A048985 -id:A048985 - cp's OEIS frontend

A048986 Home primes in base 2: primes reached when you start with n and (working in base 2) concatenate its prime factors (A048985); repeat until a prime is reached (or -1 if no prime is ever reached). Answer is written in base 10.

1, 2, 3, 31, 5, 11, 7, 179, 29, 31, 11, 43, 13, 23, 29, 12007, 17, 47, 19, 251, 31, 43, 23, 499, 4091, 4091, 127, 4091, 29, 127, 31, 1564237, 59, 4079, 47, 367, 37, 83, 61, 383, 41, 179, 43, 499, 4091, 4091, 47, 683, 127, 173, 113, 173, 53, 191, 4091

Offset: 1

Michael B Greenwald (mbgreen(AT)central.cis.upenn.edu)

a(1) = 1 by convention.

The first binary home prime that is not known is a(2295). - Ely Golden, Jan 09 2017

			4 = 2*2 -> 1010 = 10 = 2*5 ->10101 = 21 = 3*7 -> 11111 = 31 = prime.

Ely Golden, Table of n, a(n) for n = 1..2294
Patrick De Geest, Home Primes
Ely Golden, Mersenne Wiki Home Primes base 2
Ely Golden, Table of n, a(n) for n = 1..3000 (a-file)

Cf. A048985, A037274, A049065.

f[n_] := Module[{fi}, If[PrimeQ[n], n, fi = FactorInteger[n]; Table[ First[#], {Last[#]}]& /@ fi // Flatten // IntegerDigits[#, 2]& // Flatten // FromDigits[#, 2]&]]; a[1] = 1; a[n_] := TimeConstrained[FixedPoint[f, n], 1] /. $Aborted -> -1; Array[a, 55] (* Jean-François Alcover, Jan 01 2016 *)

from sympy import factorint, isprime
def f(n):
    if n == 1: return 1
    return int("".join(bin(p)[2:]*e for p, e in factorint(n).items()), 2)
def a(n):
    if n == 1: return 1
    while not isprime(n): n = f(n)
    return n
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, Oct 07 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
radix=2
index=2
while(index<=1344):
    print(str(index)+" "+str(hp(index,radix)))
    index+=1

A112417 Sequence is generated as in A048985, but primes are instead written in decreasing order before being concatenated.

Original entry on oeis.org

1, 2, 3, 10, 5, 14, 7, 42, 15, 22, 11, 58, 13, 30, 23, 170, 17, 62, 19, 90, 31, 46, 23, 234, 45, 54, 63, 122, 29, 94, 31, 682, 47, 70, 61, 250, 37, 78, 55, 362, 41, 126, 43, 186, 95, 94, 47, 938, 63, 182, 71, 218, 53, 254, 93, 490, 79, 118, 59, 378, 61, 126, 127, 2730

Offset: 1

Leroy Quet, Dec 09 2005

			12 = 3 * 2^2, with prime divisors written in descending order.
So we have 3 = 11 binary, 2 = 10 binary, 2 = 10 binary, which is (after concatenation) 111010. So a(12) is 111010 = 58 when converted from binary to decimal.

Diana Mecum, Table of n, a(n) for n = 1..847

Cf. A004676, A048985.

More terms from Diana L. Mecum, Jul 14 2007

A064795 Home primes in base 2: primes reached when you start with n and (working in base 2) concatenate its prime factors (A048985); repeat until a prime is reached (or -1 if no prime is ever reached).

Original entry on oeis.org

1, 10, 11, 11111, 101, 1011, 111, 10110011, 11101, 11111, 1011, 101011, 1101, 10111, 11101, 10111011100111, 10001, 101111, 10011, 11111011, 11111, 101011, 10111, 111110011, 111111111011, 111111111011, 1111111, 111111111011, 11101

Offset: 1

N. J. A. Sloane, Oct 31 2001

a(1) = 1 by convention.

P. De Geest, Home Primes

See A048986 for base 10 representation. Cf. A037274.

More terms from David Wasserman, Aug 15 2002

A029744 Numbers of the form 2^n or 3*2^n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152, 3145728, 4194304

