A175723 -id:A175723 - cp's OEIS frontend

A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

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

Offset: 1

N. J. A. Sloane

The initial term a(1)=1 is purely conventional: The unit 1 is not a prime number, although it has been considered so in the past. 1 is the empty product of prime numbers, thus 1 has no largest prime factor. - Daniel Forgues, Jul 05 2011
Greatest noncomposite number dividing n, (cf. A008578). - Omar E. Pol, Aug 31 2013
Conjecture: Let a, b be nonzero integers and f(n) denote the maximum prime factor of a*n + b if a*n + b <> 0 and f(n)=0 if a*n + b=0 for any integer n. Then the set {n, f(n), f(f(n)), ...} is finite of bounded size. - M. Farrokhi D. G., Jan 10 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section IV.1.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294. MR 0447086 (56 #5401).
G. Back and M. Caragiu, The greatest prime factor and recurrent sequences, Fib. Q., 48 (2010), 358-362.
A. E. Brouwer, Two number theoretic sums, Stichting Mathematisch Centrum. Zuivere Wiskunde, Report ZW 19/74 (1974): 3 pages. [Cached copy, included with the permission of the author]
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Mats Granvik, Mathematica program to check the conjectured formula, Feb 28 2021.
Nathan McNew, The Most Frequent Values of the Largest Prime Divisor Function, Exper. Math., 2017, Vol. 26, No. 2, 210-224.
OEIS Wiki, Greatest prime factor of n
H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981
David Singmaster, Letter to N. J. A. Sloane, Oct 03 1982.
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Index entries for "core" sequences

Cf. A000040, A020639 (smallest prime divisor), A034684, A028233, A034699, A053585.
See also A032742, A052126, A070087, A070089, A061395, A175723.
Cf. A046670 (partial sums), A104350 (partial products).
See A385503 for "popular" primes.

[ #f eq 0 select 1 else f[ #f][1] where f is Factorization(n): n in [1..86] ]; // Klaus Brockhaus, Oct 23 2008

with(numtheory,divisors); A006530 := proc(n) local i,t1,t2,t3,t4,t5; t1 := divisors(n); t2 := convert(t1,list); t3 := sort(t2); t4 := nops(t3); t5 := 1; for i from 1 to t4 do if isprime(t3[t4+1-i]) then return t3[t4+1-i]; fi; od; 1; end;
# alternative
A006530 := n->max(1,op(numtheory[factorset](n))); # Peter Luschny, Nov 02 2010

Table[ FactorInteger[n][[ -1, 1]], {n, 100}] (* Ray Chandler, Nov 12 2005 and modified by Robert G. Wilson v, Jul 16 2014 *)

A006530(n)=if(n>1,vecmax(factor(n)[,1]),1) \\ Edited to cover n=1. - M. F. Hasler, Jul 30 2015

from sympy import factorint
def a(n): return 1 if n == 1 else max(factorint(n))
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Aug 08 2022

def A006530(n): return list(factor(n))[-1][0] if n > 1 else 1
print([A006530(n) for n in range(1, 87)])  # Peter Luschny, Jan 07 2024

;; The following uses macro definec for the memoization (caching) of the results. A naive implementation of A020639 can be found under that entry. It could be also defined with definec to make it faster on the later calls. See http://oeis.org/wiki/Memoization#Scheme
(definec (A006530 n) (let ((spf (A020639 n))) (if (= spf n) spf (A006530 (/ n spf)))))
;; Antti Karttunen, Mar 12 2017

a(n) = A027748(n, A001221(n)) = A027746(n, A001222(n)); a(n)^A071178(n) = A053585(n). - Reinhard Zumkeller, Aug 27 2011
a(n) = A000040(A061395(n)). - M. F. Hasler, Jan 16 2015
a(n) = n + 1 - Sum_{k=1..n} (floor((k!^n)/n) - floor(((k!^n)-1)/n)). - Anthony Browne, May 11 2016
n/a(n) = A052126(n). - R. J. Mathar, Oct 03 2016
If A020639(n) = n [when n is 1 or a prime] then a(n) = n, otherwise a(n) = a(A032742(n)). - Antti Karttunen, Mar 12 2017
a(n) has average order Pi^2*n/(12 log n) [Brouwer]. See also A046670. - N. J. A. Sloane, Jun 26 2017

Edited by M. F. Hasler, Jan 16 2015

A214674 Conway's subprime Fibonacci sequence.

Original entry on oeis.org

1, 1, 2, 3, 5, 4, 3, 7, 5, 6, 11, 17, 14, 31, 15, 23, 19, 21, 20, 41, 61, 51, 56, 107, 163, 135, 149, 142, 97, 239, 168, 37, 41, 39, 40, 79, 17, 48, 13, 61, 37, 49, 43, 46, 89, 45, 67, 56, 41, 97, 69, 83, 76, 53, 43, 48, 13

Offset: 1

Wouter Meeussen, Jul 25 2012

Similar to the Fibonacci recursion starting with (1, 1), but each new nonprime term is divided by its least prime factor. Sequence enters a loop of length 18 after 38 terms on reaching (48, 13).

