A037279 -id:A037279 - cp's OEIS frontend

A037280 If n is composite replace n with the concatenation of its nontrivial divisors [ A037279 ] then divide out any factors of 2.

1, 1, 3, 1, 5, 23, 7, 3, 3, 25, 11, 1173, 13, 27, 35, 31, 17, 2369, 19, 12255, 37, 211, 23, 586703, 5, 213, 39, 12357, 29, 23561015, 31, 1551, 311, 217, 57, 117345609, 37, 219, 313, 6145255, 41, 23671421, 43, 120561, 35915, 223, 47, 2933515203, 7, 251025, 317

Offset: 1

N. J. A. Sloane

			Divisors of 12 are 1,2,3,4,6,12, so 12 -> 2346 = 2*1173 -> 1173 = a(12).

Cf. A037279.

with(numtheory):ds:=proc(s) local j: RETURN(add(s[j]*10^(j-1),j=1..nops(s))):end: a:=proc(n) options remember: local d,i,l,m: if n<3 then RETURN(1) else if not isprime(n) then d:=divisors(n): l:=nops(d): m:=ds([seq(op(convert(d[l-i+1],base,10)),i=2..l-1)]): RETURN(m/piecewise(m mod 2=1,1,2^(ifactors(m)[2][1][2]))) else RETURN(n) fi fi: end; seq(a(n),n=1..70); # C. Ronaldo

More terms from Erich Friedman

A106708 a(n) is the concatenation of its nontrivial divisors.

Original entry on oeis.org

0, 0, 0, 2, 0, 23, 0, 24, 3, 25, 0, 2346, 0, 27, 35, 248, 0, 2369, 0, 24510, 37, 211, 0, 2346812, 5, 213, 39, 24714, 0, 23561015, 0, 24816, 311, 217, 57, 234691218, 0, 219, 313, 24581020, 0, 23671421, 0, 241122, 35915, 223, 0, 23468121624, 7, 251025, 317

Offset: 1

N. J. A. Sloane, Jul 20 2007

Klaus Brockhaus, Table of n, a(n) for n=1..5000

Cf. A037278, A120712, A037279, A131983 (records), A131984 (where records occur).

Cf. A027750, A010051, A037285, A037277, A163870.

a106708 1           = 0
a106708 n
   | a010051 n == 1 = 0
   | otherwise = read $ concat $ (map show) $ init $ tail $ a027750_row n
-- Reinhard Zumkeller, May 01 2012

A106708 := proc(n) local dvs ; if isprime(n) or n = 1 then 0; else dvs := [op(numtheory[divisors](n) minus {1,n} )] ; dvs := sort(dvs) ; cat(op(dvs)) ; fi ; end: seq(A106708(n),n=1..80) ; # R. J. Mathar, Aug 01 2007

Table[If[CompositeQ[n],FromDigits[Flatten[IntegerDigits/@Rest[ Most[ Divisors[ n]]]]],0],{n,60}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 22 2020 *)

{map(n) = local(d); d=divisors(n); if(#d<3, 0, d[1]=""; eval(concat(vecextract(d, concat("..", #d-1)))))}
for(n=1,51,print1(map(n),",")) /* Klaus Brockhaus, Aug 05 2007 */

from sympy import divisors
def a(n):
  nontrivial_divisors = [d for d in divisors(n)[1:-1]]
  if len(nontrivial_divisors) == 0: return 0
  else: return int("".join(str(d) for d in nontrivial_divisors))
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Dec 31 2020

a(n) = A037279(n) * A010051(n). - R. J. Mathar, Aug 01 2007

More terms from R. J. Mathar and Klaus Brockhaus, Aug 01 2007

Name edited by Michael S. Branicky, Dec 31 2020

A120716 a(1)=1, a(p)=p if p is a prime. Otherwise, start with n and iterate the map (k -> concatenation of nontrivial divisors of k) until we reach a prime q; then a(n) = q. If we never reach a prime, a(n) = -1.

Original entry on oeis.org

1, 2, 3, 2, 5, 23, 7

Offset: 1

N. J. A. Sloane, Jul 19 2007

a(8) is currently unknown.