Offset: 1

N. J. A. Sloane

This entry is a list, and so has offset 1. WARNING: However, in this entry several comments, formulas and programs seem to refer to the original version of this sequence which had offset 0. - M. F. Hasler, Oct 06 2014
Number of necklaces with n-1 beads and two colors that are the same when turned over and hence have reflection symmetry. [edited by Herbert Kociemba, Nov 24 2016]
The subset {a(1),...,a(2k)} contains all proper divisors of 3*2^k. - Ralf Stephan, Jun 02 2003
Let k = any nonnegative integer and j = 0 or 1. Then n+1 = 2k + 3j and a(n) = 2^k*3^j. - Andras Erszegi (erszegi.andras(AT)chello.hu), Jul 30 2005
Smallest number having no fewer prime factors than any predecessor, a(0)=1; A110654(n) = A001222(a(n)); complement of A116451. - Reinhard Zumkeller, Feb 16 2006
A093873(a(n)) = 1. - Reinhard Zumkeller, Oct 13 2006
a(n) = a(n-1) + a(n-2) - gcd(a(n-1), a(n-2)), n >= 3, a(1)=2, a(2)=3. - Ctibor O. Zizka, Jun 06 2009
Where records occur in A048985: A193652(n) = A048985(a(n)) and A193652(n) < A048985(m) for m < a(n). - Reinhard Zumkeller, Aug 08 2011
A002348(a(n)) = A000079(n-3) for n > 2. - Reinhard Zumkeller, Mar 18 2012
Without initial 1, third row in array A228405. - Richard R. Forberg, Sep 06 2013
Also positions of records in A048673. A246360 gives the record values. - Antti Karttunen, Sep 23 2014
Known in numerical mathematics as "Bulirsch sequence", used in various extrapolation methods for step size control. - Peter Luschny, Oct 30 2019
For n > 1, squares of the terms can be expressed as the sum of two powers of two: 2^x + 2^y. - Karl-Heinz Hofmann, Sep 08 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 11, 22.
Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, The Binary Two-Up Sequence, arXiv:2209.04108 [math.CO], Sep 11 2022.
David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
John P. McSorley and Alan H. Schoen, Rhombic tilings of (n,k)-ovals, (n, k, lambda)-cyclic difference sets, and related topics, Discrete Math., 313 (2013), 129-154. - From _N. J. A. Sloane_, Nov 26 2012
Index entries for linear recurrences with constant coefficients, signature (0,2).
Index entries for sequences related to necklaces

Cf. A056493, A038754, A063759. Union of A000079 and A007283.
First differences are in A016116(n-1).
Cf. A082125, A094958, A048739, A048985, A193652, A048673, A064216, A246360, A354785.
Row sums of the triangle in sequence A119963. - John P. McSorley, Aug 31 2010
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent. There may be minor differences from (s(n)) at the start, and a shift of indices. A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A060482 (s(n)-3); A136252 (s(n)-3); A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4);   A354785 (3*s(n)), A061776 (3*s(n)-6); A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

a029744 n = a029744_list !! (n-1)
a029744_list = 1 : iterate
   (\x -> if x `mod` 3 == 0 then 4 * x `div` 3 else 3 * x `div` 2) 2
-- Reinhard Zumkeller, Mar 18 2012

1,seq(op([2^i,3*2^(i-1)]),i=1..100); # Robert Israel, Sep 23 2014

CoefficientList[Series[(-x^2 - 2*x - 1)/(2*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
Function[w, DeleteCases[Union@ Flatten@ w, k_ /; k > Max@ First@ w]]@ TensorProduct[{1, 3}, 2^Range[0, 22]] (* Michael De Vlieger, Nov 24 2016 *)
LinearRecurrence[{0,2},{1,2,3},50] (* Harvey P. Dale, Jul 04 2017 *)

PARI
```
a(n)=if(n%2,3/2,2)<<((n-1)\2)\1
```

def A029744(n):
    if n == 1: return 1
    elif n % 2 == 0: return 2**(n//2)
    else: return 3 * 2**((n-3)//2) # Karl-Heinz Hofmann, Sep 08 2022

(define (A029744 n) (cond ((<= n 1) n) ((even? n) (expt 2 (/ n 2))) (else (* 3 (expt 2 (/ (- n 3) 2)))))) ;; Antti Karttunen, Sep 23 2014

a(n) = 2*A000029(n) - A000031(n).
For n > 2, a(n) = 2*a(n - 2); for n > 3, a(n) = a(n - 1)*a(n - 2)/a(n - 3). G.f.: (1 + x)^2/(1 - 2*x^2). - Henry Bottomley, Jul 15 2001, corrected May 04 2007
a(0)=1, a(1)=1 and a(n) = a(n-2) * ( floor(a(n-1)/a(n-2)) + 1 ). - Benoit Cloitre, Aug 13 2002
(3/4 + sqrt(1/2))*sqrt(2)^n + (3/4 - sqrt(1/2))*(-sqrt(2))^n. a(0)=1, a(2n) = a(n-1)*a(n), a(2n+1) = a(n) + 2^floor((n-1)/2). - Ralf Stephan, Apr 16 2003 [Seems to refer to the original version with offset=0. - M. F. Hasler, Oct 06 2014]
Binomial transform is A048739. - Paul Barry, Apr 23 2004
E.g.f.: (cosh(x/sqrt(2)) + sqrt(2)sinh(x/sqrt(2)))^2.
a(1) = 1; a(n+1) = a(n) + A000010(a(n)). - Stefan Steinerberger, Dec 20 2007
u(2)=1, v(2)=1, u(n)=2*v(n-1), v(n)=u(n-1), a(n)=u(n)+v(n). - Jaume Oliver Lafont, May 21 2008
For n => 3, a(n) = sqrt(2*a(n-1)^2 + (-2)^(n-3)). - Richard R. Forberg, Aug 20 2013
a(n) = A064216(A246360(n)). - Antti Karttunen, Sep 23 2014
a(n) = sqrt((17 - (-1)^n)*2^(n-4)) for n >= 2. - Anton Zakharov, Jul 24 2016
Sum_{n>=1} 1/a(n) = 8/3. - Amiram Eldar, Nov 12 2020
a(n) = 2^(n/2) if n is even. a(n) = 3 * 2^((n-3)/2) if n is odd and for n>1. - Karl-Heinz Hofmann, Sep 08 2022

