A177904 -id:A177904 - cp's OEIS frontend

A178179 Like A177904, but start with a(1)=0, a(2)=a(3)=1.

0, 1, 1, 2, 2, 5, 3, 5, 13, 7, 5, 5, 17, 3, 5, 5, 13, 23, 41, 11, 5, 19, 7, 31, 19, 19, 23, 61, 103, 17, 181, 43, 241, 31, 7, 31, 23, 61, 23, 107, 191, 107, 5, 101, 71, 59, 11, 47, 13, 71, 131, 43, 7, 181, 11, 199, 23, 233, 13, 269, 103, 11, 383, 71, 31, 97, 199, 109, 5, 313, 61, 379, 251, 691, 1321, 73, 139, 73, 19, 11, 103, 19, 19, 47, 17, 83, 7, 107, 197, 311, 41, 61, 59, 23, 13, 19, 11, 43, 73, 127, 3, 29, 53, 17, 11, 3, 31

Offset: 1

N. J. A. Sloane, Dec 19 2010

nonn

After 7 steps, enters a cycle of length 100. The complete cycle, ending at 31, is shown.

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

Cf. A177904.

a[1] = 0; a[2] = a[3] = 1; a[n_] := a[n] = FactorInteger[a[n - 1] + a[n - 2] + a[n - 3]][[-1, 1]]; Array[a, 107] (* Robert G. Wilson v, Nov 17 2014 *)
nxt[{a_,b_,c_}]:={b,c,FactorInteger[a+b+c][[-1,1]]}; NestList[nxt,{0,1,1},110][[All,1]] (* Harvey P. Dale, Jul 17 2017 *)

A175723 a(1)=a(2)=1; thereafter a(n) = gpf(a(n-1)+a(n-2)), where gpf = "greatest prime factor".

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 16 2010

nonn

Rapidly enters a loop with period 3,5,2,7.

More generally, if a(1) and a(2) are distinct positive numbers with a(1)+a(2) >= 2, the sequence eventually enters the cycle {7,3,5,2} [Back and Caragiu].

G. Back and M. Caragiu, The greatest prime factor and recurrent sequences, Fib. Q., 48 (2010), 358-362.

Cf. A000045, A006530, A020639.

Similar or related sequences: A177904, A177923, A178094, A178095, A178174, A178179, A180101, A180107, A221183.

nxt[{a_,b_}]:={b,FactorInteger[a+b][[-1,1]]}; Transpose[NestList[nxt,{1,1},120]][[1]] (* or *) PadRight[{1,1,2},130,{5,2,7,3}] (* Harvey P. Dale, Feb 24 2015 *)

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 *)

A214320 a(0)=a(1)=a(2)=1; thereafter a(n) = gpf(a(n-1)+a(n-3)), where gpf = "greatest prime factor".

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 22 2012

nonn

Suggested by A214551 and A177904.

Cf. A000930, A006530, A177904, A214320.

a214320 n = a214320_list !! n
a214320_list = 1 : 1 : 1 : (map a006530 $
   zipWith (+) a214320_list (drop 2 $ a214320_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;
f:=proc(n) option remember; if n <= 2 then 1 else A006530(f(n-1)+f(n-3)); fi; end;
[seq(f(n),n=0..120)];

nxt[{a_,b_,c_}]:={b,c,FactorInteger[c+a][[-1,1]]}; NestList[nxt,{1,1,1},120][[All,1]] (* or *) PadRight[{1,1,1,2,3,2,2},130,{3,2,3,3,5,2,5,5,7}] (* Harvey P. Dale, Jul 08 2017 *)

After 7 terms, cycles with period 9.

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 *)

A282098 a(n) = A006530(a(n-1)) + A006530(a(n-2)) + A006530(a(n-3)) with a(0) = a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 11, 19, 33, 41, 71, 123, 153, 129, 101, 161, 167, 291, 287, 305, 199, 301, 303, 343, 151, 259, 195, 201, 117, 93, 111, 81, 71, 111, 111, 145, 103, 169, 145, 145, 71, 129, 143, 127, 183, 201, 255, 145, 113, 159, 195, 179, 245, 199, 385, 217, 241, 283, 555, 561, 337, 391, 377, 389, 441

