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%I #168 Nov 30 2023 23:34:02
%S 1,2,4,6,8,12,16,24,30,32,36,48,60,64,72,96,120,128,144,180,192,210,
%T 216,240,256,288,360,384,420,432,480,512,576,720,768,840,864,900,960,
%U 1024,1080,1152,1260,1296,1440,1536,1680,1728,1800,1920,2048,2160,2304,2310
%N Least integer of each prime signature A124832; also products of primorial numbers A002110.
%C All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n, sorted.
%C A111059 is a subsequence. - _Reinhard Zumkeller_, Jul 05 2010
%C Choie et al. (2007) call these "Hardy-Ramanujan integers". - _Jean-François Alcover_, Aug 14 2014
%C The exponents k1, k2, ... can be read off Abramowitz & Stegun p. 831, column labeled "pi".
%C 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
%C The prime signature corresponding to a(n) is given in row n of A124832. - _M. F. Hasler_, Jul 17 2019
%H Franklin T. Adams-Watters, <a href="/A025487/b025487.txt">Table of n, a(n) for n = 1..10001</a> (first 291 terms from Will Nicholes)
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
%H Kevin Broughan, <a href="https://doi.org/10.1017/9781108178228">Equivalents of the Riemann Hypothesis, Vol. 1: Arithmetic Equivalents</a>, Cambridge University Press, 2017. See section 8.2, "Hardy-Ramanujan Numbers".
%H YoungJu Choie, Nicolas Lichiardopol, Pieter Moree and Patrick Solé, <a href="https://doi.org/10.5802/jtnb.591">On Robin's criterion for the Riemann hypothesis</a>, Journal de théorie des nombres de Bordeaux, Vol. 19, No. 2 (2007), pp. 357-372. See section 5, p. 367.
%H Asaf Cohen Antonir and Asaf Shapira, <a href="https://arxiv.org/abs/2207.09410">An Elementary Proof of a Theorem of Hardy and Ramanujan</a> (2022). arXiv:2207.09410 [math.NT]
%H Michael De Vlieger, <a href="/A025487/a025487.png">Relations of A025487 to A002110, A002182, and A002201</a>.
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020, pp. 9-10.
%H G. H. Hardy and S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper34/page1.htm">Asymptotic formulae for the distribution of integers of various types</a>, 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.
%H Jeffery Kline, <a href="https://doi.org/10.1016/j.laa.2019.09.022">On the eigenstructure of sparse matrices related to the prime number theorem</a>, Linear Algebra and its Applications (2020) Vol. 584, 409-430.
%H L. B. Richmond, <a href="https://doi.org/10.1016/0022-314X(76)90085-8">Asymptotic results for partitions (I) and the distribution of certain integers</a>, Journal of Number Theory, Vol. 8, No. 4 (1976), pp. 372-389. See page 388.
%F What can be said about the asymptotic behavior of this sequence? - _Franklin T. Adams-Watters_, Jan 06 2010
%F 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
%F From _Antti Karttunen_, Jan 18 & Dec 24 2019: (Start)
%F A085089(a(n)) = n.
%F A101296(a(n)) = n [which is the first occurrence of n in A101296, and thus also a record.]
%F A001221(a(n)) = A061395(a(n)) = A061394(n).
%F A007814(a(n)) = A051903(a(n)) = A051282(n).
%F a(A101296(n)) = A046523(n).
%F a(A306802(n)) = A002182(n).
%F a(n) = A108951(A181815(n)) = A329900(A181817(n)).
%F If A181815(n) is odd, a(n) = A283980(a(A329904(n))), otherwise a(n) = 2*a(A329904(n)).
%F (End)
%F Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A002110(n)) = A161360. - _Amiram Eldar_, Oct 20 2020
%e The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...
