A324583 -id:A324583 - 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

A324198 a(n) = gcd(n, A276086(n)), where A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 15, 1, 1, 1, 1, 5, 3, 1, 1, 1, 25, 1, 3, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 3, 5, 1, 7, 1, 1, 15, 1, 1, 1, 7, 25, 3, 1, 1, 1, 5, 7, 3, 1, 1, 1, 1, 1, 21, 1, 1, 1, 1, 1, 3, 35, 1, 1, 1, 1, 75, 1, 7, 1, 1, 5, 3, 1, 1, 7, 5, 1, 3, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 49, 3, 5, 1, 1, 1, 1, 105

Offset: 0

Antti Karttunen, Feb 25 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003989, A276086, A323878, A324284, A324350, A324351, A324384, A324577, A324580, A324646, A327858, A328323, A327168, A328316, A328323, A328386, A328584, A346242 (Dirichlet inverse), A351254.
Cf. A324583 (positions of ones), A324584 (and terms larger than one).
Cf. A371098 (odd bisection), A371099 [= a(36n+9)].
Cf. also A328231.

Array[Block[{i, m, n = #, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; GCD[#, m]] &, 106, 0] (* Michael De Vlieger, Feb 04 2022 *)

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324198(n) = gcd(n,A276086(n));

A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p,valuation(orgn,p))); n = n\p; p = nextprime(1+p)); (m); }; \\ Antti Karttunen, Oct 21 2019

a(n) = gcd(n, A276086(n)).
From Antti Karttunen, Oct 21 2019: (Start)
A000005(a(n)) = A327168(n).
a(A328316(n)) = A328323(n).
a(n) = A324580(n) / A328584(n).
(End)

A356302 The least k >= 0 such that n and A276086(n+k) are relatively prime, where A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 20, 0, 0, 0, 0, 15, 0, 0, 0, 0, 10, 3, 0, 0, 0, 5, 0, 3, 0, 0, 0, 0, 0, 3, 0, 175, 0, 0, 0, 3, 20, 0, 168, 0, 0, 15, 0, 0, 0, 161, 10, 3, 0, 0, 0, 5, 154, 3, 0, 0, 0, 0, 0, 147, 0, 0, 0, 0, 0, 3, 140, 0, 0, 0, 0, 15, 0, 2233, 0, 0, 10, 3, 0, 0, 126, 5, 0, 3, 0, 0, 0, 119, 0, 3, 0, 0, 0, 0, 112

Offset: 0

Antti Karttunen, Nov 03 2022

For all nonzero terms, adding a(n) to n in primorial base generates at least one carry. See the formula involving A329041.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A276086, A324198, A356309, A329041.
Cf. A324583 (positions of zeros), A324584 (of nonzeros), A356318 (positions where a(n) > 0 and a multiple of n), A356319 (where 0 < a(n) < n).
Cf. A358213, A358214 (conjectured positions of records and their values).
Cf. also A356303, A356304.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A356302(n) = { my(k=0); while(gcd(A276086(n+k),n)!=1,k++); (k); };

a(n) = A356309(n) - n.

If a(n) > 0, then A000035(a(n)) = A000035(n) and A329041(n, a(n)) > 1.

A356309 The least j >= n such that n and A276086(j) are relatively prime, where A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 2, 6, 4, 5, 6, 7, 8, 12, 30, 11, 12, 13, 14, 30, 16, 17, 18, 19, 30, 24, 22, 23, 24, 30, 26, 30, 28, 29, 30, 31, 32, 36, 34, 210, 36, 37, 38, 42, 60, 41, 210, 43, 44, 60, 46, 47, 48, 210, 60, 54, 52, 53, 54, 60, 210, 60, 58, 59, 60, 61, 62, 210, 64, 65, 66, 67, 68, 72, 210, 71, 72, 73, 74, 90, 76, 2310, 78

Offset: 0

Antti Karttunen, Nov 04 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 0..11550
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to primorial base

Cf. A276086, A356302, A356313.

Cf. A324583 (positions of the fixed points), A356314 (positions of the terms that are primorial numbers), A356316 (where a(n) is a multiple of n), A356318 (where a nontrivial multiple), A356319 (where n < a(n) < 2*n).

f[nn_] := Block[{m = 1, i = 1, n = nn, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m]; Array[Block[{k = #}, While[! CoprimeQ[#, f[k]], k++]; k] &, 79, 0] (* Michael De Vlieger, Nov 06 2022, after Jean-François Alcover at A276086 *)

A356309(n) = (n+A356302(n)); \\ See code in the latter sequence.

a(n) = n + A356302(n).

A324584 Numbers n that share a prime factor with A276086(n).

