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.

A360842 5-full numbers (A069492) sandwiched between twin primes.

Original entry on oeis.org

139968, 995328, 63700992, 4076863488, 17714700000, 82012500000, 98802571392, 174960000000, 445240556352, 641194278912, 889223142528, 1059917571072, 1594323000000, 1663012435968, 2348273369088, 3333709317312, 5717741400000, 16260080320512, 19144761127488, 28697814000000
Offset: 1

Views

Author

Amiram Eldar, Feb 23 2023

Keywords

Examples

			139968 = 2^6 * 3^7 is a term since it is 5-full and 139967 and 139969 are twin primes.
		

Crossrefs

Intersection of A014574 and A069492.
Subsequence of A113839, A360840 and A360841.

Programs

  • Mathematica
    Select[6*Range[2*10^5], PrimeQ[# - 1] && PrimeQ[# + 1] && Min[FactorInteger[#][[;; , 2]]] > 4 &]
  • PARI
    is(n) = isprime(n-1) && isprime(n+1) && vecmin(factor(n)[,2]) > 4;

A360843 6-full numbers (A069493) sandwiched between twin primes.

Original entry on oeis.org

139968, 98802571392, 174960000000, 889223142528, 1594323000000, 2348273369088, 19144761127488, 28697814000000, 56358560858112, 84537841287168, 150289495621632, 186624000000000, 328341017826432, 369056250000000, 392147405854848, 578415690713088, 597871125000000
Offset: 1

Views

Author

Amiram Eldar, Feb 23 2023

Keywords

Examples

			139968 = 2^6 * 3^7 is a term since it is 6-full and 139967 and 139969 are twin primes.
		

Crossrefs

Intersection of A014574 and A069493.
Subsequence of A113839, A360840, A360841 and A360842.

Programs

  • Mathematica
    Select[6*Range[10^5], PrimeQ[# - 1] && PrimeQ[# + 1] && Min[FactorInteger[#][[;; , 2]]] > 5 &]
  • PARI
    is(n) = isprime(n-1) && isprime(n+1) && vecmin(factor(n)[,2]) > 5;

A360844 a(n) is the least k-full number that is sandwiched between twin primes.

Original entry on oeis.org

4, 432, 2592, 139968, 139968, 174960000000, 56358560858112, 84537841287168, 578415690713088, 578415690713088, 1141260857376768, 61628086298345472, 61628086298345472, 61628086298345472, 322850407500000000000000000000, 322850407500000000000000000000, 62518864539857068333550694039552
Offset: 2

Views

Author

Amiram Eldar, Feb 23 2023

Keywords

Comments

k-full number is a number m such that if a prime p divides m then so does p^k. All the exponents in the canonical prime factorization of a k-full number are not smaller than k.
a(2)-a(15) are the terms below 3*10^19. Except for a(7) = 174960000000, they are all 3-smooth numbers (A003586, and thus they are terms of A027856). Are there other terms that are not 3-smooth?
a(168) = 2^176 * 3^173 * 7^168 is the first term that is not 5-smooth. - Bert Dobbelaere, Feb 24 2023

Examples

			The first 3 terms, their factorizations and the corresponding twin primes are:
  n |   a(n)  prime factorization  A051904(a(n))  {a(n)-1, a(n)+1}
  ----------------------------------------------------------------
  2 |     4                  2^2              2             {3, 5}
  3 |   432            2^4 * 3^3              3         {431, 433}
  4 |  2592            2^5 * 3^4              4       {2591, 2593}
		

Crossrefs

Extensions

More terms from Bert Dobbelaere, Feb 24 2023
Showing 1-3 of 3 results.