (1), (2), (1,1), (3)}, whence a(3) = 5. (End)

Ray Chandler, Table of n, a(n) for n = 0..182 (first 151 terms from T. D. Noe)

A025487(a(n)) = 2^n.
Partial sums of A056099.
Cf. A085089, A124832.

A025488(n)=A085089(2^n) \\ For illustrative purpose, n not too large. - M. F. Hasler, Jul 16 2019

a(n) = Sum_{k=0..n} A056099(k). - M. F. Hasler, Jul 16 2019

a(n) = A085089(2^n). - M. F. Hasler, Jul 17 2019

Name edited by M. F. Hasler, Jul 16 2019

A124829 Table of exponents of prime factorizations in A055932.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 4, 1, 2, 3, 1, 1, 1, 1, 5, 2, 2, 4, 1, 1, 3, 2, 1, 1, 6, 3, 2, 1, 2, 1, 5, 1, 2, 3, 3, 1, 1, 7, 4, 2, 1, 1, 2, 1, 4, 2, 2, 1, 6, 1, 1, 1, 1, 1, 3, 3, 4, 1, 1, 8, 1, 3, 1, 5, 2, 2, 1, 2, 2, 4, 3, 2, 1, 7, 1, 2, 1, 1, 1, 4, 3, 1, 2, 2, 5, 1, 1, 1, 5, 9, 2, 3, 1, 6, 2, 3, 1, 2, 1, 2, 1, 1, 3, 4

Offset: 1

Franklin T. Adams-Watters, Nov 09 2006

This is an enumeration of all compositions. This sequence contains all finite sequences of positive integers.

			From _Michael De Vlieger_, Feb 06 2020: (Start)
Table begins:
   n   A055932(n+1)  row n
   ---------------------
   1    2            1;
   2    4            2;
   3    6            1, 1;
   4    8            3;
   5   12            2, 1;
   6   16            4;
   7   18            1, 2;
   8   24            3, 1;
   9   30            1, 1, 1;
  10   32            5;
  11   36            2, 2;
  12   48            4, 1;
  13   54            1, 3;
  14   60            2, 1, 1;
  15   64            6;
  ...  (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10382 (rows 1 <= n <= 2500).

Cf. A055932, A124830 (row lengths), A124831 (row sums), A124832, A066099.

Map[FactorInteger[#][[All, -1]] &, Select[Range[10^3], Last[#] == Length[#] &@ PrimePi@ FactorInteger[#][[All, 1]] &]] // Flatten (* Michael De Vlieger, Feb 06 2020 *)

A055932(n) = Product_k Prime(k)^T(n,k).

A304886 Irregular triangle where row n contains indices k where the product of A002110(k) = A025487(n).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 1, 1, 1, 1, 1, 2, 4, 2, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2

Offset: 1

Michael De Vlieger, May 21 2018

Row n consists of terms k such that A025487(n) = the product of primorials p_k#, the k in row n written least to greatest k.
For m = A025487(n) in A000079 (i.e., m is an integer power of 2), row n contains A000079(m) 1s.
For m = A025487(n) in A002110 (i.e., m is a primorial) row n contains a single term k that is the index of m in A002110.

			Triangle begins as in rightmost column, which lists the terms that occur on row n. Maximum value of each row is given by A061394(n).
   n  A025487(n)   Row n
--------------------------------
   1        1      0
   2        2      1
   3        4      1,1
   4        6      2
   5        8      1,1,1
   6       12      1,2
   7       16      1,1,1,1
   8       24      1,1,2
   9       30      3
  10       32      1,1,1,1,1
  11       36      2,2
  12       48      1,1,1,2
  13       60      1,3
  14       64      1,1,1,1,1,1
  15       72      1,2,2
  16       96      1,1,1,1,2
  17      120      1,1,3
  18      128      1,1,1,1,1,1,1
  19      144      1,1,2,2
  20      180      2,3
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..8600
Michael De Vlieger, Concordance of A025487, A051282, A061394, and A304886
Michael De Vlieger, Indices of primorials whose product is highly composite
Michael De Vlieger, Indices of primorials whose product is superabundant

Cf. A025487, A051282 (row lengths), A061394 (row maximum), A124832, A181815.

Cf. also A307056.

(* Simple (A025487(n) < 10^5): *)
{{0}}~Join~Map[With[{w = #}, Reverse@ Array[Function[k, Count[w, _?(# >= k &)] ], Max@ w]] &, Select[Array[{#, FactorInteger[#][[All, -1]]} &, 400], Times @@ Boole@ {#1 == Times @@ MapIndexed[Prime[First@ #2]^#1 &, #3], #2 == #3} == 1 & @@ {#1, #2, Sort[#2, Greater]} & @@ # &][[All, -1]] ]
(* Efficient (A025487(n) < 10^23): *)
f[n_] := Block[{ww, g, h},
  g[x_] := Apply[Times,
    MapIndexed[Prime[First@ #2]^#1 &, x]];
  h[x_] := Reverse@
    Array[Function[k, Count[x, _?(# >= k &)] ], Max@ x];
  ww = NestList[Append[#, 1] &, {1}, # - 1] &[-2 +
     Length@ NestWhileList[NextPrime@ # &, 1,
     Times @@ {##} <= n &, All] ];
  Map[h, SortBy[Flatten[#, 1], g]] &@
   Map[Block[{w = #, k = 1},
      Apply[
         Join, {{ConstantArray[1, Length@ w]},
           If[Length@ # == 0, #, #[[1]]] }] &@ Reap[
         Do[
          If[# < n,
            Sow[w]; k = 1,
             If[k >= Length@ w, Break[], k++]] &@
               g@ 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]]; {{0}}~Join~f@ 400

