A123581 -id:A123581 - cp's OEIS frontend

A367624 2-adic valuation of A123581(2*n).

1, 2, 2, 1, 1, 3, 1, 3, 1, 2, 2, 1, 4, 1, 4, 1, 1, 2, 1, 2, 1, 3, 1, 2, 3, 1, 2, 1, 1, 5, 1, 2, 5, 1, 2, 1, 3, 1, 3, 1, 1, 2, 1, 2, 1, 4, 1, 2, 4, 1, 2, 1, 3, 1, 1, 3, 1, 2, 2, 1, 6, 1, 1, 6, 1, 2, 1, 2, 1, 3, 3, 1, 2, 1, 1, 2, 1, 4, 1, 4, 1, 2, 1, 3, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1

Offset: 1

N. J. A. Sloane, Dec 06 2023

nonn

A123581(n) is odd if n is odd.

N. J. A. Sloane, Table of n, a(n) for n = 1..5000

Cf. A007814, A123581.

With[{nmax=100},IntegerExponent[NestList[#+FactorInteger[#][[-1,1]]&,3,2nmax-1][[2;;-1;;2]],2]] (* Paolo Xausa, Dec 07 2023 *)

A070229 Next m>n such that m is divisible by lpf(n), lpf=A006530 largest prime factor.

Original entry on oeis.org

2, 4, 6, 6, 10, 9, 14, 10, 12, 15, 22, 15, 26, 21, 20, 18, 34, 21, 38, 25, 28, 33, 46, 27, 30, 39, 30, 35, 58, 35, 62, 34, 44, 51, 42, 39, 74, 57, 52, 45, 82, 49, 86, 55, 50, 69, 94, 51, 56, 55, 68, 65, 106, 57, 66, 63, 76, 87, 118, 65, 122

Offset: 1

Reinhard Zumkeller, May 07 2002

Or, for n>1, n + (largest prime divisor of n). [Anne Robinson, daughter of Herman P. Robinson, Oct 08 1981]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981

Cf. A036441.
Cf. A006530, A061228, A256393.
Iterations: A076271 (start=1), A036441 (start=2), A123581 (start=3).

a070229 n = n + a006530 n
-- Reinhard Zumkeller, Nov 08 2015, Nov 14 2011

Table[n + Last@(First/@FactorInteger[n]), {n, 70}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)

vector(100, n, if(n>1, vecmax(factor(n)[, 1]), 1) + n) \\ Altug Alkan, Nov 13 2015

a(n) = (n/lpf(n)+1)*lpf(n).

a(n) = n+lpf(n) where lpf=A006530 largest prime factor. For example, a(14)=14+7=21 (instead of ((14/7)+1)*7). - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Jun 14 2007

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

Original entry on oeis.org

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

A367504 a(1) = 2; for n > 1, a(n) = a(n-1) + 2*gpf(a(n-1)), where gpf(k) = A006530(k) = greatest prime dividing k.

Original entry on oeis.org

2, 6, 12, 18, 24, 30, 40, 50, 60, 70, 84, 98, 112, 126, 140, 154, 176, 198, 220, 242, 264, 286, 312, 338, 364, 390, 416, 442, 476, 510, 544, 578, 612, 646, 684, 722, 760, 798, 836, 874, 920, 966, 1012, 1058, 1104, 1150, 1196, 1242, 1288, 1334, 1392, 1450, 1508, 1566, 1624, 1682, 1740, 1798

Offset: 1

Scott R. Shannon, Nov 21 2023

nonn

Conjecture: with the requirement that the prime factorization of each term is written so that the primes are ordered from smallest to largest, the sequence is the lexicographically earliest infinite sequence of distinct positive numbers such that gpf(a(n-1)) * lpf(a(n)) = |a(n) - a(n-1)|, where gpf(k) = A006530(k) = greatest prime factor of k and lpf(k) = A020639(k) = least prime factor of k. In this way the sequence is the ordered prime factorization version of the 'Commas sequence', A121805. One can show that, for such a sequence to be infinite, no odd number can appear. Although for many terms a lower even number can be chosen for the following term, which can lead to even lower numbers for further terms, it is conjectured all such choices will ultimately halt the sequence as a number is eventually reached for which no unused next number exists which follows the required rule for the difference between the terms. Therefore all terms must be larger than the previous, and the earliest such infinite sequence is the given sequence.

See A367465 for the sequence when the requirement that the primes in the factorization of each term must be in order is removed.

			a(7) = 40 as a(6) = 30 = 2*3*5, thus A006530(30) = 5 and a(7) = a(6) + 2*5 = 30 + 2*5 = 40.

Cf. A027746, A006530, A020639, A367465, A121805, A123581, A180107, A036441, A076271.

cp's OEIS Frontend

A367624 2-adic valuation of A123581(2*n).

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A070229 Next m>n such that m is divisible by lpf(n), lpf=A006530 largest prime factor.

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

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A367504 a(1) = 2; for n > 1, a(n) = a(n-1) + 2*gpf(a(n-1)), where gpf(k) = A006530(k) = greatest prime dividing k.

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