A288818 -id:A288818 - cp's OEIS frontend

A288820 Indices of records in A288818.

1, 2, 23, 223, 233, 237, 2237, 2337, 22337, 23137, 23337, 23373, 23797, 223373, 223797, 231373, 233137, 233797, 2233797, 2313797, 2331373, 2333797, 2337379, 2337397, 2353797, 22353797, 22373797, 23137397, 23173797, 23313797, 23353797, 23373797

Offset: 1

Hans Havermann, Jun 17 2017

If one assumes that larger terms begin with 22 or 23, then a(54)-a(64) are: 223373733797, 223537373797, 231337373797, 231353673797, 231353733797, 231373733797, 233137337397, 233537373797, 233753733797, 235313733797, and 235373733797. - Hans Havermann, Jul 31 2017

Giovanni Resta, Table of n, a(n) for n = 1..53

Cf. A288818, A288819 (records' value), A080670.

ric[d_, lp_] := Block[{p, e, i, j, n = Length@d}, If[n == 0, cnt++, If[d[[1]] > 0, Do[p = FromDigits@ Take[d, i]; If[p > lp && PrimeQ@p, ric[Take[d, i - n], p]; Do[e = Take[d, {i + 1, j}]; If[e[[1]] > 0 && e != {1}, ric[Take[d, j - n], p]], {j, i+1, n}]], {i, n}]]]]; a[n_] := (cnt = 0; ric[ IntegerDigits@ n, 1]; cnt); L = {1}; k = 1; bst = 0; While[Length@L < 18, v = a[++k]; If[v > bst, AppendTo[L, k]; bst = v]]; L (* Giovanni Resta, Jun 19 2017 *)

A288819 Record maxima in A288818.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 12, 13, 17, 18, 21, 22, 23, 25, 30, 34, 41, 45, 54, 58, 60, 62, 66, 75, 80, 82, 83, 88, 104, 110, 124, 131, 147, 163, 166, 173, 194, 220, 228, 233, 257, 290, 313, 334, 350, 359, 365, 412, 468, 477, 503, 520

Offset: 1

Hans Havermann, Jun 17 2017

If one assumes that larger records occur at A288818 indices beginning with 22 or 23, then a(54)-a(64) are: 528, 534, 538, 542, 636, 657, 687, 728, 764, 768, and 847. - Hans Havermann, Jul 31 2017

Cf. A288818, A288820 (records' position), A080670.

ric[d_, lp_] := Block[{p, e, i, j, n = Length@d}, If[n == 0, cnt++, If[d[[1]] > 0, Do[p = FromDigits@ Take[d, i]; If[p > lp && PrimeQ@p, ric[Take[d, i - n], p]; Do[e = Take[d, {i + 1, j}]; If[e[[1]] > 0 && e != {1}, ric[Take[d, j - n], p]], {j, i+1, n}]], {i, n}]]]]; a[n_] := (cnt = 0; ric[ IntegerDigits@ n, 1]; cnt); Union@ FoldList[Max, Array[a, 10^6]] (* Michael De Vlieger, Jun 19 2017, after Giovanni Resta at A288818 *)

a(46)-a(53) from Giovanni Resta, Jun 20 2017

A080670 Literal reading of the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 22, 5, 23, 7, 23, 32, 25, 11, 223, 13, 27, 35, 24, 17, 232, 19, 225, 37, 211, 23, 233, 52, 213, 33, 227, 29, 235, 31, 25, 311, 217, 57, 2232, 37, 219, 313, 235, 41, 237, 43, 2211, 325, 223, 47, 243, 72, 252, 317, 2213, 53, 233, 511, 237, 319, 229, 59, 2235

Offset: 1

Jon Perry, Mar 02 2003

Exponents equal to 1 are omitted and therefore this sequence differs from A067599.
Here the first duplicate (ambiguous) term appears already with a(8)=23=a(6), in A067599 this happens only much later. - M. F. Hasler, Oct 18 2014
The number n = 13532385396179 = 13·53^2·3853·96179 = a(n) is (maybe the first?) nontrivial fixed point of this sequence, making it the first known index of a -1 in A195264. - M. F. Hasler, Jun 06 2017

			8=2^3, which reads 23, hence a(8)=23; 12=2^2*3, which reads 223, hence a(12)=223.

