A037274 -id:A037274 - cp's OEIS frontend

A118756 a(n) = smallest prime p such that p is the home prime (cf. A037274) of exactly n natural numbers.

2, 23, 211, 379, 773, 3389, 23251, 3761, 178069, 77773, 379811, 378997, 747521, 2337691, 3789293, 3574657

Offset: 1

William Lindgren (william.lindgren(AT)sru.edu), Sep 04 2007, corrected Sep 15 2007

a(17) > 10^7, a(18) = 3784463. - Robert G. Wilson v, Oct 01 2007

			23 is the home prime of both 6 and 23, thus a(2) = 23; 211 is the home prime of 4, 22 and 211, thus a(3) = 211.
More compactly: (2) -> 2, (6,23) -> 23, (4,22,211) -> 211,
(42,74,237,379) -> 379, (10,25,55,511,773) -> 773,
(118, 259, 737, 801, 1167, 3389) -> 3389,
{250, 1506, 2555, 3865, 5773, 6502, 23251} -> 23251,
{140, 332, 514, 566, 1281, 2257, 2283, 3761} -> 3761,
{4206, 7402, 10786, 16123, 23701, 25393, 67379, 137173, 178069} -> 178069,
{3786, 7262, 10078, 14513, 18417, 23631, 25039, 32449, 37877, 77773} -> 77773,
{41933, 50161, 56598, 103487, 192207, 206031, 216959, 239433, 307369, 363007, 379811} -> 379811,
{1798, 5982, 22931, 23997, 41315, 53033, 58263, 181293, 184102, 292051, 319421, 378997} -> 378997,
{722, 4938, 5718, 7646, 18929, 21919, 23823, 23953, 91277, 97941, 171409, 332647, 747521} -> 747521,
{87066, 128055, 138438, 153402, 175611, 226146, 358537, 465734, 588041, 675382, 866893, 1792003, 1564051, 2337691} -> 2337691,
{8691, 9725, 23585, 31437, 32897, 55389, 67491, 144995, 168163, 337499, 547617, 964849, 1875153, 3303841, 3789293} -> 3789293,
{5978, 10654, 27761, 40307, 47985, 111855, 156657, 172371, 202881, 250519, 357457, 379661, 407507, 488985, 1723971, 3574657} -> 3574657, etc. - _Robert G. Wilson v_, Oct 01 2007

Cf. A037274, A133956, A133958, A133960, A133962, A133964, A133966, A133968, A133970, A133972, A133974, A133976, A133978, A215408.

lst = {}; f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n, 2]; h[n_] := NestWhileList[f@# &, n, !PrimeQ@# &, 1, 28]; Do[p = h[n][[ -1]]; If[ PrimeQ@p && p < 10^7 && p != n, Print[{n, p}]; AppendTo[lst, p]], {n, 2, 10^7}];
lst = Sort@ lst; Table[d = n - 2; lsu = {}; Do[If[lst[[n]] == lst[[n + d]] && lst[[n - 1]] != lst[[n]] && lst[[n]] != lst[[n + d + 1]], AppendTo[lsu, lst[[n]] ]], {n, 188004 - d - 1}]; First@ Union@ lsu, {n, 3, 16}] (* Robert G. Wilson v, Oct 01 2007 *)

a(n) = A037274(A215408(n)). - Jonathan Sondow, Aug 09 2012

a(7)-a(16) from Robert G. Wilson v, Sep 27 2007, Oct 01 2007

A133957 Form the list of home primes A037274(c) for c composite, and sort into increasing order.

Original entry on oeis.org

23, 37, 211, 223, 227, 229, 233, 241, 257, 271, 277, 283, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 523, 541, 547, 557, 571, 577, 719, 743, 761, 773, 797, 1117, 1123, 1129, 1153, 1171, 1319, 1361, 1367, 1373, 1723, 1741, 1747

Offset: 1

Robert G. Wilson v, Sep 30 2007

The old name was "Home primes the result of composite numbers."
Number of terms < 10^n: 0, 2, 37, 274, 2087, 15472, 123261, ....
Increasing sequence of all prime numbers which are concatenations of at least two primes ordered in nondecreasing order (e.g., 227=2.2.7, 1319=13.19). - Bartlomiej Pawlik, Aug 06 2023

			The home primes corresponding to the first few composite numbers c are as follows:
   c            A037274(c)
   4            211
   6            23
   8            3331113965338635107
   9            311
   10           773
   12           223
   14           13367
   15           1129
   16           31636373
   18           233
   20           3318308475676071413
   21           37
   ...          ...

