User:JrandWP/sandbox/Numbers/doc

This page is the list of numbers with prime factors and levels.

Note before reading this page: Numbers is the number of prime number prevention in the prime factors, and the level is the most base of the number in the prime factors.^[1] Seeing the introduction of notes: Click on the notes on the note column, you will see the defintion of numbers describing on the row. In the references section, click on the name references to see the (number)-smallest number has this defintion.^[2] For the example, click on the citation 7-24 (prime number citation), you will show the 24th prime number (note that the smallest number has 0 starting) and the 24th prime number is 97 (note that the 0 prime number is 2). You will also see the OEIS index of the list of number after clicking the ref.^[3] For a complete list of starting 0 references we use in the page, see A025487 (OEIS) and the prime signature level A124832.

There are two references for each numbers, the first are numbers on the manot (the type of prime signature), and a second references determines that what number in the section. Where the manot is, the templates that.^[4]

Note when using references section[edit]

Using the list of references to determine the lists of numbers, but prime number (citation 7) is a sequence A000040 in the OEIS is seen on this page. When seeing the references, a list-defined references will listing above. Note that:

The list only see the numbers, not see the prime factors.^[5] If you want to see it, you should use this page.
1 is not a prime number, not a composite, but it appears in OEIS.^[6]^[7]
In A025487, all the starting is an even number, so odd numbers be appear different.
On prime number dictionary (citation 7), all the numbers are odd, except 2.^[8]
We can click on the integers to see the varity of numbers.
There are many integers, so we can't make enough numbers to use on this page. So it is always incomplete.
2 is a smallest prime number, and it's the only even prime numbers.^[8]^[9]
4 is a smallest composite number.^[9]
6 is a smallest composite which is not squares.^[9]^[10]
9 is a smallest odd composite number.^[9]
15 is a smallest odd composite which is not squares.^[9]^[10]
2(n) is an smallest and only even number in their signatures.^[11]

Now, go around!

Also, we can use A046523 (OEIS) to choose the number rightly outside the name, but we can find it on this page easily though the references.^[12] We are also create references for easier going for the posibities for level index of numbers, for easier for seeing the OEIS index (we will update later, and something we can't have!...

Scroll down to references section

Multi-page renders[edit]

This page is used in many sequences in OEIS, shows content and group in that.

More further docs[edit]

Documentaion of A025487, which used for this page
Sections	Defintion
OFFSET	1,2
COMMENTS	All numbers of the form $2^{x}3^{y}...*p^{z}$ , where $k1>=k2>=...>=k_{n}$ , sorted. A111059 is a subsequence. - Reinhard Zumkeller, Jul 05 2010. 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`
LINKS	`Will Nicholes and Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10001 (Will Nicholes supplied the first 291 terms.)` `M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.` `Michael De Vlieger, Relations of A025487 to A002110, A002182, and A002201.` `G. H. Hardy and S. Ramanujan, Asymptotic formulas concerning the distribution of integers of various types, Proc. London Math. Soc, Ser. 2, Vol. 16 (1917), pp. 112-132.` `Jeffery Kline, On the eigenstructure of sparse matrices related to the prime number theorem, Linear Algebra and its Applications (2020) Vol. 584, 409-430.`
FORMULA	`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)`
EXAMPLE	`The first few terms are 1, 2, 2^2, 23, 2^3, 2^23, 2^4, 2^33, 23*5, ...`
MAPLE	`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`
MATHEMATICA	`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 *)`
PROG	`(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` `(PARI) factfollow(n)={local(fm, np, n2);` `fm=factor(n); np=matsize(fm)[1];` `if(np==0, return([2]));` `n2=nnextprime(fm[np, 1]+1);` `if(np==1\|\|fm[np, 2]<fm[np-1, 2], [nfm[np, 1], n2], [n2])}` `al(n) = {local(r, ms); r=vector(n);` `ms=[1];` `for(k=1, n,` `r[k]=ms[1];` `ms=vecsort(concat(vector(#ms-1, j, ms[j+1]), factfollow(ms[1]))));` `r} /* Franklin T. Adams-Watters, Dec 01 2011 /` `(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` `(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` `(PARI)` `\\ 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(2lista[i] <= u, listput(lista, 2lista[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` `(Haskell)` `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` `(Sage) def sharp_primorial(n): return sloane.A002110(prime_pi(n))` `def p(n, k): return sharp_primorial(factor(n)[k][0])^factor(n)[k][1];` `N=2310; nmax=2^floor(log(N, 2)); sorted([k for k in [prod(p(n, k) for k in range (0, len(factor(n)))) for n in (1..nmax)] if k<=N]) # Giuseppe Coppoletta, Jan 26 2015`
CROSSREFS	`Cf. A025488, A051282, A036041, A051466, A061394, A124832, 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).` `Subsequence of A055932, of A191743 and of A324583.` 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.` `Sequence in context: A323508 A324850 A095810 * A279537 A325238 A070175` `Adjacent sequences: A025484 A025485 A025486 * A025488 A025489 A025490`
KEYWORD	`nonn,easy,nice,core`
AUTHOR	`David W. Wilson`
EXTENSIONS	`Offset corrected by Matthew Vandermast, Oct 19 2008` `Minor correction by Charles R Greathouse IV, Sep 03 2010`
STATUS	`approved`

References[edit]

^ When in the docs page it will show a bunch of references.
^ See: Main Page to see whatever cites looks.
^ Note that: All of the list in OEIS are only shows a few number of first element, so should use this page.
^ On that's exampleity.
^ Note that all prime factors are used in an optical way.
^ Not a prime and no level
^ One is a smallest number, so it comes first.
^ ^a ^b That's ok because even numbers are divisible by 2.
^ ^a ^b ^c ^d ^e Information about numbers, p. 3
^ ^a ^b Note that square are unique.
^ Use that right signature!
^ "A046523 - OEIS". oeis.org. Retrieved 2020-03-16.

External links[edit]

"A000040 - OEIS". oeis.org. Retrieved 3 June 2024.{{cite web}}: CS1 maint: url-status (link)
"A025487 - OEIS". oeis.org. Retrieved 3 June 2024.{{cite web}}: CS1 maint: url-status (link)

[1] When in the docs page it will show a bunch of references.

[2] See: Main Page to see whatever cites looks.

[3] Note that: All of the list in OEIS are only shows a few number of first element, so should use this page.

[4] On that's exampleity.

[5] Note that all prime factors are used in an optical way.

[:8-6] Not a prime and no level

[7] One is a smallest number, so it comes first.

[:0-8] That's ok because even numbers are divisible by 2.

[:1-9] Information about numbers, p. 3

[:2-10] Note that square are unique.

[11] Use that right signature!

[12] "A046523 - OEIS". oeis.org. Retrieved 2020-03-16.

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Note when using references section[edit]

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More further docs[edit]

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