The sequence continues from a(8)-a(100): >10^50, 3, 5, 11, >10^50, 13, 313, 1129, >10^50, 17, >10^50, 19, >10^50, 37, 211, 23, >10^50, 5, 3251, 313, >10^50, 29, >10^50, 31, >10^50, 311, >10^50, 1129, >10^50, 37, 373, 313, >10^50, 41, >10^50, 43, >10^50, >10^50, 223, 47, >10^50, 7, >10^50, 317, >10^50, 53, 23691827, 773, >10^50, 1129, 229, 59, >10^50, 61, >10^50, 378593, >10^50, >10^5", >10^50, 67, >10^50, 39191573, >10^50, 71, >10^50, 73, 379, >10^50, >10^50, 3979237, 236132639, 79, >10^50, >10^50, 241, 83, >10^50, 3137, >10^50, 3983249, >10^50, 89, >10^50, >10^50, >10^50, 331, 1319, 36389, >10^50, 97, >10^50, 391133, >10^50. - Robert Price, Mar 27 2019

			4 -> 2, prime, so a(4) = 2.
6 -> 2,3 -> 23, prime, so a(6) = 23.
8 -> 2,4 -> 24 -> 2,3,4,6,8,12 -> 2346812 -> 2,4,13,26,52,45131,90262,180524,586703,1173406 -> 2413265245131902621805245867031173406 -> ? (see link for the continuation)
9 -> 3, prime, so a(9) = 3.
21 -> 3,7 -> 37, prime, so a(21) = 37.

N. J. A. Sloane et al., Notes on the attempt to find a(8)

Cf. A120712, A120713, A037274, A037279, A130139 (a bisection), A130140, A130141, A130142.

A120716[n_] := Module[{x},
   If[n == 1, Return[1]];
   If[PrimeQ[n], Return[n]];
   x = FromDigits[Flatten[IntegerDigits[Rest[Most[Divisors[n]]]]]];
   If[x > 10^50, Return[">10^50"], A120716[x]]];
Table[A120716[n], {n, 1, 100}] (* Robert Price, Mar 27 2019 *)

Edited by Michel Marcus, Mar 09 2023

A120712 Numbers k with the property that the concatenation of the nontrivial divisors of k (i.e., excluding 1 and k) is a prime.

Original entry on oeis.org

4, 6, 9, 21, 22, 25, 33, 39, 46, 49, 51, 54, 58, 78, 82, 93, 99, 111, 115, 121, 133, 141, 142, 147, 153, 154, 159, 162, 166, 169, 174, 177, 186, 187, 189, 201, 205, 219, 226, 235, 237, 247, 249, 253, 262, 267, 274, 286, 289, 291, 294, 301, 318

Offset: 1

Eric Angelini, Jul 19 2007

			   k |    divisors    | concatenation
  ---+----------------+--------------
   4 | (1) 2      (4) |        2
   6 | (1) 2, 3   (6) |       23
   9 | (1) 3      (9) |        3
  21 | (1) 3, 7  (21) |       37
  22 | (1) 2, 11 (22) |      211
  25 | (1) 5     (25) |        5
  33 | (1) 3, 11 (33) |      311
  39 | (1) 3, 13 (39) |      313

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A037274, A037278, A037279, A106708, A120713, A120716.

Cf. A130139, A130140, A130141, A130142.

with(numtheory):
for k from 2 to 1000 do:
v0:=divisors(k):
nn:=nops(v0):
if nn > 2 then
v1:=[seq(v0[j],j=2..nn-1)]:
v2:=cat(seq(convert(v1[n],string),n=1..nops(v1))):
v3:=parse(v2):
if isprime(v3) = true then lprint(k,v3) fi:
fi:
od: # Simon Plouffe

fQ[n_] := PrimeQ@ FromDigits@ Most@ Rest@ Divisors@ n; Select[ Range[2, 320], fQ]

from sympy import divisors, isprime
def ok(n):
    s = "".join(str(d) for d in divisors(n)[1:-1])
    return s != "" and isprime(int(s))
print([k for k in range(319) if ok(k)]) # Michael S. Branicky, Oct 01 2024

Name edited by Michel Marcus, Mar 09 2023

A130139 Let f denote the map that replaces k with the concatenation of its nontrivial divisors, written in increasing order, each divisor being written in base 10 in the normal way. Then a(n) = prime reached when starting at 2n+1 and iterating f.

Original entry on oeis.org

1, 3, 5, 7, 3, 11, 13, 1129, 17, 19, 37, 23, 5, 313, 29, 31, 311, 1129, 37, 313, 41, 43

Offset: 0

N. J. A. Sloane, Jul 30 2007

If 2n+1 is 1 or a prime, set a(n) = 2n+1. If no prime is ever reached, set a(n) = -1.

			n = 7: 2n+1 = 15 = 3*5 -> 35 = 5*7 -> 57 = 3*19 -> 319 = 11*29 -> 1129, prime, so a(7) = 1129.