Patrick De Geest, Home Primes < 100 and Beyond.
Eric Weisstein's World of Mathematics, Home Prime.

Cf. A037274, A118756, A133959, A133961, A133963, A133965, A133967, A133969, A133971, A133973, A133975, A133977, A133979.

lst = {}; f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n, 2]; h[n_] := NestWhileList[f@# &, n, !PrimeQ@# &, 1, 28]; Do[p = h[n][[ -1]]; If[ PrimeQ@p && p < 10^7 && p != n, Print[{n, p}]; AppendTo[lst, p]], {n, 2, 1000}]; Union@ lst

Entry revised by N. J. A. Sloane, Mar 24 2021

A037275 Subsequence of record holders in A037274.

Original entry on oeis.org

1, 2, 3, 211, 3331113965338635107, 6161791591356884791277

Offset: 1

N. J. A. Sloane, Jeff Burch

P. De Geest, Home Primes
Eric Weisstein's World of Mathematics, Home Primes

Cf. A037271-A037274.

De Geest's web site has many more terms.

A238579 Home prime sequence (see A037274) starting at 146.

Original entry on oeis.org

146, 273, 3713, 4779, 333359, 1325643, 3111717139, 29177760383, 69142225413, 3471134339561, 7980350584141, 1324115101168677, 33147123900129853, 1941131324815763997, 37816317113233982621, 291304010934939102849, 333777134924210136703397, 7409854792211363875345439

Offset: 1

J. Lowell, Mar 30 2014

Second sequence of this kind with as yet unknown trajectory (see A037274, A056938). - J. Lowell, Apr 27 2017

			146 = 2*73 so next term is 273; 273 = 3*7*13 so next term is 3713.

Sean A. Irvine, Table of n, a(n) for n = 1..100
Wikipedia, Home prime

Cf. A037274 (home primes), A056938.

NestList[FromDigits@ Flatten@ Map[IntegerDigits, FactorInteger[#] /. {p_, e_} /; p >= 1 :> If[p == 1, 1, ConstantArray[p, e]]] &, 146, 17] (* Michael De Vlieger, Apr 27 2017 *)

lista(nn) = {n = 146; for (i=1 , nn, print1(n, ", "); f = factor(n); s = ""; for (j=1, #f~, for (k=1, f[j, 2], s = concat(s, Str(f[j, 1])););); n = eval(s););} \\ Michel Marcus, Mar 31 2014

One more term added by J. Lowell, Mar 30 2014
More terms from Jon E. Schoenfield, Mar 30 2014
One further term from N. J. A. Sloane, Mar 31 2014

A215408 Smallest number whose home prime (cf. A037274) is the home prime of exactly n natural numbers.

Original entry on oeis.org

2, 6, 4, 42, 10, 118, 250, 140, 4206, 3786, 41933, 1798, 722, 87066, 8691, 5978

Offset: 1

Jonathan Sondow, Aug 09 2012

The home primes of the sequence are A118756.

			The home prime of the four numbers 42, 74, 237, and 379 is 379 (for example, 42 = 2*3*7 -> 237 = 3*79 -> 379 which is prime), so a(4) = 42.

J. Heleen, Family Numbers: Constructing Primes By Prime Factor Splicing, J. Rec. Math. 28 (1996-97), 116-119.

P. de Geest, Home Primes < 100 and Beyond
Eric Weisstein's World of Mathematics, Home Prime.
Wikipedia, Home prime

Cf. A037274, A118756, and their cross-references.

A037274(a(n)) = A118756(n).

A037276 Start with 1; for n>1, replace n with the concatenation of its prime factors in increasing order.

Original entry on oeis.org

1, 2, 3, 22, 5, 23, 7, 222, 33, 25, 11, 223, 13, 27, 35, 2222, 17, 233, 19, 225, 37, 211, 23, 2223, 55, 213, 333, 227, 29, 235, 31, 22222, 311, 217, 57, 2233, 37, 219, 313, 2225, 41, 237, 43, 2211, 335, 223, 47, 22223, 77, 255, 317, 2213, 53, 2333

Offset: 1

N. J. A. Sloane

			If n = 2^3*5^5*11^2 = 3025000, a(n) = 222555551111 (n=2*2*2*5*5*5*5*5*11*11, then remove the multiplication signs).