%p isA025487 := proc(n)
%p local pset,omega ;
%p pset := sort(convert(numtheory[factorset](n),list)) ;
%p omega := nops(pset) ;
%p if op(-1,pset) <> ithprime(omega) then
%p return false;
%p end if;
%p for i from 1 to omega-1 do
%p if padic[ordp](n,ithprime(i)) < padic[ordp](n,ithprime(i+1)) then
%p return false;
%p end if;
%p end do:
%p true ;
%p end proc:
%p A025487 := proc(n)
%p option remember ;
%p local a;
%p if n = 1 then
%p 1 ;
%p else
%p for a from procname(n-1)+1 do
%p if isA025487(a) then
%p return a;
%p end if;
%p end do:
%p end if;
%p end proc:
%p seq(A025487(n),n=1..100) ; # _R. J. Mathar_, May 25 2017
%t 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 *)
%t (* Second program: generate all terms m <= A002110(n): *)
%t f[n_] := {{1}}~Join~
%t Block[{lim = Product[Prime@ i, {i, n}],
%t ww = NestList[Append[#, 1] &, {1}, n - 1], dec},
%t dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]];
%t Map[Block[{w = #, k = 1},
%t Sort@ Prepend[If[Length@ # == 0, #, #[[1]]],
%t Product[Prime@ i, {i, Length@ w}] ] &@ Reap[
%t Do[
%t If[# < lim,
%t Sow[#]; k = 1,
%t If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w,
%t If[k == 1,
%t MapAt[# + 1 &, w, k],
%t PadLeft[#, Length@ w, First@ #] &@
%t Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]],
%t {i, Infinity}] ][[-1]]
%t ] &, ww]]; Sort[Join @@ f@ 13] (* _Michael De Vlieger_, May 19 2018 *)
%o (PARI) 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
%o (PARI) factfollow(n)={local(fm, np, n2);
%o fm=factor(n); np=matsize(fm)[1];
%o if(np==0,return([2]));
%o n2=n*nextprime(fm[np,1]+1);
%o if(np==1||fm[np,2]<fm[np-1,2], [n*fm[np,1], n2], [n2])}
%o al(n) = {local(r, ms); r=vector(n);
%o ms=[1];
%o for(k=1, n,
%o r[k]=ms[1];
%o ms=vecsort(concat(vector(#ms-1, j, ms[j+1]), factfollow(ms[1]))));
%o r} /* _Franklin T. Adams-Watters_, Dec 01 2011 */
%o (PARI) 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
%o (PARI) 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
%o (PARI)
%o \\ For fast generation of large number of terms, use this program:
%o 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
%o 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.
%o v025487 = A025487list(101);
%o A025487(n) = v025487[n];
%o for(n=1,#v025487,print1(A025487(n), ", ")); \\ _Antti Karttunen_, Dec 24 2019
%o (Haskell)
%o import Data.Set (singleton, fromList, deleteFindMin, union)
%o a025487 n = a025487_list !! (n-1)
%o a025487_list = 1 : h [b] (singleton b) bs where
%o (_ : b : bs) = a002110_list
%o h cs s xs'@(x:xs)
%o | m <= x = m : h (m:cs) (s' `union` fromList (map (* m) cs)) xs'
%o | otherwise = x : h (x:cs) (s `union` fromList (map (* x) (x:cs))) xs
%o where (m, s') = deleteFindMin s
%o -- _Reinhard Zumkeller_, Apr 06 2013
%o (Sage)
%o def sharp_primorial(n): return sloane.A002110(prime_pi(n))
%o N = 2310
%o nmax = 2^floor(log(N,2))
%o 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])
%o # _Giuseppe Coppoletta_, Jan 26 2015
%Y Subsequence of A055932, A191743, and A324583.
%Y Cf. A025488, A051282, A036041, A051466, A061394, A124832, A161360, A166469, A181815, A181817, A283980, A306802, A322584, A322585 (characteristic function), A329897, A329898, A329899, A329900, A329904, A330683.
%Y Cf. A085089, A101296 (left inverses).
%Y Equals range of values taken by A046523.
%Y Cf. A178799 (first differences), A247451 (squarefree kernel), A146288 (number of divisors).
%Y 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.
%Y Rearrangements of this sequence include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822, A322827, A329886, A329887.
%Y Cf. also array A124832 (row n = prime signature of a(n)) and A304886, A307056.
%K nonn,easy,nice,core
%O 1,2
%A _David W. Wilson_
%E Offset corrected by _Matthew Vandermast_, Oct 19 2008
%E Minor correction by _Charles R Greathouse IV_, Sep 03 2010
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