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 1-8 of 8 results.

A354217 Primes p such that the squarefree kernel of the product of the composite numbers between p and the next prime after p (A076978) sets a new record.

Original entry on oeis.org

3, 5, 7, 13, 19, 23, 31, 47, 53, 73, 83, 89, 113, 199, 211, 293, 467, 523, 887, 1129, 1327, 2971, 3271, 4297, 4831, 5591, 8467, 9551, 12853, 14107, 15683, 19609, 25471, 31397, 89689, 107377, 134513, 155921, 265621, 338033, 360653, 370261, 492113, 1098847, 1349533
Offset: 1

Views

Author

Hugo Pfoertner, May 19 2022

Keywords

Crossrefs

Cf. A007947, A076978, A354218.

Programs

  • Mathematica
    s = Array[Times @@ FactorInteger[Times @@ Range[#1 + 1, #2 - 1]][[All, 1]] & @@ Map[Prime, # + {0, 1}] &, 2^15]; Prime@ Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]] (* Michael De Vlieger, May 20 2022 *)
  • PARI
    a354217(limit) = {my(mp=0,pp=3); forprime(p=5,limit,my(L=List()); for(j=pp+1,p-1, my(f=factor(j),nf=#f~); for(k=1,nf, listput(~L,f~[1,k]))); listsort(L,1); my(mpp=prod(k=1,#L,L[k])); if(mpp>mp,print1(pp,", ");mp=mpp); pp=p)};
a354217(1500000)

A354218 Records in A076978, divided by 2.

Original entry on oeis.org

1, 3, 15, 105, 1155, 1365, 19635, 23205, 636405, 10555545, 24484845, 869107785, 14797252681546335, 92442344345566215, 40334203530676690635, 1451417351374223318085, 6087255082617244520985, 469256253416014832182075245585, 1519012498286389398934397206158552189345
Offset: 1

Views

Author

Hugo Pfoertner, May 19 2022

Keywords

Links

Crossrefs

Cf. A007947, A076978, A354217.

Programs

  • Mathematica
    1/2*Union@ FoldList[Max, Array[Times @@ FactorInteger[Times @@ Range[#1 + 1, #2 - 1]][[All, 1]] & @@ Map[Prime, # + {0, 1}] &, 180, 2]] (* Michael De Vlieger, May 20 2022 *)
  • PARI
    a354218(maxterm) = {my(mp=0,pp=3); forprime(p=5,oo, my(L=List());for(j=pp+1,p-1, my(f=factor(j), nf=#f~); for(k=1,nf, listput(~L,f~[1,k]))); listsort(L,1); my(mpp=prod(k=1,#L,L[k])); if(mpp>mp, if(mpp<2*maxterm, print1(mpp/2,", "); mp=mpp, return)); pp=p)};
a354218(10^75)
  • Python
    from sympy import sieve as p, primefactors
def A076978_halfen(n): # for all integers n > 1
    result = 1
    for composites in range(p[n]+1, p[n+1]):
        for primefactor in primefactors(composites):
            if result % primefactor != 0: result *= primefactor
    return result//2
A354218 = [1]
for n in range(2,180):
    if  A076978_halfen(n) > A354218[-1]: A354218.append(A076978_halfen(n))
print(A354218) # Karl-Heinz Hofmann, May 20 2022

A074168 Duplicate of A076978.

Original entry on oeis.org

1, 2, 6, 30, 6, 210, 6, 2310, 2730, 30, 39270, 7410, 42, 7590, 46410, 1272810, 30, 930930, 82110, 6, 21111090, 1230, 48969690
Offset: 0

Views

Author

Keywords

A276378 Numbers k such that 6*k is squarefree.

Original entry on oeis.org

1, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 173, 179, 181, 185, 187, 191, 193, 197, 199, 203, 205, 209, 211
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Sep 02 2016

Keywords

Comments

These are the numbers from A005117 that are not divisible by 2 and 3.
Squarefree numbers coprime to 6. - Robert Israel, Sep 02 2016
Numbers k such that A008588(k) is in A005117. - Felix Fröhlich, Sep 02 2016
The asymptotic density of this sequence is 3/Pi^2 (A104141). - Amiram Eldar, May 22 2020
From Peter Munn, Nov 20 2020: (Start)
The products generated from each subset of A215848 (primes greater than 3).
Closed under the commutative binary operation A059897(.,.), forming a subgroup of the positive integers under A059897. (End)
Multiplied by 6 we have 6, 30, 42, 66, 78, 102, ..., the values that may appear in A076978 after the 1, 2. [Don Reble, Dec 02 2020] - R. J. Mathar, Dec 15 2020
By the von Staudt-Clausen theorem, denominators of Bernoulli numbers are of the form 6*a(n) for some n. - Charles R Greathouse IV, May 16 2024

Examples

			5 is in this sequence because 6*5 = 30 = 2*3*5 is squarefree.