For row n > 1, Product_{k=1..A051282(n)} A000040(T(n,k)) = A181815(n). [Product of primes indexed by nonzero terms of row n is equal to A181815(n)] - Antti Karttunen, Dec 28 2019

A369168 Numbers k such that A000005(k) = A000688(k).

Original entry on oeis.org

1, 16, 81, 625, 1296, 2401, 10000, 14641, 23040, 28561, 32256, 38400, 38416, 50625, 50688, 59904, 75264, 78336, 83521, 87552, 89600, 105984, 125440, 130321, 133632, 140800, 142848, 166400, 170496, 185856, 188928, 194481, 198144, 216576, 217600, 234256, 243200

Offset: 1

Amiram Eldar, Jan 15 2024

nonn

The asymptotic density of this sequence is 0 (Ivić, 1983).

If k is a term, then every number with the same prime signature (A124832) as k is a term. The least term of each prime signature is given in A369169.

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 73.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137.

Cf. A000005, A000688, A124832.
Subsequence of A369170.
A369169 is a subsequence.

Select[Range[250000], DivisorSigma[0, #] == FiniteAbelianGroupCount[#] &]

is(n) = {my(e = factor(n)[,2]); vecprod(apply(x -> x+1, e)) == vecprod(apply(numbpart, e));}

x * log(log(x))/log(x) << N(x) << x / log(x)^(1-eps) for every 0 < eps < 1, where N(x) is the number of terms not exceeding x (Ivić, 1983).

A369169 Terms k of A025487 such that A000005(k) = A000688(k).

Original entry on oeis.org

1, 16, 1296, 23040, 810000, 7257600, 16934400, 283852800, 1437004800, 1944810000, 13970880000, 30735936000, 232475443200, 852409958400, 1765360396800, 3269185920000, 7192209024000, 8029628006400, 28473963210000, 97893956160000, 181803061440000, 1086822696960000

Offset: 1

Amiram Eldar, Jan 15 2024

nonn

Since both A000005(k) and A000688(k) depend only on the prime signature of k (A124832), if k is a term of this sequence then every number m such that A046523(m) = k is a term of A369168.
From David A. Corneth, Jan 15 2024: (Start)
16 | a(n) for n > 1.
This sequence contains A002110(n)^4. (End)

			16 is in the sequence as 16 has 5 divisors (1, 2, 4, 8, 16) and 5 factorizations into prime powers (16 = 2*8 = 4*4 = 2*2*4 = 2*2*2*2).

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 73.

Amiram Eldar, Table of n, a(n) for n = 1..377
Aleksandar Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137.

Cf. A000005, A000688, A002110, A046523, A124832.

Intersection of A025487 and A369168.

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; Select[lps, DivisorSigma[0, #] == FiniteAbelianGroupCount[#] &]

A212175 List of exponents >= 2 in canonical prime factorization of A025487(n) (first integer of each prime signature), in nonincreasing order, or 0 if no such exponent exists.

Original entry on oeis.org

0, 0, 2, 0, 3, 2, 4, 3, 0, 5, 2, 2, 4, 2, 6, 3, 2, 5, 3, 7, 4, 2, 2, 2, 6, 0, 3, 3, 4, 8, 5, 2, 3, 2, 7, 2, 4, 3, 5, 9, 6, 2, 4, 2, 8, 3, 5, 3, 2, 2, 2, 6, 10, 3, 3, 7, 2, 2, 2, 4, 4, 5, 2, 9, 4, 6, 3, 3, 2, 2, 7, 11, 4, 3, 8, 2, 0, 3, 2, 5, 4, 6, 2, 10, 5, 7, 3

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of row n equals A212178(n) if A212178(n) is positive, or 1 if A212178(n) = 0.

Row n of table represents second signature of A025487(n) (cf. A212172). The use of 0 in the table to represent numbers with no exponents >=2 in their prime factorization accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers is represented as { }.

			240 = 2^4*3*5 has 1 exponent in its canonical prime factorization that equals or exceeds 2 (namely, 4). Hence, 240's second signature is {4}. Since 240 = A025487(24), row 24 of the table represents the second signature {4}.

Will Nicholes, Prime Signatures

Cf. A025487, A212172, A212176, A212178.

A124832 gives all positive exponents in prime factorization of A025487(n) for n > 1.

a(n) = A212172(A025487(n)).

