A120716 -id:A120716 - cp's OEIS frontend

A037274 Home primes: for n >= 2, a(n) = the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached (a(n) = -1 if no prime is ever reached).

1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, 31, 241271, 311, 31397, 1129, 71129, 37, 373, 313, 3314192745739, 41, 379, 43, 22815088913, 3411949, 223, 47, 6161791591356884791277

Offset: 1

N. J. A. Sloane, Jeff Burch

The initial 1 could have been omitted.
Probabilistic arguments give exactly zero for the chance that the sequence of integers starting at n contains no prime, the expected number of primes being given by a divergent sequence. - J. H. Conway
After over 100 iterations, a(49) is still composite - see A056938 for the latest information.
More terms:
a(50) to a(60) are 3517, 317, 2213, 53, 2333, 773, 37463, 1129, 229, 59, 35149;
a(61) to a(65) are 61, 31237, 337, 1272505013723, 1381321118321175157763339900357651;
a(66) to a(76) are 2311, 67, 3739, 33191, 257, 71, 1119179, 73, 379, 571, 333271.
This is different from A195264. Here 8 = 2^3 -> 222 -> ... -> 3331113965338635107 (a prime), whereas in A195264 8 = 2^3 -> 23 (a prime). - N. J. A. Sloane, Oct 12 2014

			9 = 3*3 -> 33 = 3*11 -> 311, prime, so a(9) = 311.
The trajectory of 8 is more interesting:
8 ->
2 * 2 * 2 ->
2 * 3 * 37 ->
3 * 19 * 41 ->
3 * 3 * 3 * 7 * 13 * 13 ->
3 * 11123771 ->
7 * 149 * 317 * 941 ->
229 * 31219729 ->
11 * 2084656339 ->
3 * 347 * 911 * 118189 ->
11 * 613 * 496501723 ->
97 * 130517 * 917327 ->
53 * 1832651281459 ->
3 * 3 * 3 * 11 * 139 * 653 * 3863 * 5107
and 3331113965338635107 is prime, so a(8) = 3331113965338635107.

Jeffrey Heleen, Family Numbers: Mathemagical Black Holes, Recreational and Educational Computing, 5:5, pp. 6, 1990.
Jeffrey Heleen, Family numbers: Constructing Primes by Prime Factor Splicing, J. Recreational Math., Vol. 28 #2, 1996-97, pp. 116-119.

Christian N. K. Anderson, Table of known values of n, # of steps to reach a(n), and a(n) or NA if a(n) has 30 digits or more. Also, the trajectory, with factors separated by a |, terminated by either "(end)" or "-> ?" if a(n) has 30 digits or more.
Patrick De Geest, Home Primes < 100 and Beyond
M. Herman and J. Schiffman, Investigating home primes and their families, Math. Teacher, 107 (No. 8, 2014), 606-614.
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)
Eric Weisstein's World of Mathematics, Home Prime.
Wikipedia, Home prime

Cf. A006919, A037271, A037272, A037273, A037275, A037276, A037919-A037941, A048986, A056938.
Cf. A195264 (use exponents instead of repeating primes).
Cf. A084318 (use only one copy of each prime), A248713 (Fermi-Dirac analog: use unique representation of n>1 as a product of distinct terms of A050376).
Cf. also A120716 and related sequences.

b:= n-> parse(cat(sort(map(i-> i[1]$i[2], ifactors(n)[2]))[])):
a:= n-> `if`(isprime(n) or n=1, n, a(b(n))):
seq(a(n), n=1..48);  # Alois P. Heinz, Jan 09 2021

f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], { #[[2]] }] & /@ FactorInteger@n, 2]; g[n_] := NestWhile[ f@# &, n, !PrimeQ@# &]; g[1] = 1; Array[g, 41] (* Robert G. Wilson v, Sep 22 2007 *)

step(n)=my(f=factor(n),s="");for(i=1,#f~,for(j=1,f[i,2],s=Str(s,f[i,1]))); eval(s)
a(n)=if(n<4,return(n)); while(!isprime(n), n=step(n)); n \\ Charles R Greathouse IV, May 14 2015

from sympy import factorint, isprime
def f(n): return int("".join(str(p)*e for p, e in factorint(n).items()))
def a(n):
    if n == 1: return 1
    fn = n
    while not isprime(fn): fn = f(fn)
    return fn
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Jul 11 2022

def digitLen(x,n):
    r=0
    while(x>0):
        x//=n
        r+=1
    return r
def concatPf(x,n):
    r=0
    f=list(factor(x))
    for c in range(len(f)):
        for d in range(f[c][1]):
            r*=(n**digitLen(f[c][0],n))
            r+=f[c][0]
    return r
def hp(x,n):
    x1=concatPf(x,n)
    while(x1!=x):
        x=x1
        x1=concatPf(x1,n)
    return x
#example: prints the home prime of 8 in base 10
print(hp(8,10))

Corrected and extended by Karl W. Heuer, Sep 30 2003

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

A130142 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 with its digits in reverse order. Then a(n) = first 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, 9343, 29, 31, 113, -1, 37, 313, 41, 43

Offset: 0

Adam L. Buchsbaum (alb(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.

