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 1-2 of 2 results.

A096529 Numbers whose divisors can be permuted so that all sums of triple adjacent divisors are primes.

Original entry on oeis.org

4, 8, 9, 10, 12, 14, 15, 16, 20, 21, 24, 25, 26, 27, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 52, 55, 56, 57, 58, 60, 63, 65, 68, 75, 76, 77, 81, 84, 85, 86, 88, 92, 93, 99, 100, 104, 105, 111, 115, 117, 119, 123, 124, 125, 129, 132, 135, 136, 140, 143, 145, 147
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 23 2004

Keywords

Comments

Square of terms of A053182 are in this sequence. - Michel Marcus, May 08 2014
From Amiram Eldar, Nov 08 2024: (Start)
The possible values of the number of even divisors of even terms of this sequence is restricted by the number of odd divisors.
Let k be a term and d_odd(k) = A001227(k) and d_even(k) = A183063 be its number of odd divisors and number of even divisors, respectively. When k is even, in a valid permutation of its divisors there must be two even divisors between two odd divisors, at most 2 before the first odd divisor, and at most 2 after the last odd divisor.
Therefore, d_even(k) - 2*(d_odd(k) - 1) <= 4. Let d(k) = A000005(k) = d_odd(k) + d_even(k), and let e = A007814(k) and m = A000265(k). Then, k = 2^e * m, d(k) = (e+1) * d(m) = (e+1) * d_odd(k), so d_even(k) = e * d_odd(k), and |e-2| * d_odd(k) <= 2.
If m = 1, then d_odd(k) = 1 and e <= 4, so 16 = 2^4 is the largest power of 2 in this sequence.
If m = p is a prime, then d_odd(k) = 2 and e <= 3, and therefore only terms of the form 2*p, 4*p or 8*p are possible. 2*p is a term if and only if p is a term of A106067.
If m is composite, then d_odd(k) > 2 and e <= 2, and therefore k is not divisible by 8. (End)

Examples

			Divisors of 24 are {1,2,3,4,6,8,12,24}: [2,8,3,12,4,1,24,6] -> (2+8+3,8+3+12,3+12+4,12+4+1,4+1+24,1+24+6) = (13,23,19,17,29,31): therefore 24 is a term.

Links

Crossrefs

Complement of A096530.
Cf. A053182, A096527.
Cf. A000005, A000265, A001227, A007814, A106067, A183063.

Programs

  • PARI
    isok(p) = {my(n = #p); if(n < 3, return(0)); for(k = 1, n-2, if(!isprime(p[k]+p[k+1]+p[k+2]), return(0))); 1;}
is2(n) = {my(d = divisors(n)); forperm(d, p, if(isok(p), return(1))); 0;}
is1(k) = {my(e = valuation(k,2), o = k >> e); (e == 0) || (o == 1 && e <= 4) || (abs(e-2) * numdiv(o) <= 2);}
is(k) = is1(k) && is2(k); \\ Amiram Eldar, Nov 08 2024

Formula

A096527(a(n)) > 0.

Extensions

a(30)-a(51) from Michel Marcus, May 03 2014
a(52) onwards from Amiram Eldar, Nov 08 2024

A329973 Smallest prime p such that both 2*prime(n+1)+p and p*prime(n+1)+2 are primes.

Original entry on oeis.org

5, 3, 3, 7, 3, 3, 3, 7, 3, 5, 23, 67, 3, 7, 7, 13, 5, 5, 7, 5, 5, 67, 3, 3, 37, 17, 43, 5, 13, 3, 7, 127, 3, 19, 5, 17, 53, 3, 3, 43, 5, 19, 23, 3, 3, 101, 17, 3, 41, 37, 13, 17, 7, 7, 37, 3, 59, 23, 31, 257, 7, 47, 31, 5, 7, 11, 3, 67, 3, 3, 43, 23
Offset: 1

Views

Author

Ivan N. Ianakiev, Jun 08 2020

Keywords

Comments

a(n)=3 if and only if prime(n+1) is in A106067. - Robert Israel, Jul 17 2020

Links

Crossrefs

Cf. A000040, A065091, A073703, A106067.

Programs

  • Maple
    f:= proc(n) local pn,p;
 pn:= ithprime(n+1);
 p:= 1;
 do
   p:= nextprime(p);
   if isprime(2*pn+p) and isprime(p*pn+2) then return p fi
 od
end proc:
map(f, [$1..100]); # Robert Israel, Jul 17 2020
  • Mathematica
    f[n_Integer/;n>1]:=Module[{p=3},While[Or[CompositeQ[2*Prime[n]+p],CompositeQ[p*Prime[n]+2]],p=NextPrime[p]];p];f/@Range[2,100]
  • PARI
    a(n) = my(p=2,q=prime(n+1)); while(!isprime(2*q+p) || !isprime(p*q+2), p=nextprime(p+1)); p; \\ Michel Marcus, Jun 08 2020
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.