Siobhan Roberts, Genius At Play: The Curious Mind of John Horton Conway, Bloomsbury, 2015, pages xx-xxi.

Peter Kagey, Table of n, a(n) for n = 1..250
Sara Barrows, Emily Noye, Sarah Uttormark, and Matthew Wright, Three's A Crowd: An Exploration of Subprime Tribonacci Sequences, College Math. J. (2023).
Richard K. Guy, Tanya Khovanova and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099 [math.NT], 2012-2014.
Tanya Khovanova, Conway’s Subprime Fibonacci Sequences, Math Blog, July 2012.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000045, A020639, A117339, A175723, A214551, A014682, etc.
Cf. A214892-A214898, A282812, A282813, A282814.
See also A165911, A272636, A255562, etc.

guyKhoSal[{a_, b_}] := Block[{c, l, r}, c = NestWhile[(p = Tr[Take[#, -2]]; If[PrimeQ[p], q = p, q = p/Part[FactorInteger[p, FactorComplete -> False], 1, 1]]; Flatten[{#, q}]) &, {a, b}, FreeQ[Partition[#1, 2, 1], Take[#2, -2]] &, 2, 1000]; l = Length[c]; r = Tr@Position[Partition[c,2,1], Take[c,-2], 1, 1]; l-r-1; c]; guyKhoSal[{1,1}]
f[s_List] := Block[{a = s[[-2]] + s[[-1]]}, If[ PrimeQ[a], Append[s, a], Append[s, a/FactorInteger[a][[1, 1]] ]]]; Nest[f, {1, 1}, 73] (* Robert G. Wilson v, Aug 09 2012 *)

fatw(n,a=[0,1],p=[])={for(i=2,n,my(f=factor(a[i]+a[i-1])~);for(k=1,#f,setsearch(p,f[1,k])&next;f[2,k]--;p=setunion(p,Set(f[1,k]));break);a=concat(a,factorback(f~)));a}
fatw(99) /* M. F. Hasler, Jul 25 2012 */

A177904 a(1)=a(2)=a(3)=1; thereafter a(n) = gpf(a(n-1)+a(n-2)+a(n-3)), where gpf = "greatest prime factor".

Original entry on oeis.org

1, 1, 1, 3, 5, 3, 11, 19, 11, 41, 71, 41, 17, 43, 101, 23, 167, 97, 41, 61, 199, 43, 101, 7, 151, 37, 13, 67, 13, 31, 37, 3, 71, 37, 37, 29, 103, 13, 29, 29, 71, 43, 13, 127, 61, 67, 17, 29, 113, 53, 13, 179, 7, 199, 11, 31, 241, 283, 37, 17, 337, 23, 29, 389, 7, 17, 59, 83, 53, 13, 149, 43, 41, 233, 317, 197, 83, 199, 479, 761, 1439, 47, 107

Offset: 1

N. J. A. Sloane, Dec 16 2010

After 86 steps, enters a cycle of length 212 (see A177923).

N. J. A. Sloane, Table of n, a(n) for n = 1..1000
G. Back and M. Caragiu, The greatest prime factor and recurrent sequences, Fib. Q., 48 (2010), 358-362.

Cf. A006530, A175723, A178174, A178095, A214320.

a177904 n = a177904_list !! (n-1)
a177904_list = 1 : 1 : 1 : (map a006530 $ zipWith (+)
   a177904_list (tail $ zipWith (+) a177904_list $ tail a177904_list))
-- Reinhard Zumkeller, Jul 24 2012

with(numtheory, divisors); A006530 := proc(n) local i, t1, t2, t3, t4, t5; t1 := divisors(n); t2 := convert(t1, list); t3 := sort(t2); t4 := nops(t3); t5 := 1; for i from 1 to t4 do if isprime(t3[t4+1-i]) then RETURN(t3[t4+1-i]); fi; od; 1; end;
M:=1000;
t1:=[1,1,1];
for n from 4 to M do
t1:=[op(t1),A006530(t1[n-1]+t1[n-2]+t1[n-3])]; od:
t1;

nxt[{a_,b_,c_}]:={b,c,FactorInteger[a+b+c][[-1,1]]}; NestList[nxt,{1,1,1},90][[All,1]] (* Harvey P. Dale, Jul 17 2017 *)

A214892 Conway's subprime Fibonacci sequence starting with (4,1).

