A270807 -id:A270807 - cp's OEIS frontend

A270808 First differences of A270807, divided by 2.

1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 5, 3, 1, 5, 1, 8, 4, 5, 1, 2, 3, 6, 4, 2, 3, 1, 5, 1, 8, 1, 2, 6, 1, 2, 3, 1, 2, 4, 23, 1, 11, 3, 4, 2, 1, 23, 10, 2, 18, 1, 5, 1, 8, 6, 1, 8, 7, 23, 1, 2, 1, 5, 7, 2, 1, 2, 6, 18, 1, 2, 25, 2, 1, 32, 1, 17, 7, 2, 1, 2, 10, 14, 3, 7, 8

Offset: 0

N. J. A. Sloane, Apr 05 2016

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A270807.

from _future_ import division
from sympy import primefactors
A270808_list, a = [], 1
for i in range(10001):
    b = a//(max(primefactors(a)+[1])) + 1
    A270808_list.append(b//2)
    a += b # Chai Wah Wu, Apr 06 2016

A271418 Define a sequence by b(1) = n, b(m) = b(m-1) + b(m-1)/A052126(m) + 1; then a(n) is the least m such that b(m) belongs to A270807.

Original entry on oeis.org

1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 4, 1, 2, 3, 4, 1, 3, 1, 4, 2, 4, 1, 4, 1, 4, 4, 4, 3, 2, 2, 3, 2, 2, 1, 4, 2, 2, 2, 2, 1, 4, 1, 3, 3, 2, 3, 2, 2, 3, 2, 3, 1, 4, 1

Offset: 1

Cody M. Haderlie, Apr 06 2016

			the sequence where b(1) = 2 is 2, 4, 7, 9, 13, 15, ...
7 is shared with A270807
there are 3 terms in 2, 4, 7; a(2) = 3

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A052126, A269304, A270807.

f[n_] := n + n/FactorInteger[n][[-1, 1]] + 1; s = NestList[f, 1, 10^2]; Table[Length@ NestWhileList[f, n, ! MemberQ[s, #] &], {n, 120}] (* Michael De Vlieger, Apr 08 2016 *)

a(75) corrected by Chai Wah Wu, Apr 06 2016

A269304 a(n) = n + n/gpf(n) + 1, where gpf(n) is the greatest prime factor of n or 1 if n = 1.

Original entry on oeis.org

3, 4, 5, 7, 7, 9, 9, 13, 13, 13, 13, 17, 15, 17, 19, 25, 19, 25, 21, 25, 25, 25, 25, 33, 31, 29, 37, 33, 31, 37, 33, 49, 37, 37, 41, 49, 39, 41, 43, 49, 43, 49, 45, 49, 55, 49, 49, 65, 57, 61, 55, 57, 55, 73, 61, 65, 61, 61, 61, 73, 63, 65, 73, 97, 71, 73, 69

Offset: 1

Cody M. Haderlie, Feb 22 2016

a(n) is odd except when n=2.
Initially, a(n) is frequently a square or a prime.
It is conjectured that any two sequences generated with a(n)=a(n-1)+a(n-1)/gpf(a(n-1))+1 and any initial value >=1 will have a finite number of non-shared terms and an infinite number of shared terms after one initial shared term (see A270807). Example: For a(1)=314, the sequence generated is 314, 317, 319, 331, 333, 343, 393, 397, 399, 421, 423, 433, ...; for a(1)=97, the sequence generated is 97, 99, 109, 111, 115, 121, 133, 141, 145, 151, 153, 163, 165, 181, 183, 187, 199, 201, 205, 211, 213, 217, 225, 271, 273, 295, 301, 309, 313, 315, 361, 381, 385, 421, 423, 433, ...; these sequences have respectively 9 and 33 terms not shared with the other until both reach 421; the following terms of both sequences are identical.

			For n=18765, a(n)=18901.
For n=196, a(n)=225 (225 is a square).
For n=103156, a(n)=105673 (105673 is prime).

Cody M. Haderlie, Table of n, a(n) for n = 1..10000

Cf. A006530, A052126, A270807.

Table[n+n/FactorInteger[n][[-1,1]]+1,{n,100}]

gpf(n)=if(n>1, my(f=factor(n)[,1]); f[#f], 1)
a(n)=n + n/gpf(n) + 1 \\ Charles R Greathouse IV, Feb 22 2016

a(n) = n + n/A006530(n) + 1.
a(n) = n + A052126(n) + 1.
a(p) = p+2 for p prime.

A330437 Length of trajectory of n under the map n -> n - 1 + n/gpf(n) or 0 if no fixed point is reached, where gpf(n) is the greatest prime factor of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 5, 4, 2, 1, 4, 1, 2, 4, 4, 3, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 3, 2, 2, 3, 1, 2, 2, 3, 2, 2, 1, 2, 1, 3, 2, 4, 3, 2, 1, 2, 2, 2, 1, 4, 1, 3, 2, 2, 2, 2, 1, 4, 2, 2, 1, 4, 2, 3, 2, 4, 1, 2, 2, 4, 4, 4, 3, 2, 1, 3, 2, 4

Offset: 1

Elijah Beregovsky, Feb 16 2020

nonn

The table of trajectories of n under is given in A329288.
All fixed points, besides 1, are prime.
Conjecture: every number appears in the sequence infinitely many times.
Conjecture: all terms are nonzero, i.e., every trajectory eventually reaches a prime.

			For n = 26 the trajectory is (26, 27, 35, 39, 41) so a(26) = 5.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006530 (greatest prime factor), A329288, A330704 (greedy inverse).

Cf. A270807, A120685, A331410.

g:= n -> n - 1 + n/max(numtheory:-factorset(n)):
f:= proc(n) option remember;
    if isprime(n) then 1 else 1+ procname(g(n)) fi
end proc:
f(1):= 1:
map(f, [$1..200]); # Robert Israel, May 01 2020

Clear[f,it,order,seq]; f[n_]:=f[n]=n-1+n/FactorInteger[n][[-1]][[1]]; it[k_,n_]:=it[k,n]=f[it[k,n-1]]; it[k_,1]=k; order[n_]:=order[n]=SelectFirst[Range[1,100], it[n,#]==it[n,#+1]&]; Print[order/@Range[1,100]];

apply( {a(n,c=1)=n>1&&while(nM. F. Hasler, Feb 19 2020

a(p) = 1 for any prime number p.

cp's OEIS Frontend

A270808 First differences of A270807, divided by 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

A271418 Define a sequence by b(1) = n, b(m) = b(m-1) + b(m-1)/A052126(m) + 1; then a(n) is the least m such that b(m) belongs to A270807.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A269304 a(n) = n + n/gpf(n) + 1, where gpf(n) is the greatest prime factor of n or 1 if n = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A330437 Length of trajectory of n under the map n -> n - 1 + n/gpf(n) or 0 if no fixed point is reached, where gpf(n) is the greatest prime factor of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula