A120712 -id:A120712 - cp's OEIS frontend

A120713 Primes arising in A120712.

2, 23, 3, 37, 211, 5, 311, 313, 223, 7, 317, 23691827, 229, 236132639, 241, 331, 391133, 337, 523, 11, 719, 347, 271, 372149, 391751, 2711142277, 353, 236918275481, 283, 13, 236295887, 359, 236316293, 1117, 379212763, 367, 541

Offset: 1

Eric Angelini, Jul 19 2007

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

Cf. A120712, A107374.

A037279 If n is composite, replace n with the concatenation of its nontrivial divisors, 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, 24510, 37, 211, 23, 2346812, 5, 213, 39, 24714, 29, 23561015, 31, 24816, 311, 217, 57, 234691218, 37, 219, 313, 24581020, 41, 23671421, 43, 241122, 35915, 223, 47, 23468121624, 7

Offset: 1

N. J. A. Sloane

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A037278, A120712, A106708.

A037279 := proc(n) local dvs ; if isprime(n) or n = 1 then n; else dvs := [op(numtheory[divisors](n) minus {1,n} )] ; dvs := sort(dvs) ; cat(op(dvs)) ; fi ; end: seq(A037279(n),n=1..80) ; # R. J. Mathar, Jul 23 2007

f[n_]:=If[PrimeQ[n],n,FromDigits[Flatten[IntegerDigits/@Rest[ Most[ Divisors[n]]]]]]; Join[{1},Array[f,50,2]] (* Harvey P. Dale, Sep 24 2012 *)

More terms from Erich Friedman

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

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

A130846 Replace n with the concatenation of its anti-divisors.

Original entry on oeis.org

2, 3, 23, 4, 235, 35, 26, 347, 237, 58, 2359, 349, 2610, 311, 235711, 45712, 2313, 3813, 2614, 345915, 235915, 716, 2371017, 3417, 2561118, 3581119, 2319, 41220, 237921, 35791321, 2561322, 3423, 23101423, 824, 2351525, 3457111525, 2671126, 391627

Offset: 3

Jonathan Vos Post, Jul 20 2007, Jul 22 2007

Number of anti-divisors concatenated to form a(n) is A066272(n). We may consider prime values of the concatenated anti-divisor sequence and we may iterate it, i.e. n, a(n), a(a(n)), a(a(a(n))) which leads to questions of trajectory, cycles, fixed points.
See A066272 for definition of anti-divisor.
Primes in this sequence are at n=3,4,5,10,14,16,40,46,100,145,149,... - R. J. Mathar, Jul 24 2007

			3: 2, so a(3) = 2.
4: 3, so a(4) = 3.
5: 2, 3, so a(5) = 23.
6: 4, so a(6) = 4.
7: 2, 3, 5, so a(7) = 235.
17: 2, 3, 5, 7, 11, so a(17) = 235711

Jon Perry, The Anti-Divisor, cached copy.
Jonathan Vos Post, Factors of first 62 terms

Cf. A037278, A066272, A120712, A106708, A130799.

antiDivs := proc(n) local resul,odd2n,r ; resul := {} ; for r in ( numtheory[divisors](2*n-1) union numtheory[divisors](2*n+1) ) do if n mod r <> 0 and r> 1 and r < n then resul := resul union {r} ; fi ; od ; odd2n := numtheory[divisors](2*n) ; for r in odd2n do if ( r mod 2 = 1) and r > 2 then resul := resul union {2*n/r} ; fi ; od ; RETURN(resul) ; end: A130846 := proc(n) cat(op(antiDivs(n))) ; end: seq(A130846(n),n=3..80) ; # R. J. Mathar, Jul 24 2007

from sympy.ntheory.factor_ import antidivisors
def A130846(n): return int(''.join(str(s) for s in antidivisors(n))) # Chai Wah Wu, Dec 08 2021

More terms from R. J. Mathar, Jul 24 2007

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

A191859 The primes created by concatenation of anti-divisors in A191647.

Original entry on oeis.org

2, 3, 23, 347, 349, 311, 391627, 3471331, 384067, 2310175897, 239111323273399, 23167, 3784097136227, 235983149249, 3428116271, 37111677121283, 23293, 3471949133311, 231314228398154359, 378112153101159371, 2379127163381

Offset: 1

Klaus Brockhaus, Jun 18 2011

a(n) is the concatenation of the anti-divisors of A191647(n).

			A191647(6) = 16, the anti-divisors of 16 are 3, 11. Hence a(6) = 311.
A191647(8) = 46, the anti-divisors of 46 are 3, 4, 7, 13, 31. Hence a(8) = 3471331.

Klaus Brockhaus, Table of n, a(n) for n = 1..1000

Cf. A191647, A066272, A130846, A120712.

Antidivisors:=func< n | [ d: d in [2..n-1] | n mod d ne 0 and ( (IsEven(d) and 2*n mod d eq 0) or (IsOdd(d) and ((2*n-1) mod d eq 0 or (2*n+1) mod d eq 0)) ) ] >; CAD:=function(n); A:=Antidivisors(n); S:=[]; for k in [1..#A] do S:= Intseq(A[k]) cat S; end for; p:=Seqint(S); return p; end function; A191859List:=func< m | [ p: n in [1..m] | IsPrime(p) where p is CAD(n) ] >; A191859List(600);

A166733 Numbers n with the property that the concatenation of the trivial divisors of n (i.e., 1 and n) is a prime.

Original entry on oeis.org

3, 7, 9, 13, 27, 31, 37, 39, 49, 51, 57, 63, 67, 73, 79, 81, 91, 93, 97, 99, 103, 109, 117, 123, 129, 151, 153, 163, 171, 181, 187, 193, 201, 213, 217, 223, 229, 231, 237, 249, 259, 277, 279, 283, 289, 291, 297, 301, 303, 307, 319, 321, 327, 361, 367, 373, 381

Offset: 1

Juri-Stepan Gerasimov, Oct 20 2009

Essentially the same as A032664. - R. J. Mathar, Oct 23 2009

			3 -> 13; 7 -> 17; 9 -> 19; 13 -> 113.

Cf. A120712.

Edited by Charles R Greathouse IV, Apr 24 2010

cp's OEIS Frontend

A120713 Primes arising in A120712.

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A037279 If n is composite, replace n with the concatenation of its nontrivial divisors, otherwise a(n) = n.

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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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A130846 Replace n with the concatenation of its anti-divisors.

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

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A191859 The primes created by concatenation of anti-divisors in A191647.

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A166733 Numbers n with the property that the concatenation of the trivial divisors of n (i.e., 1 and n) is a prime.

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