A025488 -id:A025488 - cp's OEIS frontend

A056099 First differences of A025488 (binary order for least prime signatures).

1, 1, 1, 2, 2, 3, 4, 4, 7, 7, 8, 11, 12, 17, 18, 21, 26, 28, 34, 41, 44, 54, 58, 69, 79, 87, 103, 113, 129, 145, 160, 188, 204, 231, 258, 282, 319, 350, 393, 437, 475, 529, 581, 646, 716, 776, 861, 935, 1030, 1133, 1236, 1358, 1476, 1605, 1763, 1904, 2089, 2267, 2460

Offset: 0

Alford Arnold, Jul 29 2000

a(n) counts entries of A025487 starting a new row after each power of two.

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

Cf. A025487, A025488.

More terms from Sascha Kurz, Mar 25 2002

Extended by Ray Chandler, Aug 11 2010

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

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

A366884 Number of branching factorizations of the least integer of each prime signature (A025487).

Original entry on oeis.org

0, 1, 2, 3, 5, 11, 15, 45, 19, 51, 62, 195, 113, 188, 345, 873, 645, 731, 1890, 911, 3989, 207, 2405, 3585, 2950, 10221, 6525, 18483, 1709, 15775, 19569, 12235, 54718, 43545, 86515, 12405, 99215, 9332, 105447, 51822, 55885, 290611, 17753, 120075, 277203, 408105, 83505, 605135, 80565, 562739, 223191, 432975, 1533670

Offset: 1

Antti Karttunen, Jan 02 2024

nonn

Sequence appears to be injective, but can it be proved? This would prove also the conjectures given in A277120 and A366377.

Of the first 21001 terms, there are 701 terms ending with digit "0", 614 with "1", 68 with "2", 570 with "3", 0 with "4", 17795 with "5", 0 with "6", 550 with "7", 67 with "8", and 636 with "9". Why such an overrepresentation (~ 85% of the total) of the terms of form 10k+5? Do any terms of the form 10k+4 or 10k+6 exist? See also the comments in A052886.

Antti Karttunen, Table of n, a(n) for n = 1..21001
Michael De Vlieger, Plot k = a(n) mod 10 at (x,y) = (n mod 144, 1 + floor(n/144)), n = 1..20736, showing 0 in black, 1 in red, 2 in orange, 3 in yellow, 4 = dark green, 5 = bright green, 6 = cyan, 7 = light blue, 8 = dark blue, and 9 in purple.

Cf. A007317, A025487, A025488, A052886, A098719, A181815, A277120.

Permutation of A366377.

a(n) = A277120(A025487(n)).
a(n) = A366377(A181815(n)).
For all n >= 1, a(A025488(n)) = A007317(n), a(A098719(n)) = A052886(n).

A037088 Triangle read by rows: T(n,k) is number of numbers x, 2^n <= x < 2^(n+1), with k prime factors (counted with multiplicity).

Original entry on oeis.org

2, 2, 2, 2, 4, 2, 5, 4, 5, 2, 7, 12, 6, 5, 2, 13, 20, 17, 7, 5, 2, 23, 40, 30, 20, 8, 5, 2, 43, 75, 65, 37, 21, 8, 5, 2, 75, 147, 131, 81, 41, 22, 8, 5, 2, 137, 285, 257, 173, 91, 44, 22, 8, 5, 2, 255, 535, 536, 344, 199, 96, 46, 22, 8, 5, 2, 464, 1062, 1033, 736, 403, 215, 99, 47

Offset: 1

Alford Arnold

Sequence A092097 gives the limiting behavior at the end of the rows. - T. D. Noe, Feb 22 2008

			The triangular array begins 2; 2,2; 2,4,2; 5,4,5,2; 7,12,6,5,2; ...
a(7) = 5 because the 3-almost primes between 16 and 32 are (18,20,27,28,30).

T. D. Noe, Rows n=1..24 of triangle, flattened

A001222 counts factors of n. A000040, A001358, A014612-A014614 are special cases. A036378 and A025488 are applications of binary order A029837. Leading diagonal is essentially A036378 and has partial sums A007053.

t[n_, k_] := Count[Range[2^n, 2^(n+1)-1], x_ /; Total[FactorInteger[x][[All, 2]]] == k]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 07 2013 *)

More terms from Naohiro Nomoto, Jun 18 2001

A369137 Inverse permutation to A369136.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 11, 7, 8, 9, 51, 10, 28, 14, 12, 15, 1602, 13, 194, 16, 18, 60, 307, 17, 23, 35, 20, 21, 681, 19

Offset: 1

Pontus von Brömssen, Jan 14 2024

A284456(a(n)) is the smallest number whose prime tower factorization tree has Matula-Göbel number n.

After a(31), the sequence continues 25, 71, 1726, 31, 22, 1304, 221, 44, 24, ?, 26, ?, 79, 27, 343, ?, 29, 47, 33, 1867, 50, ?, 32, 98, 34, 250, 739, ?, 30, ?, ?, 37, 42, 66, 91, ?, 1935, 381, 41, ... .

Index entries for sequences related to Matula-Göbel numbers.

Cf. A007097, A014221, A025488, A098719, A284456, A369015, A369099, A369136, A369138.