Original entry on oeis.org

4, 1, 5, 3, 4, 7, 11, 9, 10, 19, 29, 24, 53, 11, 32, 43, 25, 34, 59, 31, 45, 38, 83, 11, 47, 29, 38, 67, 35, 51, 43, 47, 45, 46, 13, 59, 36, 19, 11, 15, 13, 14, 9, 23, 16, 13, 29, 21, 25, 23, 24, 47, 71, 59, 65, 62, 127, 63, 95, 79, 87, 83, 85, 84, 13, 97, 55, 76, 131, 69, 100, 13, 113

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with (4, 1), but each new nonprime term is divided by its least prime factor. Sequence enters a loop of length 136 after 8 terms on reaching (11, 9).

Wouter Meeussen, Table of n, a(n) for n = 1..144
Richard K. Guy, Tanya Khovanova and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099 [math.NT], 2012-2014.

Cf. A000045, A020639, A175723, A214551, A014682, A214674, A214892-A214898.

(* see A214674 *)
a[1] = 4; a[2] = 1; a[n_] := a[n] = If[an = a[n-2]+a[n-1]; PrimeQ[an], an, an/FactorInteger[an][[1, 1]]]; Array[a, 80] (* Jean-François Alcover, Nov 17 2018 *)

A214898 Conway's subprime Fibonacci sequence, largest loop elements.

Original entry on oeis.org

2, 827, 607, 239, 191, 5693, 347

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with a pair of positive integers, but each new nonprime term is divided by its least prime factor. Recursion enters a loop of length A214897(n), of which the largest element a(n) is prime (this sequence).

Richard K. Guy, Tanya Khovanova and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099v1 [math.NT]

Cf. A000045, A020639, A175723, A214551, A014682, A214674, A214892-A214898.

Mathematica
```
see A214674
```

A177923 a(1)=19, a(2)=13, a(3)=37; thereafter a(n) = gpf(a(n-1)+a(n-2)+a(n-3)), where gpf = "greatest prime factor".

Original entry on oeis.org

19, 13, 37, 23, 73, 19, 23, 23, 13, 59, 19, 13, 13, 5, 31, 7, 43, 3, 53, 11, 67, 131, 19, 31, 181, 11, 223, 83, 317, 89, 163, 569, 821, 1553, 109, 191, 109, 409, 709, 409, 509, 1627, 509, 23, 127, 659, 809, 29, 499, 191, 719, 1409, 773, 967, 67, 139, 23, 229, 23, 11, 263, 11, 19, 293, 19, 331, 643, 331, 29, 59, 419, 13, 491, 71, 23, 13, 107, 13, 19, 139, 19, 59, 31, 109, 199, 113, 421, 733, 181, 89, 59, 47, 13, 17, 11, 41, 23, 5, 23, 17, 5, 5, 3, 13, 7, 23, 43, 73, 139, 17, 229, 11, 257, 71, 113, 7, 191, 311, 509, 337

Offset: 1

N. J. A. Sloane, Dec 18 2010

nonn

This is the periodic part of A177904 - it is periodic with period 212.
A smaller start is a(1)=3, a(2)=13, a(3)=7, but that would not produce the terms in the order of their first appearance in A177904.
There are several open questions concerning this class of sequences - see the Back-Caragiu reference in A177904.

N. J. A. Sloane, Table of n, a(n) for n = 1..212

Cf. A006530, A177904, A175723.

nxt[{a_,b_,c_}]:={b,c,FactorInteger[a+b+c][[-1,1]]}; NestList[nxt,{19,13,37},120][[All,1]] (* Harvey P. Dale, Dec 11 2018 *)

A178095 a(1)=a(2)=a(3)=1; thereafter a(n) = lpf(a(n-1)+a(n-2)+a(n-3)), where lpf = "least prime factor".

Original entry on oeis.org

1, 1, 1, 3, 5, 3, 11, 19, 3, 3, 5, 11, 19, 5, 5, 29, 3, 37, 3, 43, 83, 3, 3, 89, 5, 97, 191, 293, 7, 491, 7, 5, 503, 5, 3, 7, 3, 13, 23, 3, 3, 29, 5, 37, 71, 113, 13, 197, 17, 227, 3, 13, 3, 19, 5, 3, 3, 11, 17, 31, 59, 107, 197, 3, 307, 3, 313, 7, 17, 337, 19, 373, 3, 5, 3, 11, 19, 3, 3, 5, 11, 19, 5, 5, 29, 3, 37, 3, 43, 83, 3, 3, 89, 5, 97, 191, 293, 7, 491, 7

Offset: 1

N. J. A. Sloane, Dec 16 2010

nonn

Has period 69, starting with the fourth term: 3, 5, 3, 11, 19, 3, 3, 5, 11, ...

