A146288 -id:A146288 - 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

A146290 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A061394(n)), giving the number of divisors of A025487(n) with m distinct prime factors.

Original entry on oeis.org

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

Offset: 1

Matthew Vandermast, Nov 11 2008

The formula used in obtaining the A025487(n)th row (see below) also gives the number of divisors of the k-th power of A025487(n).
Every row that appears in A146289 appears exactly once in the table. Rows appear in order of first appearance in A146289.
T(n,0)=1.

			Rows begin:
  1;
  1,1;
  1,2;
  1,2,1;
  1,3;
  1,3,2;
  1,4;
  1,4,3;...
36's 9 divisors include 1 divisor with 0 distinct prime factors (1); 4 with 1 (2, 3, 4 and 9); and 4 with 2 (6, 12, 18 and 36). Since 36 = A025487(11), the 11th row of the table therefore reads (1, 4, 4). These are the positive coefficients of the polynomial equation 1 + 4k + 4k^2 = (1 + 2k)(1 + 2k), derived from the prime factorization of 36 (namely, 2^2*3^2).

Anonymous?, Polynomial calculator
Eric Weisstein's World of Mathematics, Distinct Prime Factors
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

For the number of distinct prime factors of n, see A001221.
Row sums equal A146288(n). T(n, 1)=A036041(n) for n>1. T(n, A061394(n))=A052306(n).
Row A098719(n) of this table is identical to row n of A007318.
Cf. A146289. Also cf. A146291, A146292.

If A025487(n)'s canonical factorization into prime powers is Product p^e(p), then T(n, m) is the coefficient of k^m in the polynomial expansion of Product_p (1 + ek).

A146292 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A036041(n)), giving the number of divisors of A025487(n) with m prime factors (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 2, 2, 2, 2, 1, 1, 3, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 3, 4, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 2, 2, 2, 2

Offset: 1

Matthew Vandermast, Nov 11 2008

All rows are palindromic. T(n, 0) = T(n, A036041(n)) = 1.

Every row that appears in A146291 appears exactly once in the table. Rows appear in order of first appearance in A146291.

			Rows begin:
  1;
  1,1;
  1,1,1;
  1,2,1;
  1,1,1,1;
  1,2,2,1;
  1,1,1,1,1;...
36's 9 divisors include 1 divisor with 0 total prime factors (1);, 2 with 1 (2 and 3); 3 with 2 (4, 6 and 9); 2 with 3 (12 and 18); and 1 with 4 (36). Since 36 = A025487(11), the 11th row of the table therefore reads (1, 2, 3, 2, 1). These are the positive coefficients of the polynomial 1 + 2k + 3k^2 + 2k^3 + (1)k^4 = (1 + k + k^2)(1 + k + k^2), derived from the prime factorization of 36 (namely, 2^2*3^2).

Anonymous?, Polynomial calculator
Eric Weisstein's World of Mathematics, Roundness
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

For the number of prime factors of n counted with multiplicity, see A001222.
Row sums equal A146288(n). T(n, 1) = A061394(n) for n>1.
Row A098719(n) of this table is identical to row n of A007318.
Cf. A146291. Also cf. A146289, A146290.

If A025487(n)'s canonical factorization into prime powers is the product of p^e(p), then T(n, m) is the coefficient of k^m in the polynomial expansion of Product_p (sum_{i=0..e} k^i).

A375195 Numbers k such that A025487(k) and A025487(k+1) have an equal number of divisors.

Original entry on oeis.org

4, 8, 15, 22, 69, 116, 122, 134, 135, 168, 208, 278, 400, 453, 538, 584, 718, 1019, 1409, 1671, 1799, 2035, 2417, 2541, 2595, 2783, 3424, 3809, 3860, 4415, 5628, 6267, 6672, 6745, 6872, 6873, 7277, 9436, 9845, 10182, 10191, 10936, 11272, 11472, 12105, 16139, 16277

Offset: 1

Amiram Eldar, Aug 04 2024

nonn

Numbers k such that A146288(k) = A146288(k+1).
The corresponding values of A146288(k) are 4, 8, 12, 16, 48, 48, 96, 80, 80, ... .
The corresponding values of A025487(k) are 6, 24, 72, 210, 5400, ... (A375196).
Numbers k such that A146288(k) = A146288(k+1) = A146288(k+2) are 134, 6872, 6699401, 12421946, ... .