Cf. A037279, A130140, A130141, A130142. A bisection of A120716.

from sympy import divisors, isprime
def f(n): return int("".join(str(d) for d in divisors(n)[1:-1]))
def a(n):
    if n == 0: return 1
    fn, c = 2*n + 1, 0
    while not isprime(fn):
        fn, c = f(fn), c+1
    return fn
print([a(n) for n in range(22)]) # Michael S. Branicky, Jul 11 2022

Name edited by Michel Marcus, Mar 09 2023

A191647 Numbers n with property that the concatenation of their anti-divisors is a prime.

Original entry on oeis.org

3, 4, 5, 10, 14, 16, 40, 46, 100, 145, 149, 251, 340, 373, 406, 424, 439, 466, 539, 556, 571, 575, 617, 619, 628, 629, 655, 676, 689, 724, 760, 779, 794, 899, 901, 941, 970, 989, 1019, 1055, 1070, 1076, 1183, 1213, 1226, 1231, 1258, 1270, 1285, 1331, 1340

Offset: 1

Paolo P. Lava, Jun 10 2011

Similar to A120712 which uses the proper divisors of n.

			The anti-divisors of 40 are 3, 9, 16, 27, and 391627 is prime, hence 40 is in the sequence.

Klaus Brockhaus Table of n, a(n) for n = 1..1000 (first 250 terms from Paolo P. Lava)

Cf. A037279, A066272, A106708, A120712, A120716, A191648.

P:=proc(i) local a,b,c,d,k,n,s,v; v:=array(1..200000);
for n from 3 by 1 to i do k:=2; b:=0;
while k0 and (2*n mod k)=0 then b:=b+1; v[b]:=k; fi;
     else
     if (n mod k)>0 and (((2*n-1) mod k)=0 or ((2*n+1) mod k)=0) then
b:=b+1; v[b]:=k; fi; fi; k:=k+1; od; a:=v[1];
for s from 2 to b do a:=a*10^floor(1+evalf(log10(v[s])))+v[s]; od;
if isprime(a) then print(n); fi;
od; end: P(10^6);

antiDivisors[n_Integer] := Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)]; a191647[n_Integer] := Select[Range[n],
PrimeQ[FromDigits[Flatten[IntegerDigits /@ antiDivisors[#]]]] &]; a191647[1350] (* Michael De Vlieger, Aug 09 2014, "antiDivisors" after Harvey P. Dale at A066272 *)

from sympy import isprime
[n for n in range(3,10**4) if isprime(int(''.join([str(d) for d in range(2,n) if n%d and 2*n%d in [d-1,0,1]])))] # Chai Wah Wu, Aug 08 2014

a(618) corrected in b-file by Paolo P. Lava, Feb 28 2018

A037281 Number of iterations of transformation in A037280 needed to reach 1 or a prime, or -1 if no such number exists.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0

Offset: 1

N. J. A. Sloane

Continuation from a(12): -1, 0, 3, 4, 1, 0, -1, 0, -1, 1, 1, 0, -1, 1, 4, 2, -1, 0, -1, 0, 2, 1, -1, 3, -1, 0, 2, 1, -1, 0, -1, 0, -1, -1, 1, 0, -1, 1, -1, 1, -1, 0, 1, 2, -1, 2, 1, 0, -1; where all the -1's are conjectural. - Franklin T. Adams-Watters, Jul 28 2011

Cf. A037279.

with(numtheory):ds:=proc(s) local j: RETURN(add(s[j]*10^(j-1),j=1..nops(s))):end: a:=proc(n) options remember: local d,i,l,m: if n<3 then RETURN(1) else if not isprime(n) then d:=divisors(n): l:=nops(d): m:=ds([seq(op(convert(d[l-i+1],base,10)),i=2..l-1)]): RETURN(m/piecewise(m mod 2=1,1,2^(ifactors(m)[2][1][2]))) else RETURN(n) fi fi: end; for n from 1 to 11 do m:=n: for i from 0 while(not isprime(m) and m<>1) do m:=a(m) od: printf("%d, ",i) od: # C. Ronaldo

Corrected by C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 29 2004

A361580 If n is composite, replace n with the concatenation of its nontrivial divisors, written in decreasing order, each divisor being written in base 10 with its digits in normal order, otherwise a(n) = n.

