A130142 -id:A130142 - cp's OEIS frontend

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.

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

A130140 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 with its digits in reverse order. 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

Offset: 0

David Applegate, 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 -> 391 = 17*23 -> 7132.
Then 7132 has nontrivial divisors 2, 4, 1783, 3566, so we get 2438716653.
Then 2438716653 has nontrivial divisors 3, 9, 27, 81, 243, 1453, 4359, 6907, 13077, 20721, 39231, 62163, 117693, 186489, 353079, 559467, 1678401, 10035871, 30107613, 90322839, 270968517, 812905551, so we get
397218342354195347096770311270213293361263967119846819703537649551048761178530013167010393822309715869072155509218 = 2*3^4*1217*317539*1211548321*33378971294653*8960783431807*17509226460292689821646170308388500174366980857582533580184934929433.

Cf. A130139, A130141, A130142, A120716.

Edited by Michel Marcus, Mar 09 2023

A130141 Let f denote the map that replaces k with the concatenation of its nontrivial divisors, written in decreasing 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, 53, 17, 19, 73, 23, 5, 313, 29, 31, 113

Offset: 0

Carsten Lund (lund(AT)research.att.com), Jul 30 2007, Aug 01 2007

If 2n+1 is 1 or a prime, set a(n) = 2n+1. If no prime is ever reached, set a(n) = -1.
The value of a(17) is currently unknown.
From Klaus Brockhaus, Aug 01 2007: (Start)
"The sequence, starting from the beginning but writing 0 when a number with more than 90 digits is reached, is:
1,3,5,7,3,11,13,53,17,19,73,23,5,313,29,31,113,0,37,197,41,43,0,47,7,173,53,0,193,
59,61,0,0,67,233,71,73,0,391393,79,9313991471335749211973,83,0,293,89,137,313,
0,97,0,101,103,0,107,109,373,113,0,391393,593,11,1993,0,127,433,131,197,0,137,139,
0,0,0,0,149,151,511793,0,157,1141931201,6113,163,0,167,13,0,173,0,593,179,181,
613,12575251553,0,0,191,193,0,197,199,673,..." (End)
From Andrew Carter (acarter09(AT)newarka.edu), Dec 16 2008: (Start)
If "when starting at 2n+1" in the definition were replaced by "when starting at n", all even-indexed terms except for a(2) and a(4) would be -1.
A simple proof is that an even number's smallest nontrivial divisor is 2, and the result of iterating f for any even number except for 2 and 4 would have factors in front, so the resulting number would be of the form 10m+2 where m is an integer greater than 1, and therefore an even number ending in 2 other than two itself, so no prime would ever be reached. (End)

			n = 13: 2n+1 = 27 has nontrivial divisors 3 and 9, so we get 93, which has proper divisors 3 and 31, so we get 313, prime. So a(13) = 313.

Cf. A130139, A130140, A130142, A120716.

Edited by 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

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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A130140 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 with its digits in reverse order. Then a(n) = prime reached when starting at 2n+1 and iterating f.

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A130141 Let f denote the map that replaces k with the concatenation of its nontrivial divisors, written in decreasing 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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Extensions

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.

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Python