A075365 -id:A075365 - cp's OEIS frontend

A175856 a(n) = a(n-1) - 1 if n is composite, a(n) = n otherwise.

1, 2, 3, 2, 5, 4, 7, 6, 5, 4, 11, 10, 13, 12, 11, 10, 17, 16, 19, 18, 17, 16, 23, 22, 21, 20, 19, 18, 29, 28, 31, 30, 29, 28, 27, 26, 37, 36, 35, 34, 41, 40, 43, 42, 41, 40, 47, 46, 45, 44, 43, 42, 53, 52, 51, 50, 49, 48, 59, 58, 61, 60, 59, 58, 57, 56, 67, 66

Offset: 1

Jaroslav Krizek, Sep 29 2010

a(n) = n for noncomposite n, a(n) = 2 * (previous prime (n)) - n for composite n.
See A175859 - absent positive integers (pairs of consecutive numbers) in sequence: 8, 9, 14, 15, 24, 25, 32, 33, ...
a(n) = A075365(n) except at n = 1 and n = 10. - Alexandre Herrera, Nov 11 2023

			a(7) = 7 (7 is a noncomposite number), a(8) = 2*7 - 8 = 6 (8 is composite).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002808, A008578, A175857, A175858, A175859, A175860, A175861, A175862, A075365.

a:= proc(n) option remember;
      `if`(n=1 or isprime(n), n, a(n-1)-1)
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jun 22 2021

a[1] := 1; a[n_] := a[n] = If[PrimeQ[n], n, a[n - 1] - 1] (* Ben Branman, Jan 02 2011 *)
nxt[{n_,a_}]:={n+1,If[CompositeQ[n+1],a-1,n+1]}; NestList[nxt,{1,1},70][[All,2]] (* Harvey P. Dale, Jan 01 2023 *)

a(n)=if(n==1, 1, 2*precprime(n)-n) \\ Charles R Greathouse IV, Apr 22 2022~

Corrected by Jaroslav Krizek, Oct 01 2010

A075366 Smallest product (n+1)(n+2)...(n+k) that is divisible by the product of all the primes up to n.

Original entry on oeis.org

1, 12, 120, 30, 30240, 5040, 17297280, 2162160, 240240, 360360, 28158588057600, 2346549004800, 64764752532480000, 4626053752320000, 308403583488000, 19275223968000, 830034394580628357120000, 46113021921146019840000

Offset: 1

Amarnath Murthy, Sep 20 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..300

Cf. A075365, A075367, A075368.

Cf. A000040.

a075366 n = a075366_list !! (n-1)
a075366_list = 1 : f 2 1 a000040_list where
   f x pp ps'@(p:ps)
     | p <= x    = f x (p * pp) ps
     | otherwise = g $ dropWhile (< pp) $ scanl1 (*) [x+1, x+2 ..]
     where g (z:zs) | mod z pp == 0 = z : f (x + 1) pp ps'
                    | otherwise     = g zs
-- Reinhard Zumkeller, May 18 2015

a75365[n_] := Module[{div, k, pr}, div=Times@@Prime/@Range[PrimePi[n]]; For[k=0; pr=1, True, k++; pr*=n+k, If[Mod[pr, div]==0, Return[k]]]]; a[n_] := Times@@Range[n+1, n+a75365[n]]

If p <= n < q, where p and q are consecutive primes, then a(n) = (2p)!/n!, unless n=10.

Edited by Dean Hickerson, Oct 28 2002

A075367 Smallest value of lcm(n+1, n+2, ..., n+k) (for k >= 0) that is divisible by the product of all the primes up to n.

Original entry on oeis.org

1, 12, 60, 30, 2520, 2520, 360360, 180180, 60060, 60060, 232792560, 232792560, 26771144400, 26771144400, 26771144400, 13385572200, 144403552893600, 144403552893600, 5342931457063200, 5342931457063200, 5342931457063200

Offset: 1

Amarnath Murthy, Sep 20 2002

T. D. Noe, Table of n, a(n) for n = 1..200

Cf. A075365, A075366, A075368.

a75365[n_] := Module[{div, k, pr}, div=Times@@Prime/@Range[PrimePi[n]]; For[k=0; pr=1, True, k++; pr*=n+k, If[Mod[pr, div]==0, Return[k]]]]; a[1]=1; a[n_] := LCM@@Range[n+1, n+a75365[n]]

a(n) = lcm(n+1, ..., n + A075365(n)).

Edited by Dean Hickerson, Oct 28 2002

A075368 Smallest integer value of lcm(n+1, n+2, ..., n+k) (for k >= 0) divided by the product of all the primes up to n.

Original entry on oeis.org

1, 6, 10, 5, 84, 84, 1716, 858, 286, 286, 100776, 100776, 891480, 891480, 891480, 445740, 282861360, 282861360, 550835280, 550835280, 550835280, 550835280, 42222721680, 42222721680, 8444544336, 8444544336, 2814848112, 2814848112

Offset: 1

Amarnath Murthy, Sep 20 2002

			a(3) = 10 as the product of primes <= (n = 3) is 6 and the smallest integer of the form lcm(3+1, 3+2, ..., 3+k) = lcm(4, 5, 6) = 60 giving a(3) = 60/6 = 10. - _David A. Corneth_, Dec 05 2023

David A. Corneth, Table of n, a(n) for n = 1..2236

Cf. A075365, A075366, A075367.

a75365[n_] := Module[{div, k, pr}, div=Times@@Prime/@Range[PrimePi[n]]; For[k=0; pr=1, True, k++; pr*=n+k, If[Mod[pr, div]==0, Return[k]]]]; a[1]=1; a[n_] := LCM@@Range[n+1, n+a75365[n]]/Times@@Prime/@Range[PrimePi[n]]

a(n) = {if(n==1, return(1));
	my(pp = vecprod(primes(primepi(n))), l = n+1);
	for(k = n+2, 2*n,
		l = lcm(l, k);
		if(l%pp == 0,
			return(l\pp)
		)
	)
} \\ David A. Corneth, Dec 05 2023

a(n) = A075367(n)/A034386(n).

Edited by Dean Hickerson, Oct 28 2002

cp's OEIS Frontend

A175856 a(n) = a(n-1) - 1 if n is composite, a(n) = n otherwise.

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A075366 Smallest product (n+1)(n+2)...(n+k) that is divisible by the product of all the primes up to n.

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A075367 Smallest value of lcm(n+1, n+2, ..., n+k) (for k >= 0) that is divisible by the product of all the primes up to n.

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Mathematica

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A075368 Smallest integer value of lcm(n+1, n+2, ..., n+k) (for k >= 0) divided by the product of all the primes up to n.

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Author

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Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions