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.

A262723 Products of three distinct primes that form an arithmetic progression.

Original entry on oeis.org

105, 231, 627, 897, 935, 1581, 1729, 2465, 2967, 4123, 4301, 4715, 5487, 7685, 7881, 9717, 10707, 11339, 14993, 16377, 17353, 20213, 20915, 23779, 25327, 26331, 26765, 29341, 29607, 32021, 33335, 40587, 40807, 42911, 48635, 49321, 54739, 55581, 55637, 59563, 60297, 63017
Offset: 1

Views

Author

Antonio Roldán, Sep 28 2015

Keywords

Comments

This sequence is subsequence of A046389, A088595, A187073, A203614 and A229094.
Obviously, the most repeated prime divisor for values of a(n) is 3. - Altug Alkan, Sep 30 2015
These are numbers 3(2k + 3)(4k + 3) where 2k + 3 and 4k + 3 are prime, together with numbers p(p - 6d)(p + 6d) where p, p - 6d, and p + 6d are prime. - Charles R Greathouse IV, Mar 16 2018

Examples

			627 is in this sequence because 627=3*11*19, and 3, 11, 19 form an arithmetic progression (11-3 = 19-11).

Links

Crossrefs

Cf. A046389, A088595, A187073, A203614, A229094.

Programs

  • Mathematica
    Select[Range@ 64000, And[SquareFreeQ@ #, PrimeOmega@ # == 3, Subtract @@ Differences[First /@ FactorInteger@ #] == 0] &] (* Michael De Vlieger, Sep 30 2015 *)
  • PARI
    for(i=2,10^5,if(issquarefree(i)&&omega(i)==3,f=factor(i);if(f[1, 1]+f[3, 1]==2*f[2,1],print1(i,", "))))
  • PARI
    list(lim)=my(v=List()); lim\=1; forstep(d=6,sqrtint(lim\10),6, forprime(p=d+5, solve(x=sqrtn(lim,3),d*sqrtn(lim,3), x^3-d^2*x-lim)+.5, if(isprime(p-d) && isprime(p+d), listput(v, p*(p-d)*(p+d))))); forprime(p=5,(sqrt(24*lim+81)-27)/12+3.5, if(isprime(2*p-3), listput(v,p*(2*p-3)*3))); Set(v) \\ Charles R Greathouse IV, Mar 16 2018

Extensions

New name from Peter Munn, Aug 27 2022

A088949 Composite numbers k such that (A006530(k) + A020639(k))/2 is an integer that divides k; special terms of A088948.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 105, 121, 125, 128, 169, 231, 243, 256, 289, 315, 343, 361, 512, 525, 529, 625, 627, 693, 729, 735, 841, 897, 935, 945, 961, 1024, 1155, 1331, 1369, 1575, 1581, 1617, 1681, 1729, 1849, 1881, 2048, 2079, 2187, 2197, 2205, 2209
Offset: 1

Views

Author

Labos Elemer, Nov 20 2003

Keywords

Examples

			k = 315 = 3*3*5*7 (composite); (3 + 7)/2 = 5, which divides k.

Links

Crossrefs

Cf. A006530, A020639, A088948, A088595, A046687.

Programs

  • Mathematica
    q[k_] := CompositeQ[k] && Module[{p = FactorInteger[k][[;;, 1]], m}, m = (p[[1]] + p[[-1]]); EvenQ[m] && Divisible[k, m/2]]; Select[Range[2500], q] (* Amiram Eldar, Mar 01 2025 *)
  • PARI
    lista(nn) = {forcomposite(n=1, nn, my(f = factor(n)[,1], x = (vecmin(f) + vecmax(f))/2); if ((denominator(x)==1) && !(n % x), print1(n, ", ")););} \\ Michel Marcus, Jul 09 2018

Extensions

Edited by Jon E. Schoenfield, Jul 08 2018
More terms from Michel Marcus, Jul 09 2018

A283105 Numbers that are an integer multiple of the mean of their smallest and largest nontrivial divisors.

Original entry on oeis.org

4, 9, 12, 25, 45, 49, 121, 169, 289, 361, 529, 637, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 13357, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481, 37249, 38809, 39601, 44521
Offset: 1

Views

Author

Emmanuel Vantieghem, Feb 28 2017

Keywords

Comments

No prime is in the sequence since there are no nontrivial divisors of a prime.
The sequence includes every number that is the square of a prime.
It is easy to show that the other terms are of the form (2p-1)*p^2 where p and 2p-1 are prime. Therefore, the mean of the two divisors in question is always an integer.

Examples

			4 is in the sequence because its smallest nontrivial divisor is 2, its largest nontrivial divisor is 2, and their mean is 2.
45 is in the sequence because its smallest nontrivial divisor is 3, its largest nontrivial divisor is 15, and their mean is 9, a divisor of 45.
10 is not in the sequence because it is not an integral multiple of 7/2, the mean of 2 and 5.

Links

Crossrefs

Cf. A005382, A088595.

Programs

  • Mathematica
    mslndQ[n_]:=Module[{d=Divisors[n]},Divisible[n,Mean[{d[[2]],d[[-2]]}]]]; Select[Range[2,50000],mslndQ] (* Harvey P. Dale, Jul 24 2017 *)
  • PARI
    is(n) = my(d=divisors(n), m=(d[2]+d[#d-1])/2); if(n%m==0, 1, 0) \\ Felix Fröhlich, Feb 28 2017
Showing 1-3 of 3 results.

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