Cf. A178094, A020639, A175723, A177904.

nxt[{a_,b_,c_}]:={b,c,FactorInteger[a+b+c][[1,1]]}; Transpose[ NestList[ nxt,{1,1,1},100]][[1]] (* Harvey P. Dale, Aug 19 2014 *)

A180101 a(0)=0, a(1)=1; thereafter a(n) = largest prime factor of sum of all previous terms.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41

Offset: 0

N. J. A. Sloane, Jan 16 2011

nonn

More precisely, a(n) = A006530 applied to sum of previous terms.
Inspired by A175723.
Except for initial terms, same as A076272, but the simple definition warrants an independent entry.

Cf. A006530, A076272, A175723, A180107 (partial sums).

For the purposes of this paragraph, regard 0 as the (-1)st prime and 1 as the 0th prime. Conjectures: All primes appear; the primes appear in increasing order; the k-th prime p(k) appears p(k+1)-p(k-1) times (cf. A031131); and p(k) appears for the first time at position A164653(k) (sums of two consecutive primes). These assertions are stated as conjectures only because I have not written out a formal proof, but they are surely true.

A178094 a(1)=a(2)=1; thereafter a(n) = lpf(a(n-1)+a(n-2)), where lpf = "least prime factor".

Original entry on oeis.org

1, 1, 2, 3, 5, 2, 7, 3, 2, 5, 7, 2, 3, 5, 2, 7, 3, 2, 5, 7, 2, 3, 5, 2, 7, 3, 2, 5, 7, 2, 3, 5, 2, 7, 3, 2, 5, 7, 2, 3, 5, 2, 7, 3, 2, 5, 7, 2, 3, 5, 2, 7, 3, 2, 5, 7, 2, 3, 5, 2, 7, 3, 2, 5, 7, 2, 3, 5, 2, 7, 3, 2, 5, 7, 2, 3, 5, 2, 7, 3, 2, 5, 7, 2, 3, 5, 2, 7, 3, 2, 5, 7, 2, 3, 5, 2, 7, 3, 2, 5, 7, 2, 3, 5, 2, 7, 3, 2, 5, 7

Offset: 1

N. J. A. Sloane, Dec 16 2010

nonn

Cycles with a period of length 9.

Cf. A000045, A175723, A020639, A178095.

nxt[{a_,b_}]:={b,FactorInteger[a+b][[1,1]]}; NestList[nxt,{1,1},110][[;;,1]] (* or *) PadRight[ {1,1},110,{5,7,2,3,5,2,7,3,2}] (* Harvey P. Dale, May 29 2023 *)

A178174 a(1)=a(2)=a(3)=a(4)=1; thereafter a(n) = gpf(a(n-1)+a(n-2)+a(n-3)+a(n-4)), where gpf is the greatest prime factor.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 3, 11, 7, 13, 17, 3, 5, 19, 11, 19, 3, 13, 23, 29, 17, 41, 11, 7, 19, 13, 5, 11, 3, 2, 7, 23, 7, 13, 5, 3, 7, 7, 11, 7, 2, 3, 23, 7, 7, 5, 7, 13, 2, 3, 5, 23, 11, 7, 23, 2, 43, 5, 73, 41, 3, 61, 89, 97, 5, 7, 11, 5, 7, 5, 7, 3, 11, 13, 17, 11, 13, 3, 11, 19, 23, 7, 5, 3, 19, 17, 11, 5, 13, 23, 13, 3, 13, 13, 7, 3, 3, 13, 13, 2, 31, 59, 7, 11, 3, 5, 13, 2, 23, 43, 3, 71, 7, 31

Offset: 1

N. J. A. Sloane, Dec 18 2010

After 133 steps, enters a cycle of length 14.

Greg Back and Mihai Caragiu, The Greatest Prime Factor and Recurrent Sequences, Fibonacci Quart. 48 (2010), no. 4, 358-362.

Cf. A006530, A175723, A177904.

nxt[{a_,b_,c_,d_}]:={b,c,d,FactorInteger[a+b+c+d][[-1,1]]}; Transpose[ NestList[ nxt,{1,1,1,1},120]][[1]] (* Harvey P. Dale, Sep 24 2013 *)

cp's OEIS Frontend

A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

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A214674 Conway's subprime Fibonacci sequence.

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A177904 a(1)=a(2)=a(3)=1; thereafter a(n) = gpf(a(n-1)+a(n-2)+a(n-3)), where gpf = "greatest prime factor".

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A214892 Conway's subprime Fibonacci sequence starting with (4,1).

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A214898 Conway's subprime Fibonacci sequence, largest loop elements.

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A177923 a(1)=19, a(2)=13, a(3)=37; thereafter a(n) = gpf(a(n-1)+a(n-2)+a(n-3)), where gpf = "greatest prime factor".

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A178095 a(1)=a(2)=a(3)=1; thereafter a(n) = lpf(a(n-1)+a(n-2)+a(n-3)), where lpf = "least prime factor".

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A180101 a(0)=0, a(1)=1; thereafter a(n) = largest prime factor of sum of all previous terms.

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