			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 133.
Then 133 has nontrivial divisors 7 and 19, so we get 917.
Then 917 has nontrivial divisors 7 and 131, so we get 1317.
Then 1317 has nontrivial divisors 3 and 439, so we get 9343, a prime and a(13) = 9343.
From _Sean A. Irvine_, Sep 11 2009: (Start)
Proof chain for a(17). The following gives the argument to f at each step, followed by its factorization.
35 factors as 5 * 7.
75 has factors 3 * 5 * 5.
525153 has factors 3 * 193 * 907.
15057112727099753913 has factors 3 * 4463 * 17215189 * 65325353.
179719996575730910515106159846737337176838928854211713151146478934050745192125561494032705883138679506795913535235676554615981512719833136443 has factors 29 * 29 * 5546454298803948416569 * 8370112457804191610629 * 13338101723922940394396774098231 * 345111672681489292530961043464303237918570147336150469919363833
765...4892 (3249 digits) is divisible by 2, and hence all subsequent steps will be divisible by 2, therefore no prime is ever reached, therefore a(17)=-1. (End)

Cf. A130139, A130140, A130141, A120716.

5 more terms (details for a(17) in example). Next term requires factoring a 1478-digit number. - Sean A. Irvine, Sep 11 2009

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

A153023 If n is 1 or prime then a(n) = n. Otherwise, start with n and iterate the map k -> A048050(k) until we reach a prime p; then a(n) = p. If we never reach a prime, a(n) = -1. A048050 gives the sum of proper divisors of k, excluding both 1 and n from the sum.

Original entry on oeis.org

1, 2, 3, 2, 5, 5, 7, 5, 3, 7, 11, 5, 13, 3, 5, 3, 17, 7, 19, 7, 7, 13, 23, 5, 5, 5, 5, 5, 29, 41, 31, 41, 3, 19, 5, 7, 37, 7, 3, 7, 41, 53, 43, 3, 41, 5, 47, -1, 7, 53, 7, 41, 53, 7, 3, 7, 13, 31, 59, 107, 61, 3, 7, 3, 7, 7, 67, 13, 5, 73, 71, 7, 73, 3, -1, 7, 7, 89, 79, 41, 3, 43, 83, 139, 13

Offset: 1

Andrew Carter (acarter09(AT)newarka.edu), Dec 16 2008

sign

			a(18) -> {2,3,6,9} -> 20 -> {2,4,5,10} -> 21 -> {3,7} -> 10 -> {2,5} -> 7 = 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001065, A048050, A120716, A153024.

f := proc(n) L := {} ; a := n ; while not isprime(a) do a := A048050(a) ; if a in L then RETURN(-1) ; fi; L := L union {a} ; od; a ; end:
A048050 := proc(n) numtheory[sigma](n)-n-1 ; end:
A153023 := proc(n) if n =1 then 1; elif isprime(n) then n; else f(n) ; fi; end: # R. J. Mathar, Dec 19 2008

Table[If[! CompositeQ[n], n, NestWhile[DivisorSigma[1, #] - (# + 1) &, n, Nor[PrimeQ@ #, # == 0] &, 1, 100] /. k_ /; CompositeQ@ k -> -1], {n, 85}] (* Michael De Vlieger, Nov 03 2017 *)

(define (A153023 n) (let loop ((n n) (visited (list n))) (let ((next (A048050 n))) (cond ((or (= 1 n) (= 1 (A010051 n))) n) ((member next visited) -1) (else (loop next (cons next visited)))))))
(define (A048050 n) (if (= 1 n) 0 (- (A001065 n) 1)))
(define (A001065 n) (- (A000203 n) n)) ;; For an implementation of A000203, see under that entry.
;; Antti Karttunen, Nov 03 2017

Extended by R. J. Mathar, Dec 19 2008

Description clarified by Antti Karttunen, Nov 03 2017

A191648 a(1)=1, a(2)=1. For n > 2, start with n and iterate the map (k -> concatenation of anti-divisors of k) until we reach a prime q; then a(n) = q. If we never reach a prime, a(n) = 0.

Original entry on oeis.org

1, 1, 2, 3, 23, 3

Offset: 1

Paolo P. Lava, Jun 10 2011

Similar to A120716, which uses the proper divisors of n. Other known values include a(10) = 347, a(14) = 349, and a(16) = 311. See also A191859.

			The anti-divisors of 5 are 2, 3, and 23 is prime, hence a(5) = 23.
The anti-divisors of 7 are 2, 3, 5, and 235 is composite; the anti-divisors of 235 are 2, 3, 7, 10, 67, 94, 157, and 237106794157 = 59*547*7346909 is composite; the anti-divisors of 237106794157 start 2, 3, 5, 15, 118, 1094, 1709, 4519, 61403, 64546, 7722971, 14693818, 104937727, but the others are unknown, hence a(7) is also unknown.
Note that the example speaks of the anti-divisors of 237106794157 being incomplete.  As of this date, those are well-established (see code of A066272).  It is the primality evaluation chain from anti-divisors for n=7 from 23515118109417094519614036454677229711469381810493772727748015786693526280375184463161423922194842717663158071196105 that is incomplete. - _Bill McEachen_, Dec 14 2022

Cf. A130846, A066272, A120716, A191647.

antidivisors := proc(n) local a, k; a := {} ; for k from 2 to n-1 do if abs((n mod k)- k/2) < 1 then a := a union {k} ; end if; end do: a ; end proc:
A130846 := proc(n) digcatL(sort(convert(antidivisors(n),list))) ; end proc:
A191648 := proc(n) if n <=2 then 1; else m := A130846(n) ; while not isprime(m) do m := A130846(m) ; end do: return m; end if; end proc: # R. J. Mathar, Jun 30 2011

cp's OEIS Frontend

A037274 Home primes: for n >= 2, a(n) = the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached (a(n) = -1 if no prime is ever reached).

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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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A130142 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 with its digits in reverse order. Then a(n) = first 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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Maple

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A153023 If n is 1 or prime then a(n) = n. Otherwise, start with n and iterate the map k -> A048050(k) until we reach a prime p; then a(n) = p. If we never reach a prime, a(n) = -1. A048050 gives the sum of proper divisors of k, excluding both 1 and n from the sum.

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