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 10000 terms from T. D. Noe]
Patrick De Geest, Home Primes
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

Cf. A037274, A048985, A067599, A080670, A084796. Different from A073646.

Cf. also A027746, A289660 (a(n)-n).

a037276 = read . concatMap show . a027746_row
-- Reinhard Zumkeller, Apr 03 2012

# This is for n>1
read("transforms") ;
A037276 := proc(n)
    local L,p ;
    L := [] ;
    for p in ifactors(n)[2] do
        L := [op(L),seq(op(1,p),i=1..op(2,p))] ;
    end do:
    digcatL(L) ;
end proc: # R. J. Mathar, Oct 29 2012

co[n_, k_] := Nest[Flatten[IntegerDigits[{#, n}]] &, n, k - 1]; Table[FromDigits[Flatten[IntegerDigits[co @@@ FactorInteger[n]]]], {n, 54}] (* Jayanta Basu, Jul 04 2013 *)
FromDigits@ Flatten@ IntegerDigits[Table[#1, {#2}] & @@@ FactorInteger@ #] & /@ Range@ 54 (* Michael De Vlieger, Jul 14 2015 *)

a(n)={ n<4 & return(n); for(i=1,#n=factor(n)~, n[1,i]=concat(vector(n[2,i],j,Str(n[1,i])))); eval(concat(n[1,]))}  \\ M. F. Hasler, Jun 19 2011

from sympy import factorint
def a(n):
    f=factorint(n)
    l=sorted(f)
    return 1 if n==1 else int("".join(str(i)*f[i] for i in l))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 23 2017

A037273 Number of steps to reach a prime under "replace n with concatenation of its prime factors", or -1 if no prime is ever reached.

Original entry on oeis.org

-1, 0, 0, 2, 0, 1, 0, 13, 2, 4, 0, 1, 0, 5, 4, 4, 0, 1, 0, 15, 1, 1, 0, 2, 3, 4, 4, 1, 0, 2, 0, 2, 1, 5, 3, 2, 0, 2, 1, 9, 0, 2, 0, 9, 6, 1, 0, 15

Offset: 1

N. J. A. Sloane, Jeff Burch

Starting with 49, no prime has been reached after 79 steps.

a(49) > 118, see A056938 and FactorDB link. - Michael S. Branicky, Nov 19 2020

			13 is already prime, so a(13) = 0.
Starting with 14 we get 14 = 2*7, 27 = 3*3*3, 333 = 3*3*37, 3337 = 47*71, 4771 = 13*367, 13367 is prime; so a(14) = 5.

Patrick De Geest, Home Primes
FactorDB, Home prime base 10
Eric Weisstein's World of Mathematics, Home Prime

Cf. A037271, A037272, A037274, A037275, A056938.

Cf. A010051, A027746.

a037273 1 = -1
a037273 n = length $ takeWhile ((== 0) . a010051) $
   iterate (\x -> read $ concatMap show $ a027746_row x :: Integer) n
-- Reinhard Zumkeller, Jan 08 2013

nxt[n_] := FromDigits[Flatten[IntegerDigits/@Table[#[[1]], {#[[2]]}]&/@ FactorInteger[n]]]; Table[Length[NestWhileList[nxt, n, !PrimeQ[#]&]] - 1, {n, 48}] (* Harvey P. Dale, Jan 03 2013 *)

row_a027746(n, o=[1])=if(n>1, concat(apply(t->vector(t[2], i, t[1]), Vec(factor(n)~))), o) \\ after M. F. Hasler in A027746
tonum(vec) = my(s=""); for(k=1, #vec, s=concat(s, Str(vec[k]))); eval(s)
a(n) = if(n==1, return(-1)); my(x=n, i=0); while(1, if(ispseudoprime(x), return(i)); x=tonum(row_a027746(x)); i++) \\ Felix Fröhlich, May 17 2021

from sympy import factorint
def a(n):
    if n < 2: return -1
    klst, f = [n], sorted(factorint(n, multiple=True))
    while len(f) > 1:
        klst.append(int("".join(map(str, f))))
        f = sorted(factorint(klst[-1], multiple=True))
    return len(klst) - 1
print([a(n) for n in range(1, 49)]) # Michael S. Branicky, Aug 02 2021

Edited by Charles R Greathouse IV, Apr 23 2010
a(1) = -1 by Reinhard Zumkeller, Jan 08 2013
Name edited by Felix Fröhlich, May 17 2021

A195264 Iterate x -> A080670(x) (replace x with the concatenation of the primes and exponents in its prime factorization) starting at n until reach 1 or a prime (which is then the value of a(n)); or a(n) = -1 if a prime is never reached.

