cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A073483 For the n-th squarefree number: the product of all primes greater than its smallest factor and less than its largest factor and not dividing it.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 15, 1, 1, 1, 5, 105, 1, 1155, 1, 1, 1, 35, 15015, 1, 1, 255255, 385, 1, 5, 1, 4849845, 1, 5005, 1, 7, 85085, 111546435, 1, 1, 3234846615, 77, 35, 1, 1616615, 3, 1, 1, 100280245065, 1, 385, 1, 3710369067405, 1, 1001
Offset: 1

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Author

Reinhard Zumkeller, Aug 03 2002

Keywords

Comments

a(n)=1 iff A073484(n)=0; a(A000040(n))=1, a(A006094(n))=1, a(A002110(n))=1.

Examples

			The 69th squarefree number is 110=2*5*11, primes between 2 and 11, not dividing 110, are 3 and 7, therefore a(69)=21.

Links

Crossrefs

Cf. A005117, A000040, A073484, A117214.

Programs

  • Haskell
    a073483 n = product $ filter ((> 0) . (mod m)) $
   dropWhile (<= a020639 m) $ takeWhile (<= a006530 m) a000040_list
   where m = a005117 n
-- Reinhard Zumkeller, Jan 15 2012
  • Mathematica
    ppg[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},Times@@Select[Prime[ Range[PrimePi[First[f]]+1,PrimePi[Last[f]]-1]],!MemberQ[f,#]&]]; ppg/@ Select[ Range[100],SquareFreeQ] (* Harvey P. Dale, Jan 16 2013 *)

Formula

a(n) = A002110(A073482(n))/(A005117(n)*A002110(A073481(n))).

Extensions

a(44) and a(49) corrected by Reinhard Zumkeller, Jan 14 2012
Definition clarified by Harvey P. Dale, Jan 16 2013

A117213 a(n) = smallest term of sequence A002110 divisible by n-th squarefree positive integer.

Original entry on oeis.org

1, 2, 6, 30, 6, 210, 30, 2310, 30030, 210, 30, 510510, 9699690, 210, 2310, 223092870, 30030, 6469693230, 30, 200560490130, 2310, 510510, 210, 7420738134810, 9699690, 30030, 304250263527210, 210, 13082761331670030, 223092870
Offset: 1

Views

Author

Leroy Quet, Mar 03 2006

Keywords

Examples

			10 is the 7th squarefree integer. And 2*3*5 = 30 is the smallest primorial number divisible by 10 = 2*5. So a(7) = 30.

Links

Crossrefs

Cf. A002110, A073482, A117214.

Programs

  • Maple
    issquarefree := proc(n::integer) local nf, ifa, lar ; nf := op(2,ifactors(n)) ; for ifa from 1 to nops(nf) do lar := op(1,op(ifa,nf)) ; if op(2,op(ifa,nf)) >= 2 then RETURN(0) ; fi ; od : RETURN(lar) ; end: primor := proc(n::integer) local resul, nepr ; resul :=2 ; nepr :=3 ; while nepr <= n do resul := resul*nepr ; nepr:=nextprime(nepr) ; od : RETURN(resul) ; end: printf("1,") ; for n from 2 to 100 do lfa := issquarefree(n) ; if lfa > 0 then printf("%a,",primor(lfa) ) ; fi ; od : # R. J. Mathar, Apr 02 2006
  • Mathematica
    Select[Array[Which[# == 1, 1, SquareFreeQ@ #, Product[Prime@ i, {i, PrimePi@ FactorInteger[#][[-1, 1]]}], True, 0] &, 50], # > 0 & ] (* Michael De Vlieger, Sep 30 2017 *)

Formula

For n >= 2, a(n) = product of the primes <= A073482(n).

Extensions

More terms from R. J. Mathar, Apr 02 2006

A362364 a(n) is the product of the first n primes that are coprime to a(n-1); a(0) = 1.

Original entry on oeis.org

1, 2, 15, 154, 3315, 67298, 2980185, 102091066, 6022953885, 319238763382, 24615812527995, 1654614510608906, 161405882746063215, 14284287070086685498, 1679105398207295625645, 166597640098421012963174, 24096841569672899523631395, 2989927846846361919650083778, 499069685749495422033929821845
Offset: 0

Views

Author

Robert Israel, Apr 18 2023

Keywords

Comments

Lexicographically first sequence of squarefree numbers such that A001222(a(n)) = n and each term is coprime to the next.

Examples

			a(0) = 1.
a(1) = 2 is the least prime coprime to a(0).
a(2) = 3*5 is the product of the two least primes coprime to a(1).
a(3) = 2*7*11 is the product of the three least primes coprime to a(2).
a(4) = 3*5*13*17 = 3315 is the product of the four least primes coprime to a(3).

Links

Crossrefs

Cf. A001222.
See the formula section for the relationships with A002110, A019565, A037481, A100112, A117214.

Programs

  • Maple
    f:= proc(n) local i;
   if n::odd then 2 * mul(ithprime(4*i)*ithprime(4*i+1),i=1..(n-1)/2)
   else mul(ithprime(4*i-2)*ithprime(4*i-1),i=1..(n/2))
   fi
end proc:
map(f, [$0..20]);
  • Python
    from math import prod
from sympy import prime
def A362364(n): return prod(prime(i)*prime(i+1) for i in range(2+((n&1)<<1),(n<<1)-1,4))<<(n&1) # Chai Wah Wu, Apr 20 2023

Formula

If n is even, a(n) = Product_{i=1..n/2} prime(4*i-2)*prime(4*i-1).
If n is odd, a(n) = 2 * Product_{i=1..(n-1)/2} prime(4*i)*prime(4*i+1).
From Peter Munn, Apr 21 2023: (Start)
a(0) = 1, for n >= 1, a(n) = A002110(2n-1)/a(n-1).
a(n) = A019565(A037481(n)).
For n >= 1, a(n-1) = A117214(A100112(a(n))).
(End)
Showing 1-3 of 3 results.

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