A369015(A284456(a(n))) = A369136(a(n)) = n.
A369099(n) = A284456(a(n)).
A369138(a(A007097(n))) = A025488(A014221(n-1)).
A369138(a(2^n)) = A098719(n+1).

A369138 Index of A284456(n) in A025487.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 12, 13, 14, 19, 20, 21, 22, 24, 29, 33, 34, 38, 39, 43, 44, 47, 54, 57, 61, 63, 66, 67, 68, 72, 73, 74, 88, 93, 94, 101, 102, 107, 108, 114, 118, 121, 128, 129, 131, 138, 140, 142, 145, 147, 149, 161, 172, 185, 186, 187, 188, 192, 198

Offset: 1

Pontus von Brömssen, Jan 14 2024

nonn

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A007097, A014221, A025487, A025488, A085089, A098719, A284456, A369137.

a(n) = A085089(A284456(n)).
A025487(a(n)) = A284456(n).
a(A369137(A007097(n))) = A025488(A014221(n-1)).
a(A369137(2^n)) = A098719(n+1).

A341351 a(n) = A048673(A181815(n)).

Original entry on oeis.org

1, 2, 5, 3, 14, 8, 41, 23, 4, 122, 13, 68, 11, 365, 38, 203, 32, 1094, 113, 18, 608, 6, 63, 95, 3281, 338, 53, 1823, 17, 188, 284, 9842, 1013, 158, 5468, 50, 563, 25, 851, 29525, 88, 3038, 28, 313, 473, 16403, 149, 1688, 74, 2552, 88574, 263, 9113, 7, 83, 938, 1418, 49208, 446, 5063, 221, 7655, 265721, 788, 27338, 20

Offset: 1

Antti Karttunen and Michael De Vlieger, Feb 10 2021

nonn

Maxima are in A007051 and appear at n in A025488, which are the indices of 2^k in A025487. 2^k is idempotent via A181815 but transformed by A003961 to 3^n, which are rendered by A048673 to (3^n + 1)/2.

Local minima are in A111333 and appear at n in A098719, which are the indices of P(k) = A002110(k) in A025487. P(k) is transformed by A181815 to p_k = A000040(k), which become p_(k+1) under A003961. Therefore these become (p_(k+1)+1)/2 via A048673.

Michael De Vlieger, Annotated logarithmic plot of a(n) for 1 <= n <= A098719(8) accentuating records in red and local minima in blue.
Michael De Vlieger, log-log plot of a(n) for 1 <= n <= A098719(16).
Michael De Vlieger, Notes on this sequence.
Index entries for sequences that are permutations of the natural numbers

Cf. A002110, A025487, A025488, A048673, A098719, A108951, A111333, A181812, A181815.
Cf. A341352 (inverse).
Cf. A007051 (record values).

a025487[n_] := {{1}}~Join~Block[{lim = Product[Prime@ i, {i, n}], ww = NestList[Append[#, 1] &, {1}, n - 1]}, 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++]] &@ Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, #]] &@ 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]]; Map[(1 + If[# == 1, 1, Apply[Times, NextPrime[#1]^#2 & @@@ FactorInteger[#]]])/2 &@ Apply[Times, Prime@ Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &@ If[# == 1, {0}, Function[f, ReplacePart[ConstantArray[0, PrimePi@ f[[-1, 1]] ], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #]] &, Union@ Flatten@ a025487@ 5] (* Michael De Vlieger, Feb 11 2021 *)

PARI
```
A341351(n) = A048673(A181815(n));
```

a(n) = A048673(A181815(n)).

For all n >= 1, A181812(a(n)) = A025487(n).

A361809 Fixed points of A181820 and A361808.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 15, 46, 58, 817, 5494, 8502

Offset: 1

Pontus von Brömssen, Mar 25 2023

Numbers k such that the partition with Heinz number k is identical to the partition given by the prime signature of A025487(k).

There are no more terms below 10177058 = A025488(143).

			4 is a term because the partition with Heinz number 4 = 2^2 = prime(1)^2 is (1,1), which is identical to the partition given by the prime signature of A025487(4) = 6 = 2^1*3^1.
15 is a term because the partition with Heinz number 15 = 3*5 = prime(2)*prime(3) is (2,3), which is identical to the partition given by the prime signature of A025487(15) = 72 = 2^3*3^2.
8502 is a term because the partition with Heinz number 8502 = 2*3*13*109 = prime(1)*prime(2)*prime(6)*prime(29) is (1,2,6,29), which is identical to the partition given by the prime signature of A025487(8502) = 68491306598400 = 2^29*3^6*5^2*7.

Cf. A025487, A025488, A181820, A361808.

A181820(a(n)) = A361808(a(n)) = a(n).

cp's OEIS Frontend

A056099 First differences of A025488 (binary order for least prime signatures).

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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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PARI

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A366884 Number of branching factorizations of the least integer of each prime signature (A025487).

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A037088 Triangle read by rows: T(n,k) is number of numbers x, 2^n <= x < 2^(n+1), with k prime factors (counted with multiplicity).

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Mathematica

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A369137 Inverse permutation to A369136.

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A369138 Index of A284456(n) in A025487.

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A341351 a(n) = A048673(A181815(n)).

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