A231815 -id:A231815 - cp's OEIS frontend

A231814 Squarefree numbers (from A005117) with prime divisors in a 2p-1 progression.

6, 15, 30, 91, 703, 1891, 2701, 12403, 18721, 38503, 49141, 51319, 79003, 88831, 104653, 146611, 188191, 218791, 226801, 269011, 286903, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1314631, 1373653, 1537381, 1755001, 1869211

Offset: 1

Jaroslav Krizek, Nov 13 2013

nonn

Squarefree numbers with k >= 2 prime factors of the form p_1 * p_2 * ... * p_k, where p_1 < p_2 < ... < p_k = primes with p_k = 2 * p_(k-1) - 1.
Each of these numbers is divisible by the arithmetic mean of its proper divisors.
Supersequence of A129521 (numbers of the form p*q, p and q prime with q=2*p-1; see A005382 and A005383).

			51319 = 19*37*73 where 37 = 2*19 - 1, 73 = 2*37 - 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A005117, A005382, A005383, A129521, A231815, A231816.

Cf. A057330 (first prime for such numbers that has n factors).

N:= 10^7: # for terms <= N
p:= 1: S:= NULL: count:= 0:
do
  p:= nextprime(p);
  if p*(2*p-1) > N then break fi;
  q:= p; x:= p;
  do
    q:= 2*q-1;
    if not isprime(q) then break fi;
    x:= x*q;
    if x > N then break fi;
    S:= S,x; count:= count+1;
  od;
od:
sort([S]); # Robert Israel, Mar 24 2023

geomQ[lst_] := Module[{x = lst - 1}, x = x/x[[1]]; Log[2, x] + 1 == Range[Length[x]]]; Select[Range[2, 1000000], ! PrimeQ[#] && SquareFreeQ[#] && geomQ[Transpose[FactorInteger[#]][[1]]] &] (* T. D. Noe, Nov 14 2013 *)

A231816 a(n) = the smallest squarefree number (A005117) with n prime factors in a 2p-1 progression.

Original entry on oeis.org

2, 6, 30, 351137972965951, 8596208716179446431, 698211042943963834650959743951, 744014385572130806167897354113929551, 901203402294977554329263775346819632824908852456695769189267773301

Offset: 1

Jaroslav Krizek, Nov 13 2013

nonn

Smallest squarefree numbers with n >= 2 prime divisors of the form p_1 * p_2 * … * p_n, where p_1 < p_2 < … < p_k = primes with p_k = 2 * p_(k-1) - 1.

Subsequence of A231814.

			8596208716179446431 = 1531*3061*6121*12241*24481, where 3061 = 2*1531 - 1, 6121 = 2*3061 - 1, 12241 = 2*6121 - 1, 24481 = 2*12241 - 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..15

Cf. A005117, A000040, A231814, A231815.

Cf. A057330 (first prime for such numbers that has n factors).

cp's OEIS Frontend

A231814 Squarefree numbers (from A005117) with prime divisors in a 2p-1 progression.

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