Original entry on oeis.org

1, 2, 3, 211, 5, 23, 7, 23, 2213, 2213, 11, 223, 13, 311, 1129, 233, 17, 17137, 19

Offset: 1

N. J. A. Sloane, Sep 14 2011, based on discussions on the Sequence Fans Mailing List by Alonso del Arte, Franklin T. Adams-Watters, D. S. McNeil, Charles R Greathouse IV, Sean A. Irvine, and others

J. H. Conway offered $1000 for a proof or disproof for his conjecture that every number eventually reaches a 1 or a prime - see OEIS50 link. - N. J. A. Sloane, Oct 15 2014
However, James Davis has discovered that a(13532385396179) = -1. This number D = 13532385396179 = (1407*10^5+1)*96179 = 13*53^2*3853*96179 is clearly fixed by the map x -> A080670(x), and so never reaches 1 or a prime. - Hans Havermann, Jun 05 2017
The number n = 3^6 * 2331961591220850480109739369 * 21313644799483579440006455257 is a near-miss for another nonprime fixed point. Unfortunately here the last two factors only look like primes (they have no prime divisors < 10), but in fact both are composite. - Robert Gerbicz, Jun 07 2017
The number D' = 13^532385396179 maps to D and so is a much larger number with a(D') = -1. Repeating this process (by finding a prime prefix of D') should lead to an infinite sequence of counterexamples to Conway's conjecture. - Hans Havermann, Jun 09 2017
The first 47 digits of D' form a prime P = 68971066936841703995076128866117893410448319579, so if Q denotes the remaining digits of 13^532385396179 then D'' = P^Q is another counterexample. - Robert Gerbicz, Jun 10 2017
This sequence is different from A037274. Here 8 = 2^3 -> 23 (a prime), whereas in A037274 8 = 2^3 -> 222 -> ... -> 3331113965338635107 (a prime). - N. J. A. Sloane, Oct 12 2014
The value of a(20) is presently unknown (see A195265).

			4 = 2^2 -> 22 =2*11 -> 211, prime, so a(4) = 211.
9 = 3^2 -> 32 = 2^5 -> 25 = 5^2 -> 52 = 2^2*13 -> 2213, prime, so a(9)=2213.

R. J. Mathar, Table of n, a(n) for n = 1..19
J. H. Conway, Five $1000 Problems, Oct 10 2014 (recorded by Bill Cheswick).
Alonso del Arte and Sean A. Irvine, Table of n, a(n) for n = 1..1000
Hans Havermann, Table of n, a(n) for n = 1..10000 (includes links to lengthy (>40) and unknown-outcome evolutions, and a list of unfactored composites in the unknowns' last step)
Hans Havermann, Table of n, a(n) for n = 1..10000 (includes links to lengthy (>40) and unknown-outcome evolutions, and a list of unfactored composites in the unknowns' last step) [Cached copy, pdf version as of May 08 2018, with permission]
Hans Havermann, 13532385396179 precursors
Hans Havermann, 13532385396179 precursors [Cached copy, pdf version as of May 08 2018, with permission]
OEIS50 DIMACS Conference on Challenges of Identifying Integer Sequences, Problem Session 2, Oct 10 2014, J. H. Conway, Five $1000 Problems (starting at about 06.44). This sequence is mentioned in the fifth problem, starting at around 19:30.
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)
Doron Zeilberger, Videos of Talks Delivered in SLOANE75/OEIS50 DIMACS Conference on Challenges of Identifying Integer Sequences (see Problem Session 2, Oct 10 2014)

A variant of the home primes, A037271. Cf. A080670, A195265 (trajectory of 20), A195266 (trajectory of 105), A230305, A084318. A230627 (base-2), A290329 (base-3)

f[1] := 1; f[n_] := Block[{p = Flatten[FactorInteger[n]]}, k = Length[p]; While[k > 0, If[p[[k]] == 1, p = Delete[p, k]]; k--]; FromDigits[Flatten[IntegerDigits[p]]]]; Table[FixedPoint[f, n], {n, 19}] (* Alonso del Arte, based on the program for A080670, Sep 14 2011 *)
fn[n_] := FromDigits[Flatten[IntegerDigits[DeleteCases[Flatten[
FactorInteger[n]], 1]]]];
Table[NestWhile[fn, n, # != 1 && ! PrimeQ[#] &], {n, 19}] (* Robert Price, Mar 15 2020 *)

a(n)={n>1 && while(!ispseudoprime(n), n=A080670(n));n} \\ M. F. Hasler, Oct 12 2014

A037271 Number of steps to reach a prime under "replace n with concatenation of its prime factors" when applied to n-th composite number, or -1 if no such number exists.

