A070229 -id:A070229 - cp's OEIS frontend

A036441 a(n+1) = next number having largest prime dividing a(n) as a factor, with a(1) = 2.

2, 4, 6, 9, 12, 15, 20, 25, 30, 35, 42, 49, 56, 63, 70, 77, 88, 99, 110, 121, 132, 143, 156, 169, 182, 195, 208, 221, 238, 255, 272, 289, 306, 323, 342, 361, 380, 399, 418, 437, 460, 483, 506, 529, 552, 575, 598, 621, 644, 667, 696, 725, 754, 783, 812, 841, 870

Offset: 1

Frederick Magata (frederick.magata(AT)uni-muenster.de)

a(n) satisfies the following inequality: (1/4)*(n^2 + 3*n + 1) <= a(n) <= (1/4)*(n+2)^2. [Corrected by M. F. Hasler, Apr 08 2015]
The present sequence is the special case a(n) = a(2,n) with a more general a(m, n) := a(m, n-1) + gpf(a(m, n-1)), a(m, 1) := m, where gpf(x) := "greatest prime factor of x" = A006530(x). Also a(a(r,k), n) = a(r,n+k-1), for all n,k in N\{0} and all r in N\{0,1}; a(prime(k), n) = a(prime(i), n + prime(k) - prime(i)), for all k,i,n in N\{0}, with k >= i, n >= prime(k-1) and with prime(x) := x-th prime.
Essentially the same as A076271 and A180107, cf. formula.

			a(2,2) = 4 because 2 + gpf(2) = 2 + 2 = 4;
a(2,3) = 6 because 4 + gpf(4) = 4 + 2 = 6.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A006530. See A076271 and A180107 for other versions.
Cf. A001043, A070229.
Cf. A123581.
Partial sums of A076973.

a036441 n = a036441_list !! (n-1)
a036441_list = tail a076271_list
-- Reinhard Zumkeller, Nov 08 2015, Nov 14 2011

f[n_]:=Last[First/@FactorInteger[n]];Join[{a=2},Table[a+=f[a],{n,2,100}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011*)
NestList[#+FactorInteger[#][[-1,1]]&,2,60] (* Harvey P. Dale, Dec 02 2012 *)

a(n)=(n+2-if(n\2+1<(p=nextprime(n\2+1))&&n+1M. F. Hasler, Apr 08 2015

a(n) = p(m)*(n+2-p(m)), where p(k) is the k-th prime and m is the smallest index such that n+2 <= p(m) + p(m+1). - Max Alekseyev, Oct 21 2008
a(n) = A076271(n+1) = A180107(n+2). - M. F. Hasler, Apr 08 2015
a(n+1) = A070229(a(n)). - Reinhard Zumkeller, Nov 07 2015

Better description from Reinhard Zumkeller, Feb 04 2002

Edited by M. F. Hasler, Apr 08 2015

A123581 a(1) = 3, a(n) = a(n-1) + greatest prime factor of a(n-1).

Original entry on oeis.org

3, 6, 9, 12, 15, 20, 25, 30, 35, 42, 49, 56, 63, 70, 77, 88, 99, 110, 121, 132, 143, 156, 169, 182, 195, 208, 221, 238, 255, 272, 289, 306, 323, 342, 361, 380, 399, 418, 437, 460, 483, 506, 529, 552, 575, 598, 621, 644, 667, 696, 725, 754, 783, 812, 841, 870

Offset: 1

Ben Paul Thurston, Nov 12 2006

			a(16) = 88 because a(15) is 77 whose largest prime factor is 11 so 77 + 11 = 88.

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

Essentially the same as A036441 and A076271.

Cf. A070229.

a123581 n = a123581_list !! (n-1)
a123581_list = iterate a070229 3  -- Reinhard Zumkeller, Nov 07 2015

A123581:= proc(n) option remember;
local t;
t:= procname(n-1);
t + max(numtheory[factorset](t));
end proc;
A123581(1):= 3;
seq(A123581(n),n=1..100); # Robert Israel, May 18 2014

a[1] = 3; a[n_] := a[n] = a[n - 1] + FactorInteger[a[n - 1]][[ -1, 1]]; Array[a, 56] (* Robert G. Wilson v *)

{print1(a=3,",");for(n=2,57,print1(a=a+vecmax(factor(a)[,1]),","))} \\ Klaus Brockhaus, Nov 19 2006

a(n+1) = A070229(a(n)). - Reinhard Zumkeller, Nov 07 2015

More terms from Robert G. Wilson v and Klaus Brockhaus, Nov 18 2006

A260648 Number of distinct prime divisors p of the n-th composite number c such that gpf(c - p) = p, where gpf = greatest prime factor (A006530).

Original entry on oeis.org

1, 2, 0, 1, 2, 1, 1, 2, 0, 1, 1, 2, 1, 0, 1, 1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 2, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 2, 1, 2, 1, 0, 1, 1, 0, 2, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 0, 2, 1, 1, 1, 2, 0, 0, 2, 0, 1, 1, 2, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 0, 0, 1, 3

Offset: 1

Gionata Neri, Nov 12 2015

nonn

a(n) gives the number of times that the n-th composite number occurs in A070229.

			a(8) = 2 since the distinct prime divisors of A002808(8) = 15 are 3 and 5, A006530(15 - 3) = 3 and A006530(15 - 5) = 5, so all prime 3 and 5 are to be considered.

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

Cf. A002808 (composite), A006530 (gpf).

N:= 1000: # to consider composites <= N
f:= proc(c) local p, t;
   if isprime(c) then return NULL fi;
   nops(select(p -> max(numtheory:-factorset(c/p-1))<=p, numtheory:-factorset(c)))
end proc:
map(f, [$4..N]); # Robert Israel, May 02 2017

a(87) corrected by Robert Israel, May 02 2017

cp's OEIS Frontend

A036441 a(n+1) = next number having largest prime dividing a(n) as a factor, with a(1) = 2.

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A123581 a(1) = 3, a(n) = a(n-1) + greatest prime factor of a(n-1).

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A260648 Number of distinct prime divisors p of the n-th composite number c such that gpf(c - p) = p, where gpf = greatest prime factor (A006530).

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Examples

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Crossrefs

Programs

Maple

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