A231966 -id:A231966 - cp's OEIS frontend

A231967 Squarefree numbers (A005117) of the form pqr with prime factors p, q, r with q = 2p + 1 and r = 2q + 1.

110, 1265, 11891, 568301, 5719229, 46203659, 371436119, 1057570169, 2978731439, 8475105539, 8777935031, 14865764009, 22397944469, 24460553171, 26008879181, 27621202391, 47549400491, 53960155829, 54994829321, 57639193331, 119010782819, 157361958899

Offset: 1

Jaroslav Krizek, Nov 16 2013

nonn

Squarefree numbers of the form p*q*r, where p < q < r = primes with q = 2*p + 1 and r = 2*q + 1; that is, r = 4*p + 3.

			5719229 = 89*179*359, 179 = 2*89 + 1, 359 = 2*179 + 1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005117, A000040, A231968, A231969, A231966. Cf. A007700 (first member of a prime triple in a 2p+1 progression).

sfQ[n_]:=Module[{q=2n+1,r},r=2q+1;AllTrue[{q,r},PrimeQ]&& SquareFreeQ[ n*q*r]]; 3#+10#^2+8#^3&/@Select[Prime[Range[400]],sfQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 26 2016 *)

A231969 a(n) = the smallest squarefree number (A005117) with n prime factors in a 2p+1 progression.

Original entry on oeis.org

2, 10, 110, 2530, 118910, 17036047229140531, 4713753689937227789548410467592773787730621935419, 4754361703029497628070972207349674154455369685904736544199583856401, 17434718204270642890620908753958444038404912529730635812020757976125828120134034469

Offset: 1

Jaroslav Krizek, Nov 16 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.

			17036047229140531 = 89*179*359*719*1439*2879, where 179 = 2*89 + 1, 359 = 2*179 + 1, 719 = 2*359 + 1, 1439 = 2*719 + 1, 2879 = 2*1439 + 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..12

Cf. A005117, A000040, A231966, A231967, A231968.

A231968 Squarefree numbers (A005117) of the form pqrs with prime factors p, q, r, s with q = 2p + 1, r = 2q + 1 and s = 2r + 1.

Original entry on oeis.org

2530, 59455, 4112125651, 66487065301, 1069364586601, 4313828719351, 146264252084551, 252626415665851, 3398860950259501, 10045301361318901, 110736954977078251

Offset: 1

Jaroslav Krizek, Nov 18 2013

nonn

			4112125651 = 89*179*359*719; 179 = 2*89 + 1; 359 = 2*179 + 1, 719 = 2*359 + 1.

Cf. A005117, A000040, A231967, A231969, A231966.

A367576 Semiprimes pq such that 2p divides q-1.

Original entry on oeis.org

10, 21, 26, 34, 39, 55, 57, 58, 74, 82, 93, 106, 111, 122, 129, 146, 155, 178, 183, 194, 201, 202, 203, 205, 218, 219, 226, 237, 253, 274, 291, 298, 301, 305, 309, 314, 327, 346, 355, 362, 381, 386, 394, 417, 453, 458, 466, 471, 482, 489, 497, 505, 514, 538

Offset: 1

Jon E. Schoenfield, Nov 24 2023

nonn

These terms and their squarefree proper multiples are the squarefree numbers that never appear as terms in A367566.

			10 is a term: 10 = 2*5, and 2*2 | 5 - 1.
33 is not a term: 33 = 3*11, but 2*3 does not divide 11 - 1.

Cf. A156592, A231966, A367566.

Subsequence of A006881.

cp's OEIS Frontend

A231967 Squarefree numbers (A005117) of the form pqr with prime factors p, q, r with q = 2p + 1 and r = 2q + 1.

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A231969 a(n) = the smallest squarefree number (A005117) with n prime factors in a 2p+1 progression.

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A231968 Squarefree numbers (A005117) of the form pqrs with prime factors p, q, r, s with q = 2p + 1, r = 2q + 1 and s = 2r + 1.

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A367576 Semiprimes pq such that 2p divides q-1.

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cp's OEIS Frontend

A231967 Squarefree numbers (A005117) of the form p*q*r with prime factors p, q, r with q = 2*p + 1 and r = 2*q + 1.

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Mathematica

A231969 a(n) = the smallest squarefree number (A005117) with n prime factors in a 2p+1 progression.

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Author

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Comments

Examples

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A231968 Squarefree numbers (A005117) of the form p*q*r*s with prime factors p, q, r, s with q = 2*p + 1, r = 2*q + 1 and s = 2*r + 1.

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A367576 Semiprimes p*q such that 2*p divides q-1.

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A231967 Squarefree numbers (A005117) of the form pqr with prime factors p, q, r with q = 2p + 1 and r = 2q + 1.

A231968 Squarefree numbers (A005117) of the form pqrs with prime factors p, q, r, s with q = 2p + 1, r = 2q + 1 and s = 2r + 1.

A367576 Semiprimes pq such that 2p divides q-1.