Original entry on oeis.org

2, 1, 13, 2, 4, 1, 5, 4, 4, 1, 15, 1, 1, 2, 3, 4, 4, 1, 2, 2, 1, 5, 3, 2, 2, 1, 9, 2, 9, 6, 1, 15

Offset: 1

Jeff Burch

a(33) is presently unknown: starting with 49, no prime has been reached after 110 steps. See A037274 for the latest information.

			Starting with 14 (the seventh composite number) we get 14=2*7, 27=3*3*3, 333=3*3*37, 3337=47*71, 4771=13*367, 13367 is prime; so a(7)=5.

Patrick De Geest, Home Primes
M. Herman and J. Schiffman, Investigating home primes and their families, Math. Teacher, 107 (No. 8, 2014), 606-614.
Eric Weisstein's World of Mathematics, Home Prime

Cf. A037272, A037273, A037274, A056938, A002808, A037276, A027746, A230305.

a037271 = length . takeWhile ((== 0) . a010051'') .
                             iterate a037276 . a002808
-- Reinhard Zumkeller, Apr 03 2012

maxComposite = 49; maxIter = 40; concat[n_] := FromDigits[ Flatten[ IntegerDigits /@ Flatten[ Apply[ Table, {#[[1]], {#[[2]]}} & /@ FactorInteger[n], {1}]]]]; composites = Select[ Range[2, maxComposite], ! PrimeQ[#] &]; a[n_] := ( lst = NestWhileList[ concat, composites[[n]], ! PrimeQ[#] &, 1, maxIter]; If[PrimeQ[ Last[lst]], Length[lst] - 1, - 1]); Table[a[n], {n, 1, Length[composites]}] (* Jean-François Alcover, Jul 10 2012 *)

A133956 Complement of A133957.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 239, 251, 263, 269, 281, 293, 307, 349, 401, 409, 419, 421, 431, 433

Offset: 1

Robert G. Wilson v, Sep 30 2007

Home primes whose homeliness is 1.

Number of terms < 10^n: Pi(10^n)- {0, 2, 37, 274, 2087, 15472, 76940, ...}.

			Only {2} -> 2, {3} -> 3, etc. Whereas {6 & 23} -> 23 thus 23 has a homeliness of 2 and therefore is not a member of this sequence.

Cf. A037274, A118756, A133957, A133958, A133960, A133962, A133964, A133966, A133968, A133970, A133972, A133974, A133976, A133978.

lst = {}; f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n, 2]; h[n_] := NestWhileList[f@# &, n, !PrimeQ@# &, 1, 28]; Do[p = h[n][[ -1]]; If[ PrimeQ@p && p < 10^7 && p != n, Print[{n, p}]; AppendTo[lst, p]], {n, 2, 1000}];
Complement[ Prime@ Range@ 100, {23, 37, 211, 223, 227, 229, 233, 241, 257, 271, 277, 283, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 523, 541}]

cp's OEIS Frontend

A118756 a(n) = smallest prime p such that p is the home prime (cf. A037274) of exactly n natural numbers.

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A133957 Form the list of home primes A037274(c) for c composite, and sort into increasing order.

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A037275 Subsequence of record holders in A037274.

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A238579 Home prime sequence (see A037274) starting at 146.

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A215408 Smallest number whose home prime (cf. A037274) is the home prime of exactly n natural numbers.

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A037276 Start with 1; for n>1, replace n with the concatenation of its prime factors in increasing order.

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A037273 Number of steps to reach a prime under "replace n with concatenation of its prime factors", or -1 if no prime is ever reached.

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A195264 Iterate x -> A080670(x) (replace x with the concatenation of the primes and exponents in its prime factorization) starting at n until reach 1 or a prime (which is then the value of a(n)); or a(n) = -1 if a prime is never reached.