A108164 -id:A108164 - cp's OEIS frontend

A108166 Semiprimes p*q where both p and q are primes of the form 6n-1 (A007528).

25, 55, 85, 115, 121, 145, 187, 205, 235, 253, 265, 289, 295, 319, 355, 391, 415, 445, 451, 493, 505, 517, 529, 535, 565, 583, 649, 655, 667, 685, 697, 745, 781, 799, 835, 841, 865, 895, 901, 913, 943, 955, 979, 985, 1003, 1081, 1111, 1135, 1165, 1177, 1189

Offset: 1

Jonathan Vos Post, Jun 13 2005

Every semiprime not divisible by 2 or 3 must be in one of these three disjoint sets:
A108164 - the product of two primes of the form 6n + 1 (A002476),
A108166 - the product of two primes of the form 6n - 1 (A007528),
A108172 - the product of a prime of the form 6n + 1 and a prime of the form 6n - 1.
The product of two primes of the form 6n - 1 is a semiprime of the form 6n + 1.

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A001358, A007528.

Module[{nn = 150, pf}, pf = Select[6Range[nn] - 1, PrimeQ]; Take[Union[Times@@@Tuples[pf, 2]], nn/2]] (* Harvey P. Dale, Dec 09 2013 *)
Select[6Range[200] + 1, PrimeOmega[#] == 2 && Mod[FactorInteger[#][[1, 1]], 6] == 5 &] (* Alonso del Arte, Aug 24 2017 *)

{a(n)} = {p*q where both p and q are in A007528}.

Edited and extended by Ray Chandler, Oct 15 2005

A112771 Semiprimes of the form 6n + 1.

Original entry on oeis.org

25, 49, 55, 85, 91, 115, 121, 133, 145, 169, 187, 205, 217, 235, 247, 253, 259, 265, 289, 295, 301, 319, 355, 361, 391, 403, 415, 427, 445, 451, 469, 481, 493, 505, 511, 517, 529, 535, 553, 559, 565, 583, 589, 649, 655, 667, 679, 685, 697, 703, 721, 745

Offset: 1

Jonathan Vos Post and Ray Chandler, Oct 15 2005

Union of A108164 and A108166.

Subsequence of A091300. - Zak Seidov, Dec 28 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

IsSemiprime:=func; [s: n in [2..150] | IsSemiprime(s) where s is 6*n + 1]; // Vincenzo Librandi, Sep 22 2012

Select[6 Range[0, 200] + 1, PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

a(n) = 6 * A112775(n) +1.

A108172 Semiprimes p*q where p is a prime of the form 6n+1 (A002476) and q is a prime of the form 6n-1 (A007528).

Original entry on oeis.org

35, 65, 77, 95, 119, 143, 155, 161, 185, 203, 209, 215, 221, 287, 299, 305, 323, 329, 335, 341, 365, 371, 377, 395, 407, 413, 437, 473, 485, 497, 515, 527, 533, 545, 551, 581, 611, 623, 629, 635, 671, 689, 695, 707, 713, 731, 737, 749, 755, 767, 779, 785

Offset: 1

Jonathan Vos Post, Jun 13 2005

Also semiprimes of the form 6n-1 (or 6n+5).
Every semiprime not divisible by 2 or 3 must be in one of these three disjoint sets:
A108164 - the product of two primes of the form 6n+1 (A002476),
A108166 - the product of two primes of the form 6n-1 (A007528),
A108172 - the product of a prime of the form 6n+1 and a prime of the form 6n-1.
The product of a prime of the form 6n+1 and a prime of the form 6n-1 is a semiprime of the form 6n-1.
There are 740 of these numbers below 10,000.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A001358, A002476, A007528, A190299.

Select[Range[15,1000,2], Last/@FactorInteger[#]=={1,1} && IntegerQ[(#-2)/3]&] (* Vladimir Joseph Stephan Orlovsky, May 07 2011 *)

list(lim)=my(v=List(),t); forprime(p=5, lim\7, if(p%6<5, next); forprime(q=7, lim\5, if(q%6>1, next); t=p*q; if(t>lim, break); listput(v, t))); Set(v) \\ Charles R Greathouse IV, Feb 08 2017

a(n) = 6 * A112776(n) + 5.