Corrected and extended by Joe Keane (jgk(AT)jgk.org), Feb 20 2000

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

A193652 A020988 and A007583 interleaved.

Original entry on oeis.org

0, 1, 2, 3, 10, 11, 42, 43, 170, 171, 682, 683, 2730, 2731, 10922, 10923, 43690, 43691, 174762, 174763, 699050, 699051, 2796202, 2796203, 11184810, 11184811, 44739242, 44739243, 178956970, 178956971, 715827882, 715827883, 2863311530, 2863311531, 11453246122

Offset: 0

Reinhard Zumkeller, Aug 08 2011

a(2*n) = A020988(n), a(2*n+1) = a(2*n) + 1 = A007583(n);

apart from initial zero, record values in A048985: a(n)=A048985(A029744(n)) and a(n)<A048985(m) for m<A029744(n).

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

A193652 := proc(n)
        if type (n,'even') then
                A020988(n/2) ;
        else
                A007583(floor(n/2)) ;
        end if;
end proc: # R. J. Mathar, Jul 20 2016

LinearRecurrence[{0, 5, 0, -4}, {0, 1, 2, 3}, 35] (* Jean-François Alcover, Dec 16 2021 *)

a(n) = 2 * (4^floor(n/2) - 1) / 3 + n mod 2.

G.f.: ( -x*(-1-2*x+2*x^2) ) / ( (x-1)*(2*x+1)*(2*x-1)*(1+x) ). - R. J. Mathar, Feb 19 2015

Terms corrected by R. J. Mathar, Feb 19 2015

A006919 Write down all the prime divisors in previous term.

Original entry on oeis.org

8, 222, 2337, 31941, 33371313, 311123771, 7149317941, 22931219729, 112084656339, 3347911118189, 11613496501723, 97130517917327, 531832651281459, 3331113965338635107, 3331113965338635107

Offset: 1

N. J. A. Sloane

H. Jaleebi, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Patrick De Geest, Home Primes

Cf. A056938 (same for a(1)=49), A037271-A037276, A048985, A048986, A049065.

g[ n_ ] := (x = n; d = {}; While[ FactorInteger[ x ] != {}, f = FactorInteger[ x, FactorComplete -> True ][ [ 1, 1 ] ]; x = x/f; AppendTo[ d, IntegerDigits[ f ] ] ]; FromDigits[ Flatten[ d ] ]); NestList[ g, 8, 15 ]
NestList[FromDigits[Flatten[IntegerDigits/@(Table[First[#],{Last[#]}]& /@ FactorInteger[#])]]&,8,15] (* Harvey P. Dale, Dec 04 2011 *)

first(N, a=8)=vector(N,i,if(i>1,a=A037276(a),a)) \\ M. F. Hasler, Oct 07 2022

a(n+1) = A037276(a(n)), a(1) = 8. - M. F. Hasler, Oct 07 2022

More terms from Robert G. Wilson v, Sep 05 2000, who remarks that sequence stabilizes at 13th term with a prime.

The value 37246812772043701411753149215934377 is the base-2 home prime for 922 and occurs after 66 steps. The value 3690727229000499480592573891534356177653018575120050845976045596834749951228879 is the base-2 home prime for 1345 and occurs after 131 steps. The next term (home prime for 2295) contains at least 124 digits. Computation of further terms involves large factorizations. - Sean A. Irvine, Aug 04 2005 [corrected Jul 17 2021]

cp's OEIS Frontend

A048986 Home primes in base 2: primes reached when you start with n and (working in base 2) concatenate its prime factors (A048985); repeat until a prime is reached (or -1 if no prime is ever reached). Answer is written in base 10.

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A112417 Sequence is generated as in A048985, but primes are instead written in decreasing order before being concatenated.

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A064795 Home primes in base 2: primes reached when you start with n and (working in base 2) concatenate its prime factors (A048985); repeat until a prime is reached (or -1 if no prime is ever reached).

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A029744 Numbers of the form 2^n or 3*2^n.

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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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A193652 A020988 and A007583 interleaved.

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Maple

Mathematica

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A006919 Write down all the prime divisors in previous term.

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Mathematica

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