			4 is a term since A146288(4) = A146288(5) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..4390

Cf. A000005, A025487, A146288, A375196.

With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, Position[Differences[DivisorSigma[0, lps]], 0] // Flatten]

A025487(a(n)) = A375196(n).

A375196 Smaller of two successive terms of A025487 that have an equal number of divisors.

Original entry on oeis.org

6, 24, 72, 210, 5400, 30720, 36960, 51840, 53760, 120120, 264600, 887040, 3991680, 6912000, 14968800, 22118400, 58198140, 319334400, 1703116800, 4151347200, 6273146880, 12247200000, 31757806080, 42343741440, 47636709120, 70572902400, 238378140000, 442810368000

Offset: 1

Amiram Eldar, Aug 04 2024

nonn

There are runs of three successive terms of A025487 that have an equal number of divisors. The smallest elements in these runs are 51840, 17149215283200, 63147292984115358771227840741376000000000, ... . Are there such runs of four successive terms?

			6 is a term since 6 and 8 are two successive terms of A025487, and they have an equal number of divisors: A000005(6) = A000005(8) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..4390

Cf. A000005, A025487, A146288, A375195.

With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, lps[[Position[Differences[DivisorSigma[0, lps]], 0] // Flatten]]]

a(n) = A025487(A375195(n)).

A356425 Sum of divisors of numbers of least prime signature: a(n) = A000203(A025487(n)).

Original entry on oeis.org

1, 3, 7, 12, 15, 28, 31, 60, 72, 63, 91, 124, 168, 127, 195, 252, 360, 255, 403, 546, 508, 576, 600, 744, 511, 819, 1170, 1020, 1344, 1240, 1512, 1023, 1651, 2418, 2044, 2880, 2520, 2821, 3048, 2047, 3600, 3315, 4368, 3751, 4914, 4092, 5952, 5080, 6045, 6120

Offset: 1

Hal M. Switkay, Dec 11 2022

nonn

This sequence is to A146288 as sigma (A000203, the sum of divisors function) is to tau (A000005, the number of divisors function).

Hal M. Switkay, Table of n, a(n) for n = 1..67

Cf. A000005, A000203, A025487, A146288.

s = {1}; Do[If[GreaterEqual @@ (f = FactorInteger[n])[[;;,2]] && PrimePi[f[[-1,1]]] == Length[f], AppendTo[s, DivisorSigma[1,n]]], {n, 2, 2000}]; s (* Amiram Eldar, Dec 12 2022 *)

A359182 Totient of numbers of least prime signature: a(n) = A000010(A025487(n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 8, 8, 16, 12, 16, 16, 32, 24, 32, 32, 64, 48, 48, 64, 48, 72, 64, 128, 96, 96, 128, 96, 144, 128, 256, 192, 192, 256, 192, 288, 240, 256, 512, 288, 384, 288, 432, 384, 512, 384, 576, 480, 512, 1024, 576, 768, 480, 576, 864, 768, 1024, 768

Offset: 1

Hal M. Switkay, Dec 18 2022

nonn

This sequence is to A146288 as phi (A000010, the totient function) is to tau (A000005, the number of divisors function).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..160 from Hal M. Switkay)

Cf. A000005, A000010, A025487, A146288.

s = {1}; Do[If[GreaterEqual @@ (f = FactorInteger[n])[[;; , 2]] && PrimePi[f[[-1, 1]]] == Length[f], AppendTo[s, EulerPhi[n]]], {n, 2, 3500}]; s (* Amiram Eldar, Dec 19 2022 *)

is(n) = {if(n==1, return(1)); my(f = factor(n));  f[#f~, 1] == prime(#f~) && vecsort(f[, 2], , 4) == f[, 2]}; \\ A025487
lista(nn) = apply(eulerphi, select(is, [1..nn])); \\ Michel Marcus, Dec 19 2022

cp's OEIS Frontend

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

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A146290 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A061394(n)), giving the number of divisors of A025487(n) with m distinct prime factors.

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A146292 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A036041(n)), giving the number of divisors of A025487(n) with m prime factors (counted with multiplicity).

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A375195 Numbers k such that A025487(k) and A025487(k+1) have an equal number of divisors.

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A375196 Smaller of two successive terms of A025487 that have an equal number of divisors.

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A356425 Sum of divisors of numbers of least prime signature: a(n) = A000203(A025487(n)).

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A359182 Totient of numbers of least prime signature: a(n) = A000010(A025487(n)).

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