Edited by Ray Chandler, Oct 15 2005

A107890 Semiprimes that are the product of two members of A007645.

Original entry on oeis.org

9, 21, 39, 49, 57, 91, 93, 111, 129, 133, 169, 183, 201, 217, 219, 237, 247, 259, 291, 301, 309, 327, 361, 381, 403, 417, 427, 453, 469, 471, 481, 489, 511, 543, 553, 559, 579, 589, 597, 633, 669, 679, 687, 703, 721, 723, 763, 793, 813, 817, 831, 849, 871

Offset: 1

Jonathan Vos Post, Jun 12 2005

Conway, J. H. and Guy, R. K., The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
Wagon, S. "Eisenstein Primes." Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Eisenstein Integer.
Eric Weisstein's World of Mathematics, Eisenstein Prime.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A007645, A108164.

N:= 1000: # for terms <= N
P:= [3,op(select(isprime, [seq(i,i=1..N/3,6)]))]:
R:= NULL:
for i from 1 while P[i]^2 <= N do
  m:= ListTools:-BinaryPlace(P,N/P[i]+1/2);
  R:= R, seq(P[i]*P[j],j=i..m);
od:
sort([R]); # Robert Israel, Aug 28 2020

{a(n)} = {p*q: p and q both elements of A007645} = {p*q: p and q both of form 3*m^2 * n^2 for integers m, n}.

Edited by Ray Chandler, Oct 15 2005

Definition corrected by N. J. A. Sloane, Feb 06 2008

A167857 Numbers whose divisors are represented by an integer polynomial.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 11, 13, 17, 19, 22, 23, 25, 29, 31, 34, 37, 41, 43, 46, 47, 49, 53, 55, 58, 59, 61, 67, 71, 73, 79, 82, 83, 85, 89, 91, 94, 97, 101, 103, 106, 107, 109, 113, 115, 118, 121, 127, 131, 133, 137, 139, 142, 145, 149, 151, 157, 163, 166, 167, 169, 171

Offset: 1

T. D. Noe, Nov 13 2009

nonn

That is, these numbers n have the property that there is a polynomial f(x) with integer coefficients whose values at x=0..tau(n)-1 are the divisors of n, where tau(n) is the number of divisors of n.

Every prime has this property, as do 1 and 9, the squares of primes of the form 6k+1, and semiprimes p*q with p and q both primes of the form 3k-1 or 3k+1. Terms of the form p^2*q also appear. We can find terms of the form p^m for any m. For example, 2311^13 is the smallest 13th power that appears. For any m, it seems that p^m appears for p a prime of the form k*m#+1, where m# is the product of the primes up to m. Are there terms with three distinct prime divisors?

			The divisors of 55 are (1, 5, 11, 55). The polynomial 1+15x-17x^2+6x^3 takes these values at x=0..3.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A108164, A108166, A112774 (forms of semiprimes)
Cf. A002476 (primes of the form 6k+1)
Cf. A132230 (primes of the form 30k+1)
Cf. A073103 (primes of the form 210k+1)
Cf. A073917 (least prime of the form k*prime(n)#+1)

Select[Range[1000], And @@ IntegerQ /@ CoefficientList[Expand[InterpolatingPolynomial[Divisors[ # ], x+1]], x] &]

is(n)=my(d=divisors(n));denominator(content(polinterpolate([0..#d-1],d))) == 1 \\ Charles R Greathouse IV, Jan 29 2016

cp's OEIS Frontend

A108166 Semiprimes p*q where both p and q are primes of the form 6n-1 (A007528).

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A112771 Semiprimes of the form 6n + 1.

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A108172 Semiprimes p*q where p is a prime of the form 6n+1 (A002476) and q is a prime of the form 6n-1 (A007528).

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A107890 Semiprimes that are the product of two members of A007645.

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Maple

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A167857 Numbers whose divisors are represented by an integer polynomial.

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