Offset: 0

Altug Alkan, Feb 06 2017

A287051 a(0) = 0, a(1) = 1; a(2n) = gpf(a(n)), a(2n+1) = a(n) + a(n+1), where gpf(a(n)) is the greatest prime dividing a(n) for a(n) >= 2 and 1 if a(n) = 1 (A006530).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 2, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 2, 5, 1, 6, 5, 7, 2, 9, 7, 10, 3, 11, 2, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 2, 11, 3, 10, 7, 9, 2, 7, 5, 6, 1, 7, 3, 11, 5, 12, 7, 9, 2, 11, 3, 16, 7, 17, 5, 13, 3, 14, 11, 13, 2, 15, 13, 18, 5, 17, 3, 19, 7, 16, 3, 11, 2, 11, 3, 16, 7

Offset: 0

Ilya Gutkovskiy, May 18 2017

nonn

A variation on Stern's diatomic sequence.

			a(0) = 0;
a(1) = 1;
a(2) = a(2*1) = gpf(a(1)) = 1;
a(3) = a(2*1+1) = a(1) + a(2) = 2;
a(4) = a(2*2) = gpf(a(2)) = 1;
a(5) = a(2*2+1) = a(2) + a(3) = 3;
a(6) = a(2*3) = gpf(a(3)) = 2, etc.

Michael Gilleland, Some Self-Similar Integer Sequences
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Eric Weisstein's World of Mathematics, Stern's Diatomic Series
Index entries for sequences related to Stern's sequences

Cf. A002487, A006530, A175723, A177904.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], FactorInteger[a[n/2]][[-1, 1]], a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 100}]

A282184 a(n) = A034699(a(n-1) + a(n-2) + a(n-3)) with a(0) = a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 17, 31, 19, 67, 13, 11, 13, 37, 61, 37, 27, 125, 27, 179, 331, 179, 53, 563, 53, 223, 839, 223, 257, 1319, 257, 47, 541, 169, 757, 163, 121, 347, 631, 157, 227, 29, 59, 9, 97, 11, 13, 121, 29, 163, 313, 101, 577, 991, 1669, 83, 211, 151, 89, 41, 281, 137, 27, 89, 23, 139, 251, 59, 449, 23, 59, 59

Offset: 0

Altug Alkan, Feb 08 2017

Sequence is cyclical, beginning with terms a(255)-a(257) = {19,13,43} with a period of 306 terms. - Michael De Vlieger, Feb 08 2017

			a(5) = 9 because A034699(1 + 3 + 5) = A034699(9) = 9.

Cf. A000213, A034699, A177904, A282098.

a = {1, 1, 1}; Do[AppendTo[a, If[# == 1, 1, Max[Power @@@ FactorInteger@ #]] &@ Total@ {a[[i - 3]], a[[i - 2]], a[[i - 1]]}], {i, 4, 72}]; a (* Michael De Vlieger, Feb 08 2017 *)

lappf(n) = my(f=factor(n)); vecmax(vector(#f[, 1], i, f[i, 1]^f[i, 2]));
lista(nn) = {x = 1; y = 1; z = 1; print1(x, ", ", y, ", ", z, ", "); for (n=4, nn, t = lappf(x+y+z); print1(t, ", "); x = y; y = z; z = t;);} \\ Michel Marcus, Feb 10 2017

cp's OEIS Frontend

A178179 Like A177904, but start with a(1)=0, a(2)=a(3)=1.

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A175723 a(1)=a(2)=1; thereafter a(n) = gpf(a(n-1)+a(n-2)), where gpf = "greatest prime factor".

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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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A214320 a(0)=a(1)=a(2)=1; thereafter a(n) = gpf(a(n-1)+a(n-3)), where gpf = "greatest prime factor".

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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.

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A282098 a(n) = A006530(a(n-1)) + A006530(a(n-2)) + A006530(a(n-3)) with a(0) = a(1) = a(2) = 1.

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A287051 a(0) = 0, a(1) = 1; a(2*n) = gpf(a(n)), a(2*n+1) = a(n) + a(n+1), where gpf(a(n)) is the greatest prime dividing a(n) for a(n) >= 2 and 1 if a(n) = 1 (A006530).

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A282184 a(n) = A034699(a(n-1) + a(n-2) + a(n-3)) with a(0) = a(1) = a(2) = 1.

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A287051 a(0) = 0, a(1) = 1; a(2n) = gpf(a(n)), a(2n+1) = a(n) + a(n+1), where gpf(a(n)) is the greatest prime dividing a(n) for a(n) >= 2 and 1 if a(n) = 1 (A006530).