Original entry on oeis.org

3, 9, 10, 15, 20, 21, 25, 27, 33, 35, 39, 40, 42, 45, 49, 50, 51, 55, 56, 57, 63, 69, 70, 75, 77, 80, 81, 84, 85, 87, 91, 93, 98, 99, 100, 105, 110, 111, 112, 115, 117, 119, 123, 126, 129, 130, 133, 135, 140, 141, 145, 147, 153, 154, 159, 160, 161, 165, 168, 170, 171, 175, 177, 182, 183, 189, 190, 195, 196, 200, 201, 203, 205

Offset: 1

Antti Karttunen, Mar 10 2019

nonn

Numbers n for which A324198(n) <> 1.

Cf. A276086, A324198.

Cf. A324583 (complement).

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324198(n) = gcd(n,A276086(n));
for(n=1, oo, if(1!=A324198(n), print1(n, ", ")));

A356311 Numbers k for which A003415(k) and A276086(k) are relatively prime, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 2, 3, 4, 5, 7, 10, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 28, 29, 30, 31, 32, 34, 37, 40, 41, 42, 43, 47, 53, 54, 56, 58, 59, 60, 61, 66, 67, 70, 71, 72, 73, 78, 79, 80, 82, 83, 84, 89, 90, 96, 97, 101, 103, 104, 105, 107, 108, 109, 113, 114, 118, 120, 124, 127, 130, 131, 132, 136, 137, 138, 139, 140, 142, 144

Offset: 1

Antti Karttunen, Nov 03 2022

nonn

Index entries for sequences related to primorial base

Positions of 1's in A327858. Positions of 0's in A356304 (for n >= 2) and in A356305.
Cf. A003415, A276086, A356310 (characteristic function), A356312 (complement).
Cf. also A324583.

isA356311(n) = A356310(n); \\ See code in A356310.

A351254 a(n) = A276085(gcd(n, A276086(n))).

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 8, 0, 0, 0, 0, 6, 2, 0, 0, 0, 12, 0, 2, 0, 0, 0, 0, 0, 2, 0, 30, 0, 0, 0, 2, 6, 0, 30, 0, 0, 8, 0, 0, 0, 30, 12, 2, 0, 0, 0, 6, 30, 2, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 2, 36, 0, 0, 0, 0, 14, 0, 30, 0, 0, 6, 2, 0, 0, 30, 6, 0, 2, 0, 0, 0, 30, 0, 2, 0, 0, 0, 0, 60, 2, 6

Offset: 0

Antti Karttunen, Feb 05 2022

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A276085, A276086, A324198, A351253.
Cf. also A351234.
Cf. A324583 (positions of the rest of zeros after a(0)=0).

A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
A351254(n) = A276085(A324198(n));

a(n) = A276085(A324198(n)) = A276085(gcd(n, A276086(n))).

a(n) = n - A351253(n).

A355821 Numbers k for which A003961(k) and A276086(k) are relatively prime, where A003961 is fully multiplicative with a(p) = nextprime(p), and A276086 is primorial base exp-function.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 33, 37, 41, 43, 47, 49, 53, 59, 61, 63, 67, 71, 73, 77, 79, 83, 89, 91, 93, 97, 101, 103, 107, 109, 113, 119, 121, 123, 127, 131, 133, 137, 139, 143, 149, 151, 153, 157, 161, 163, 167, 169, 173, 179, 181, 183, 187, 191, 193, 197, 199, 203, 209, 211, 213, 215, 221, 223, 227

Offset: 1

Antti Karttunen, Jul 18 2022

nonn

Positions of 1's in A355442 and in A355001.
Cf. A003961, A276086, A355820 (characteristic function), A355822 (complement).
Cf. also A324583.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355820(n) = (1==gcd(A003961(n), A276086(n)));
isA355821(n) = A355820(n);

from math import prod, gcd
from itertools import count, islice
from sympy import factorint, nextprime
def A355821_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        k = prod(nextprime(p)**e for p, e in factorint(n).items())
        m, p, c = 1, 2, n
        while c:
            c, a = divmod(c,p)
            m *= p**a
            p = nextprime(p)
        if gcd(k,m) == 1:
            yield n
A355821_list = list(islice(A355821_gen(),30)) # Chai Wah Wu, Jul 18 2022

cp's OEIS Frontend

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

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A324198 a(n) = gcd(n, A276086(n)), where A276086 is the primorial base exp-function.

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A356302 The least k >= 0 such that n and A276086(n+k) are relatively prime, where A276086 is the primorial base exp-function.

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A356309 The least j >= n such that n and A276086(j) are relatively prime, where A276086 is the primorial base exp-function.

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A324584 Numbers n that share a prime factor with A276086(n).

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A356311 Numbers k for which A003415(k) and A276086(k) are relatively prime, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A351254 a(n) = A276085(gcd(n, A276086(n))).

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A355821 Numbers k for which A003961(k) and A276086(k) are relatively prime, where A003961 is fully multiplicative with a(p) = nextprime(p), and A276086 is primorial base exp-function.

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