A353248 Irregular table, read by rows, where row n is the concatenation of all prime signatures leading to n divisors, in reverse lexicographic order.

Original entry on oeis.org

1, 2, 3, 1, 1, 4, 5, 2, 1, 6, 7, 3, 1, 1, 1, 1, 8, 2, 2, 9, 4, 1, 10, 11, 5, 1, 3, 2, 2, 1, 1, 12, 13, 6, 1, 14, 4, 2, 15, 7, 1, 3, 3, 3, 1, 1, 1, 1, 1, 1, 16, 17, 8, 1, 5, 2, 2, 2, 1, 18, 19, 9, 1, 4, 3, 4, 1, 1, 20, 6, 2, 21, 10, 1, 22, 23, 11, 1, 7, 2, 5, 3, 5, 1, 1, 3, 2, 1

Offset: 1

M. F. Hasler, Apr 08 2022

The number-of-divisors function d = A000005 is multiplicative with d(p^e) = e+1. Therefore, a prime signature (e1, e2, ..., eN) which yields a given number of divisors corresponds to a factorization (e1+1)*...*(eN+1) of that number. Following the usual convention, we only consider prime signatures with decreasing exponents (e1 >= e2 >= ... >= eN). Furthermore, we list them in reverse lexicographic order, i.e., largest e1 first, etc.

The sequence starts with row 1 which has length 0 (the only number having only 1 divisor is 1 which has the empty product as prime factorization), so the first term a(1) = 1 is the first element of row 2. Here and thereafter, the first element of row n is easily recognized as the first occurrence of n-1, which is the only element of the row (and therefore followed by n) iff n is prime.

			Table begins:
row n | prime signatures
   1  | ()
   2  | (1)
   3  | (2)
   4  | (3), (1,1)
   5  | (4)
   6  | (5), (2,1)
   7  | (6)
   8  | (7), (3,1), (1,1,1)
   9  | (8), (2,2)
  10  | (9), (4,1)
  11  | (10)
  12  | (11), (5,1), (3,2), (2,1,1)

Cf. A000005 (d = tau = number-of-divisors function).

Cf. A025487 (products of primorial numbers, representatives of prime signatures), A046523 (representative of prime signature of n), A118914, A212171 and A124010 (prime signature of n), A124832 (prime signatures listed in order of representatives A025487), A080577 (partitions in grad.rev.lex order), A036036 (partitions in rev.lex order).

A353248_row(n, M=n)={if(n>1, my(f=factor(n)~, m=f[1,#f], L=List()); fordiv(n, d, n < m*d && break; n > M*d || foreach(self()(d, n/d), S, listput(L,concat(n/d-1,S)))); Vec(L), [[]])}

A368448 Positive integers k such that there is no m different from k where both s(k) = s(m) and s(k+1) = s(m+1), where s(k) is the prime signature of k.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 15, 16, 24, 26, 27, 31, 32, 35, 48, 63, 64, 80, 99, 124, 127, 128, 224, 242, 243, 255, 256, 288, 343, 511, 512, 528, 575, 624, 675, 728, 783, 960, 999, 1023, 1024, 1088, 1295, 1331, 2047, 2048, 2186, 2187, 2208, 2303, 2400, 3375, 3968, 4095, 4096

Offset: 1

Jon E. Schoenfield, Dec 24 2023

nonn

In other words, numbers k that are uniquely identified by the values of the ordered pair (s(k), s(k+1)), where s(k) is the prime signature of k.
Other than the first two terms, every term <= 4096 is either a proper power (a number of the form b^e with e > 1) or one less than a proper power.
For the analogous sequence using the number of divisors rather than the prime signature, see A161460.

			The prime factorizations of k = 15 and k+1 = 16 are 3 * 5 and 2^4, respectively, so their prime signatures can be represented as [1,1] and [4], respectively. If any ordered pair of consecutive integers m and m+1 has this same ordered pair of prime signatures, then m+1 = p^4 for some prime p, so m = p^4 - 1 = (p-1)*(p+1)*(p^2+1), which is a multiple of 16 for any odd prime p, so the prime signature of m cannot be [1,1] unless the prime p is even, i.e., p = 2, so m = 2^4 - 1 = 15; there is no m other than k = 15 that yields the same pair of prime signatures, so k = 15 is a term of the sequence.
k = 125 is not a term of the sequence: 125 = 5^3 and 126 = 2 * 3^2 * 7, and the same pair of prime signatures occurs for m and m+1 at m = 67^3 = 300763; m+1 = 300764 = 2^2 * 17 * 4423.

Cf. A124832 (prime signatures), A161460.

cp's OEIS Frontend

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

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A085089 Number of distinct prime signatures arising up to n.

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A025488 Number of distinct prime signatures of the positive integers up to 2^n.

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A124829 Table of exponents of prime factorizations in A055932.

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A304886 Irregular triangle where row n contains indices k where the product of A002110(k) = A025487(n).

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A369168 Numbers k such that A000005(k) = A000688(k).

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A369169 Terms k of A025487 such that A000005(k) = A000688(k).

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