Original entry on oeis.org

1, 2, 3, 2, 5, 32, 7, 42, 3, 52, 11, 6432, 13, 72, 53, 842, 17, 9632, 19, 10542, 73, 112, 23, 1286432, 5, 132, 93, 14742, 29, 15106532, 31, 16842, 113, 172, 75, 181296432, 37, 192, 133, 20108542, 41, 21147632, 43, 221142, 15953, 232, 47, 24161286432, 7, 251052

Offset: 1

Tyler Busby, Mar 16 2023

			Nontrivial divisors of 20 are 2,4,5,10, so a(20)=10542.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A130141, A130139, A037279.

f:= proc(n) local L;
if isprime(n) then return n fi;
L:= sort(convert(numtheory:-divisors(n) minus {1,n},list),`>`);
parse(cat(op(L)))
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Mar 16 2023

Array[If[CompositeQ[#], FromDigits@ Flatten@ Map[IntegerDigits, Reverse@ Divisors[#][[2 ;; -2]] ], #] &, 50] (* Michael De Vlieger, Mar 22 2023 *)

a(n) = if (isprime(n) || (n==1), n, my(d=divisors(n)); my(s=""); for(k=2, #d-1, s=concat(Str(d[k]), s)); eval(s)); \\ Michel Marcus, Mar 16 2023

from sympy import divisors, isprime
def a(n):
    if n == 1 or isprime(n): return n
    return int("".join(str(d) for d in divisors(n)[-2:0:-1]))
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Mar 21 2023

A361320 If n is composite, replace n with the concatenation of its nontrivial divisors, written in increasing order, each divisor being written in base 10 with its digits in reverse order, otherwise a(n) = n.

Original entry on oeis.org

1, 2, 3, 2, 5, 23, 7, 24, 3, 25, 11, 2346, 13, 27, 35, 248, 17, 2369, 19, 24501, 37, 211, 23, 2346821, 5, 231, 39, 24741, 29, 23560151, 31, 24861, 311, 271, 57, 234692181, 37, 291, 331, 24580102, 41, 23674112, 43, 241122, 35951, 232, 47, 23468216142, 7, 250152

Offset: 1

Tyler Busby, Mar 09 2023

First differs from A037279 at a(20).

			Divisors of 20 are 1,2,4,5,10,20, so a(20)=24501.

Cf. A037279, A130140.

a(n) = if (isprime(n) || (n==1), return (n)); my(d=divisors(n), list=List()); for (i=2, #d-1, my(dd=digits(d[i])); forstep (j=#dd, 1, -1, listput(list, dd[j]))); fromdigits(Vec(list)); \\ Michel Marcus, Mar 09 2023

A361581 If n is composite, replace n with the concatenation of its nontrivial divisors, written in decreasing order, each divisor being written in base 10 with its digits in reverse order, otherwise a(n) = n.

Original entry on oeis.org

1, 2, 3, 2, 5, 32, 7, 42, 3, 52, 11, 6432, 13, 72, 53, 842, 17, 9632, 19, 1542, 73, 112, 23, 2186432, 5, 312, 93, 41742, 29, 51016532, 31, 61842, 113, 712, 75, 812196432, 37, 912, 313, 2018542, 41, 12417632, 43, 221142, 51953, 322, 47, 42612186432, 7, 520152

Offset: 1

Tyler Busby, Mar 16 2023

			Nontrivial divisors of 20 are 2,4,5,10, so a(20)=1542.

Cf. A130142, A130139, A037279.

a(n) = if (isprime(n) || (n==1), n, my(d=divisors(n)); my(s=""); forstep(k=#d-1, 2, -1, my(dk = digits(d[k])); for (i=1, #dk, s=concat(s, Str(dk[#dk-i+1])))); eval(s)); \\ Michel Marcus, Mar 16 2023

from sympy import divisors, isprime
def a(n):
    if n == 1 or isprime(n): return n
    return int("".join(str(d)[::-1] for d in divisors(n)[-2:0:-1]))
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Mar 21 2023

cp's OEIS Frontend

A037280 If n is composite replace n with the concatenation of its nontrivial divisors [ A037279 ] then divide out any factors of 2.

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A106708 a(n) is the concatenation of its nontrivial divisors.

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A120716 a(1)=1, a(p)=p if p is a prime. Otherwise, start with n and iterate the map (k -> concatenation of nontrivial divisors of k) until we reach a prime q; then a(n) = q. If we never reach a prime, a(n) = -1.

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A120712 Numbers k with the property that the concatenation of the nontrivial divisors of k (i.e., excluding 1 and k) is a prime.

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A130139 Let f denote the map that replaces k with the concatenation of its nontrivial divisors, written in increasing order, each divisor being written in base 10 in the normal way. Then a(n) = prime reached when starting at 2n+1 and iterating f.

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A191647 Numbers n with property that the concatenation of their anti-divisors is a prime.

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A037281 Number of iterations of transformation in A037280 needed to reach 1 or a prime, or -1 if no such number exists.

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A361580 If n is composite, replace n with the concatenation of its nontrivial divisors, written in decreasing order, each divisor being written in base 10 with its digits in normal order, otherwise a(n) = n.