A195330 -id:A195330 - cp's OEIS frontend

A195331 A080670 applied to A195330(n).

25, 26, 34, 112, 53, 27, 132, 35, 28, 172, 265, 73, 192, 353, 273, 345, 267, 235, 29, 232, 347, 54, 275, 273, 36, 283, 292, 537, 277, 312, 210, 373, 355, 254, 285, 113, 372, 236, 293, 412, 357, 573, 287, 432, 354, 211, 37, 133, 472, 74, 2355, 2254, 295, 2534, 2583, 2113, 2372, 2373, 2643, 532, 2811, 2589, 2236, 2723, 2647, 2103, 2597, 55, 2672, 2752, 2813, 2412, 3353, 2653, 2357, 2573, 2733, 592, 3443, 297, 365, 2432, 2729, 612, 2354, 2659, 3447, 2835, 2435, 2661, 2731

Offset: 1

N. J. A. Sloane, Sep 15 2011

A195330 gives the numbers k such that A080670(k) < k; this sequence gives the corresponding values of A080670(k).

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 220 terms from Robert Price)

Cf. A080670, A195330.

b:= 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:
g:= proc(n) option remember; local k; for k from
      `if`(n=1, 0, g(n-1))+1 while b(k)>=k do od; k
    end:
a:= n-> b(g(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 17 2020

A080670 = Cases[Import["https://oeis.org/A080670/b080670.txt", "Table"], {, }][[All, 2]];
A195330 = Cases[Import["https://oeis.org/A195330/b195330.txt", "Table"], {, }][[All, 2]];
Table[A080670[[n]], {n, A195330}](* Robert Price, Mar 17 2020 *)

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

A376254 Numbers k such that A376294(k) < k.

Original entry on oeis.org

32, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 864, 961, 972, 1024, 1152, 1250, 1296, 1331, 1369, 1458, 1536, 1600, 1681, 1728, 1849, 1875, 1944, 2000, 2025, 2048, 2187, 2197, 2209, 2304, 2401, 2500, 2560, 2592, 2662, 2744, 2809, 2916, 3087, 3125, 3136

Offset: 1

Haines Hoag, Sep 17 2024

There are infinitely many numbers in this sequence, since the growth of powers of small primes far outpaces the growth of their digits when concatenated.

First differs from A195330 at 320 which is a term there but not here.

			32 is a term since 32=2^5 and 25<32.
1152 is a term since 1152=2^7*3^2 and 27*32=864, and 864<1152.

Haines Hoag, Table of n, a(n) for n = 1..20000

Cf. A376294, A195330.

f[p_, e_] := 10^IntegerLength[e]*p + e; q[1] = False; q[k_] := Times @@ f @@@ FactorInteger[k] < k; Select[Range[3200], q] (* Amiram Eldar, Sep 26 2024 *)

from math import prod
from sympy import factorint
def ok(n): return prod(int(str(p)+str(e)) for p, e in factorint(n).items()) < n
print([k for k in range(1, 3200) if ok(k)]) # Michael S. Branicky, Sep 27 2024

A287763 Numbers n for which A230625(n) < n, where A230625 is the concatenation of the prime factorization written in binary, converted back to decimal.

Original entry on oeis.org

25, 27, 32, 49, 54, 64, 81, 96, 98, 121, 125, 128, 135, 147, 162, 169, 189, 192, 216, 224, 242, 243, 245, 250, 256, 289, 297, 320, 338, 343, 351, 352, 361, 363, 375, 384, 392, 405, 416, 432, 448, 486, 500, 507, 512, 513, 529, 539, 567, 576, 578, 605, 621, 625, 637, 640, 648, 675, 686, 704, 722, 729, 736, 750, 768, 775, 783, 784

Offset: 1

M. F. Hasler, Jun 21 2017

Relevant for the study of closed orbits. (This is to A230625 the analog of A195330 for A080670.) Up to a certain limit, the trajectory of all numbers, under iteration of A230625, end either in a prime (fixed point) or in one of the orbits {1007, 1269} or {1503,3751}.

See A288985 for the analog when A287874 is used instead of A230625, i.e., without converting back the concatenation of the binary strings to decimal, or rather, reading it as a decimal number.

Cf. A230625, A195330, A080670, A287874.

Select[Range@ 800, Function[n, FromDigits[#, 2] < n &@ Flatten@ Map[IntegerDigits[#, 2] &, FactorInteger[n] /. {p_, 1} :> {p}]]] (* Michael De Vlieger, Jun 23 2017 *)

PARI
```
forcomposite(n=1,1e6,A230625(n)
				
```

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A195331 A080670 applied to A195330(n).

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A080670 Literal reading of the prime factorization of n.

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A376254 Numbers k such that A376294(k) < k.

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A287763 Numbers n for which A230625(n) < n, where A230625 is the concatenation of the prime factorization written in binary, converted back to decimal.

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Mathematica

PARI