Links

Crossrefs

Numbers m such that k*m is squarefree: A005117 (k = 1), A056911 (k = 2), A261034 (k = 3), A274546 (k = 5).
Subsequence of A007310, A300957, and A339690.
Cf. A003961, A008588, A059897, A076978, A104141, A215848.

Programs

  • Magma
    [n: n in [1..230] | IsSquarefree(6*n)];
  • Maple
    select(numtheory:-issqrfree, [seq(seq(6*i+j,j=[1,5]),i=0..100)]); # Robert Israel, Sep 02 2016
  • Mathematica
    Select[Range@ 212, SquareFreeQ[6 #] &] (* Michael De Vlieger, Sep 02 2016 *)
  • PARI
    is(n) = issquarefree(6*n) \\ Felix Fröhlich, Sep 02 2016

Formula

{a(n) : n >= 1} = {A003961(A003961(A005117(n))) : n >= 1} = {A003961(A056911(n)) : n >= 1}. - Peter Munn, Nov 20 2020
Sum_{n>=1} 1/a(n)^s = (6^s)*zeta(s)/((1+2^s)*(1+3^s)*zeta(2*s)), s>1. - Amiram Eldar, Sep 26 2023

A074167 Product of prime divisors of composite numbers between consecutive primes.

Original entry on oeis.org

1, 2, 6, 60, 6, 420, 6, 4620, 32760, 30, 471240, 14820, 42, 15180, 556920, 15273720, 30, 11171160, 164220, 6, 253333080, 2460, 587636280, 625757605200, 4620, 102, 289380, 6, 170940, 26848135265397670224000, 33540, 599888520, 138, 39560762839197600, 30
Offset: 1

Views

Author

Amarnath Murthy, Aug 30 2002

Keywords

Examples

			a(4) = product of prime factors of composite numbers between 7 and 11 = 2 * 3 * (2 * 5) = 60.

Links

Crossrefs

Cf. A000040, A007947, A059960, A076978.

Programs

  • Maple
    a:= n-> mul(mul(i[1], i=ifactors(j)[2]), j=ithprime(n)+1..ithprime(n+1)-1):
seq(a(n), n=1..40);  # Alois P. Heinz, May 29 2022
  • Mathematica
    Array[Times @@ Flatten@ Map[FactorInteger[#][[All, 1]] &, Range[#1 + 1, #2 - 1]] & @@ Prime[{#, # + 1}] &, 35] (* Michael De Vlieger, May 29 2022 *)
  • PARI
    a(n) = my(p=1); forcomposite(c=prime(n), prime(n+1), p*=factorback(factorint(c)[, 1])); p; \\ Michel Marcus, May 29 2022

Formula

a(n) = 6 <=> A000040(n) in { A059960 }. - Alois P. Heinz, May 29 2022

Extensions

Corrected and extended by Joshua Zucker, May 08 2006
Offset corrected by Alois P. Heinz, May 29 2022

A079615 Product of all distinct prime factors of all composite numbers between n-th prime and next prime.

Original entry on oeis.org

2, 6, 30, 6, 210, 6, 2310, 2730, 30, 39270, 7410, 42, 7590, 46410, 1272810, 30, 930930, 82110, 6, 21111090, 1230, 48969690, 1738215570, 2310, 102, 144690, 6, 85470, 29594505363092670, 16770, 49990710, 138, 7849357706190, 30
Offset: 2

Views

Author

Reinhard Zumkeller, Jan 29 2003

Keywords

Comments

a(n) = A007947(A056831(n)), squarefree kernel of least common multiple of composite numbers between n-th prime and next prime.
Note that each term is a product of distinct primes. - T. D. Noe, May 19 2007
Equals A076978 without its first term. - R. J. Mathar, Sep 19 2008
Same for A074168. - Georg Fischer, Oct 06 2018
For n > 2, a(n) is of the form 2*3*r, where r is relatively prime to 6. Therefore, for every n > 2, a(n) is a Zumkeller number (see Corollary 5, Rao/Peng link). - Ivan N. Ianakiev, Jan 24 2020

Examples

			n=9: factorizations of numbers between 23=A000040(9) and 29=A000040(10) are 24=3*2^3, 25=5^2, 26=13*2 and 27=3^3, therefore a(9) = 2*3*5*7*13 = 2730.

Links

Crossrefs

Cf. A005117, A002110, A083207.

Programs

  • Mathematica
    a[n_] := (p = Prime[n]; s = Select[Table[k, {k, p, NextPrime[p]}], ! PrimeQ[#] &]; Times @@ ((FactorInteger /@ s // Flatten[#, 1] &)[[All, 1]] // Union)); a /@ Range[2, 35] (* Jean-François Alcover, Jul 13 2011 *)
Table[Times@@Union[Flatten[Transpose[FactorInteger[#]][[1]]&/@ (Range[ Prime[ n]+1, NextPrime[Prime[n]]-1])]],{n,2,50}] (* Harvey P. Dale, Oct 10 2011 *)

Extensions

Corrected by T. D. Noe, May 19 2007

A354219 Primes p such that the number of distinct prime factors omega of the product of the composite numbers between p and the next prime after p sets a new record.

Original entry on oeis.org

3, 5, 7, 13, 19, 31, 53, 73, 89, 113, 211, 293, 523, 887, 1129, 1327, 4297, 4831, 5351, 5591, 8467, 12853, 15683, 19609, 25471, 31397, 134513, 155921, 338033, 360653, 370261, 492113, 1349533, 1357201, 1561919, 2010733, 4652353, 8421251, 11113933, 15203977, 17051707
Offset: 1

Views

Author

Hugo Pfoertner, May 20 2022

Keywords

Examples

			a(6) = 31, because the first product of consecutive composites with 6 primes in its squarefree kernel is P = 32*33*34*35*36 with rad(P) = 2*3*5*7*11*17 = 39270, whereas the interval starting after A354217(6) = 23 leads only to 5 distinct factors, i.e., rad(24*25*26*27*28) = 2*3*5*7*13, not sufficient to beat the record set by the composites after a(5) = A354217(5) = 19 with rad(20*21*22) = 2*3*5*7*11.

Crossrefs

A354220 provides the corresponding values of omega.
Cf. A001221, A076978, A354217.

Programs

  • Mathematica
    s = Array[PrimeNu[Times @@ FactorInteger[Times @@ Range[#1 + 1, #2 - 1]][[All, 1]] & @@ Map[Prime, # + {0, 1}]] &, 10^4]; Prime@ Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]] (* Michael De Vlieger, May 20 2022 *)
DeleteDuplicates[Table[{p,Length[Union[Flatten[FactorInteger[#][[;;,1]]&/@Range[p+1,NextPrime[p]-1]]]]},{p,Prime[Range[ 2,10^6]]}], GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* The program generates the first 40 terms of the sequence, i.e., every term up to the 1 millionth prime. *) (* Harvey P. Dale, Feb 01 2025 *)

A354220 a(n) is the record setting number of distinct prime factors in the product of the composite numbers between p = A354219(n) and the next prime after p.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 20, 25, 27, 37, 38, 39, 40, 41, 45, 49, 56, 64, 66, 89, 90, 114, 120, 131, 145, 154, 164, 174, 187, 207, 217, 218, 222, 225, 266, 302, 328, 329, 330, 351, 364, 369, 377, 381, 437, 447, 449, 454, 461, 468, 497, 530, 535, 548, 551
Offset: 1

Views

Author

Hugo Pfoertner, May 20 2022

Keywords

Crossrefs

Cf. A001221, A076978, A354219.

Programs

  • Mathematica
    Union@ FoldList[Max, Array[PrimeNu[Times @@ FactorInteger[Times @@ Range[#1 + 1, #2 - 1]][[All, 1]]] & @@ Map[Prime, # + {0, 1}] &, 10^4]] (* Michael De Vlieger, May 20 2022 *)
Showing 1-8 of 8 results.

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