T. D. Noe, Table of n, a(n) for n = 1..10000
Tony Padilla and Brady Haran, 13532385396179, Numberphile Video, 2017
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)

Cf. A037276, A067599, A230305, A230625, A027748, A124010, A288818.
See A195330, A195331 for those n for which a(n) is a contraction.
See also home primes, A037271.
See A195264 for what happens when k -> a(k) is repeatedly applied to n.
Partial sums: A287881, A287882.

import Data.Function (on)
a080670 1 = 1
a080670 n = read $ foldl1 (++) $
zipWith (c `on` show) (a027748_row n) (a124010_row n) :: Integer
where c ps es = if es == "1" then ps else ps ++ es
-- Reinhard Zumkeller, Oct 27 2013

ifsSorted := proc(n)
        local fs,L,p ;
        fs := sort(convert(numtheory[factorset](n),list)) ;
        L := [] ;
        for p in fs do
                L := [op(L),[p,padic[ordp](n,p)]] ;
        end do;
        L ;
end proc:
A080670 := proc(n)
        local a,p ;
        if n = 1 then
                return 1;
        end if;
        a := 0 ;
        for p in ifsSorted(n) do
                a := digcat2(a,op(1,p)) ;
                if op(2,p) > 1 then
                        a := digcat2(a,op(2,p)) ;
                end if;
        end do:
        a ;
end proc: # R. J. Mathar, Oct 02 2011
# second Maple program:
a:= proc(n) option remember; `if`(n=1, 1, (l->
      parse(cat(seq(`if`(l[i, 2]=1, l[i, 1], [l[i, 1],
      l[i, 2]][]), i=1..nops(l)))))(sort(ifactors(n)[2])))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 17 2020

f[n_] := FromDigits[ Flatten@ IntegerDigits[ Flatten[ FactorInteger@ n /. {1 -> {}}]]]; f[1] = 1; Array[ f, 60] (* Robert G. Wilson v, Mar 02 2003 and modified Jul 22 2014 *)

A080670(n)=if(n>1, my(f=factor(n),s=""); for(i=1,#f~,s=Str(s,f[i,1],if(f[i,2]>1, f[i,2],""))); eval(s),1) \\ Charles R Greathouse IV, Oct 27 2013; case n=1 added by M. F. Hasler, Oct 18 2014

A080670(n)=if(n>1,eval(concat(apply(f->Str(f[1],if(f[2]>1,f[2],"")),Vec(factor(n)~)))),1) \\ M. F. Hasler, Oct 18 2014

import sympy
[int(''.join([str(y) for x in sorted(sympy.ntheory.factorint(n).items()) for y in x if y != 1])) for n in range(2,100)] # compute a(n) for n > 1
# Chai Wah Wu, Jul 15 2014

Edited and extended by Robert G. Wilson v, Mar 02 2003

A290385 Base-ten pandigital factorization integers. The normal factorization (primes raised to greater-than-one exponents) of these numbers uses each digit exactly once.

Original entry on oeis.org

15618090, 20824120, 22022490, 22816290, 22908090, 23294190, 23427135, 23507490, 24843120, 26104560, 26152080, 26679990, 27114690, 27687090, 28275690, 29218704, 29363320, 29447898, 29544690, 29582490, 29670378, 29688144, 29910138, 30120144

Offset: 1

Hans Havermann, Jul 28 2017

The sequence contains 14856143 terms, the largest being 7^986543210.

The corresponding zeroless sequence contains 2295201 terms, from 2992890 = 2*3*5*67*1489 to 7^98654321. - Giovanni Resta, Jul 29 2017

			20824120 is in the sequence because 2^3*5*487*1069 is pandigital.

Hans Havermann, Table of n, a(n) for n = 1..10000
Hans Havermann, Pandigital factorization integer cascades

Cf. A058909, A288818.

pop[d_, mn_] := Union @@ Table[ Select[ FromDigits /@ Flatten[ Permutations /@ Subsets[d, {k}], 1], # > mn && PrimeQ[#] &], {k, IntegerLength@ mn, Length[d]}]; ric[w_, d_, p_] := If[d == {}, cnt++; If[Max[Last /@ w] < 30 && Times @@ (Power @@@ w) <= 4*10^7, AppendTo[L, w]], Block[{pp = pop[d, p], v}, Do[v = Complement[d, IntegerDigits@ x]; ric[Append[w, {x, 1}], v, x]; Do[If[e > 1, ric[Append[w, {x, e}], Complement[v, IntegerDigits@e], x]], {e, Union[ FromDigits /@ Flatten[ Permutations /@ Subsets[v, {1, Length@v}], 1]]}], {x, pp}]]]; Monitor[cnt = 0; L = {}; ric[{}, Range[0, 9], 1];, cnt]; Print["cnt = ", cnt]; Sort[(Times @@ (Power @@@ #)) & /@ L] (* Giovanni Resta, Jul 29 2017 *)

cp's OEIS Frontend

A288820 Indices of records in A288818.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A288819 Record maxima in A288818.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A080670 Literal reading of the prime factorization of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Extensions

A290385 Base-ten pandigital factorization integers. The normal factorization (primes raised to greater-than-one exponents) of these numbers uses each digit exactly once.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica