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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.

User: Sílvia Casacuberta Puig

Sílvia Casacuberta Puig's wiki page.

Sílvia Casacuberta Puig has authored 2 sequences.

A290290 Numbers m having greatest prime power divisor d such that d is smaller than the difference between m and the largest prime smaller than m and d is smaller than the difference between m and twice the largest prime smaller than m/2.

Original entry on oeis.org

3432, 5980, 12870, 12880, 13090, 14280, 14586, 20475, 28272, 28275, 31416, 31450, 34580, 35650, 39270, 45045, 45220, 72072, 76076, 96135, 97812, 106080, 106590, 120120, 121992, 125580, 132804, 139230, 173420, 181350, 185640, 191400, 195624, 202275, 203112, 215050, 216315, 222768, 232254, 240240, 266475
Offset: 1

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Author

Sílvia Casacuberta Puig, Jul 26 2017

Keywords

Comments

It is conjectured that for all integers m there exist two primes p and r such that all the binomial coefficients (m,k) with 1 <= k <= m-1 are divisible by either p or r. Using Lucas's Theorem we can prove that the conjecture is true for integers m such that the difference between m and the largest prime smaller than m is smaller than the greatest prime power divisor of m. In A290203 we examine the integers that do not satisfy this property. However, when this property is not satisfied, if the difference between m and twice the largest prime smaller than m/2 is smaller than the greatest prime power divisor of m we can prove the conjecture for odd numbers and some cases of even numbers. Note that we take the difference between m and twice the largest prime smaller than m/2 because this is the last digit of the base-p representation of m, where p refers to the largest prime smaller than m/2. Therefore, this list examines the numbers that satisfy neither of the two properties and hence are the difficult cases for the proof of the conjecture.

Examples

			The first number of the sequence is 3432. 3432 = 2^3*3*11*13, so the greatest prime power divisor of 3432 is 13. The largest prime smaller than 3432 is 3413, and the difference between these two numbers is 19. Because 19 is greater than 13 the first property is not satisfied. 3432/2 = 1716, and the closest prime to 1716 is 1709. The difference between m and 2*1709 = 3418 is 14. 14 is also greater than 13, which means that the second property is also not satisfied. Thus 3432 is in the sequence.

Links

Crossrefs

Cf. A034699.

Programs

  • Mathematica
    okQ[m_] := m > 2 && Module[{d = Max[Power @@@ FactorInteger[m]]}, d < m - NextPrime[m, -1] && d < m - 2 NextPrime[(m-1)/2, -1]];
Reap[For[m = 1, m <= 300000, m++, If[okQ[m], Print[m]; Sow[m]]]][[2, 1]] (* Jean-François Alcover, Feb 18 2019 *)
  • PARI
    a034699(n) = my(d = Vecrev(divisors(n))); for (k=1, #d, if (isprimepower(d[k]), return (d[k])););
isok(n) = my(thed = a034699(n)); (thed < (n - precprime(n-1))) && (thed < (n - 2*precprime((n-1)/2))); \\ Michel Marcus, Aug 08 2017
  • PARI
    is(n)=if(n<9, return(0)); my(f=factor(n),d=f[1,1]^f[1,2],t); for(i=2,#f~, t=f[i,1]^f[i,2]; if(t>d, d=t)); d < n-precprime(n-1) && d < n - 2*precprime((n-1)/2) \\ Charles R Greathouse IV, Aug 10 2017
  • PARI
    allUnitaryDivisorsLessThan(n,sz)=my(t,e); if(t==1, return(sz>1)); forprime(p=2,sz-1, e=valuation(n,p); t=p^e; if(t>1, if(t>=sz, return(0)); n/=t; if(n==1, return(1)))); 0
is(n)=n>9 && allUnitaryDivisorsLessThan(n, min(n - 2*precprime((n-1)/2), n-precprime(n-1))) \\ Charles R Greathouse IV, Aug 10 2017

A290203 Numbers m having greatest prime power divisor d such that d is smaller than the difference between m and the largest prime smaller than m.

Original entry on oeis.org

126, 210, 330, 630, 1144, 1360, 2520, 2574, 2992, 3432, 3960, 4199, 4620, 5544, 5610, 5775, 5980, 6006, 6930, 7280, 8008, 8415, 9576, 10005, 10032, 12870, 12880, 13090, 14280, 14586, 15708, 15725, 16182, 17290, 18480, 18837, 19635, 19656, 20475, 20592, 22610
Offset: 1

Views

Author

Sílvia Casacuberta Puig, Jul 24 2017

Keywords

Comments

It is conjectured that for all integers m there exist two primes p and r such that all the binomial coefficients (m,k) with 1 <= k <= m-1 are divisible by either p or r. Using Lucas's Theorem we can prove that the conjecture is true for integers m such that the difference between m and the largest prime smaller than m is smaller than the greatest prime power divisor of m. Therefore this list examines the numbers m that do not satisfy this property.
Also numbers m such that m - A007917(m) > A034699(m). - David A. Corneth, Jul 24 2017

Examples

			The first number of the sequence is 126. The prime factorization of 126 is 2*3^2*7. Therefore, the greatest prime power divisor is 9. The largest prime smaller than 126 is 113, and then the difference between 126 and 113 is 13. Then 13 is larger than 9 and therefore 126 is part of the sequence.

Links

Crossrefs

Cf. A007917, A034699, A081805.

Programs

  • Mathematica
    Reap[For[k = 3, k < 30000, k++, If[k - NextPrime[k, -1] > Max[Power @@@ FactorInteger[k]], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jul 24 2017 *)
  • PARI
    isok(n)=my(f = factor(n)); n - precprime(n) > vecmax(vector(#f~, k, f[k,1]^f[k,2])); \\ Michel Marcus, Jul 24 2017
  • PARI
    list(lim)=my(v=List(),p=2,f); forfactored(n=3,lim\1, f=n[2]; if(f[,2]==[1]~, p=n[1]; next); if(n[1]-p > vecmax(vector(#f~, i, f[i,1]^f[i,2])), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 24 2017

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