A007528 -id:A007528 - 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

A334479 Decimal expansion of Product_{k>=1} (1 + 1/A007528(k)^3).

Original entry on oeis.org

1, 0, 0, 9, 1, 3, 4, 5, 0, 8, 6, 3, 8, 4, 7, 4, 4, 7, 8, 0, 7, 1, 1, 3, 7, 5, 3, 9, 5, 8, 9, 2, 0, 5, 5, 8, 8, 1, 7, 4, 5, 6, 4, 7, 8, 5, 2, 9, 5, 2, 5, 5, 9, 9, 3, 0, 7, 2, 3, 6, 2, 0, 8, 1, 4, 8, 7, 9, 6, 2, 8, 3, 5, 9, 1, 6, 3, 6, 0, 3, 2, 1, 1, 9, 3, 2, 6, 6, 4, 3, 5, 2, 6, 4, 0, 4, 9, 6, 5, 9, 7, 5, 6, 1, 6

Offset: 1

Vaclav Kotesovec, May 02 2020

In general, for s > 0, Product_{k>=1} (1 + 1/A007528(k)^(2*s+1)) / (1 - 1/A007528(k)^(2*s+1)) = (1 - 1/2^(2*s + 1)) * (3^(2*s + 1) - 1) * (2*s)! * zeta(2*s + 1) / (sqrt(3) * A002114(s) * Pi^(2*s + 1)).
For s > 1, Product_{k>=1} (1 + 1/A007528(k)^s) / (1 - 1/A007528(k)^s) = (2^s - 1) * (3^s - 1) * zeta(s) / (zeta(s, 1/6) - zeta(s, 5/6)).
For s > 1, Product_{k>=1} (1 + 1/A002476(k)^s) * (1 + 1/A007528(k)^s) = 6^s * zeta(s) / ((2^s + 1) * (3^s + 1) * zeta(2*s)).

			1.0091345086384744780711375395892055881745647852...

Cf. A007528, A175646, A334424, A334426, A334480, A334482.

A334479 / A334480 = 91*sqrt(3)*zeta(3)/(6*Pi^3).

A334477 * A334479 = 810*zeta(3)/Pi^6.

More digits from Vaclav Kotesovec, Jun 27 2020

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

A334480 Decimal expansion of Product_{k>=1} (1 - 1/A007528(k)^3).

Original entry on oeis.org

9, 9, 0, 8, 8, 4, 1, 4, 5, 5, 2, 5, 2, 1, 3, 3, 5, 6, 5, 6, 3, 4, 0, 3, 1, 7, 3, 5, 5, 9, 4, 3, 2, 7, 5, 1, 6, 4, 3, 4, 8, 3, 1, 2, 1, 7, 5, 0, 0, 7, 6, 1, 3, 3, 0, 4, 8, 6, 7, 7, 4, 7, 8, 4, 9, 4, 3, 1, 7, 8, 8, 8, 2, 5, 7, 6, 7, 4, 3, 1, 7, 7, 5, 2, 7, 6, 3, 4, 5, 2, 1, 7, 8, 9, 8, 8, 9, 2, 9, 2, 1, 3, 5, 4, 6, 7

Offset: 0

Vaclav Kotesovec, May 02 2020

In general, for s > 0, Product_{k>=1} (1 + 1/A007528(k)^(2*s+1)) / (1 - 1/A007528(k)^(2*s+1)) = (1 - 1/2^(2*s + 1)) * (3^(2*s + 1) - 1) * (2*s)! * zeta(2*s + 1) / (sqrt(3) * A002114(s) * Pi^(2*s + 1)).
For s > 1, Product_{k>=1} (1 + 1/A007528(k)^s) / (1 - 1/A007528(k)^s) = (2^s - 1) * (3^s - 1) * zeta(s) / (zeta(s, 1/6) - zeta(s, 5/6)).
For s > 1, Product_{k>=1} (1 - 1/A002476(k)^s) * (1 - 1/A007528(k)^s) = 6^s / ((2^s - 1)*(3^s - 1)*zeta(s)).

			0.990884145525213356563403173559432751643483121750... = 1/1.0091997177631243951237...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26 (case 6 5 3 = 1/A334480).

Cf. A007528, A175646, A334425, A334427, A334479.

A334479 / A334480 = 91*sqrt(3)*zeta(3)/(6*Pi^3).

A334478 * A334480 = 108/(91*zeta(3)).

More digits from Vaclav Kotesovec, Jun 27 2020

A334482 Decimal expansion of Product_{k>=1} (1 + 1/A007528(k)^2).

Original entry on oeis.org

1, 0, 5, 8, 7, 6, 0, 2, 0, 1, 7, 8, 2, 5, 4, 5, 4, 9, 1, 3, 1, 5, 8, 9, 5, 4, 5, 4, 5, 7, 2, 1, 5, 3, 3, 3, 6, 7, 3, 4, 7, 1, 2, 6, 6, 3, 2, 4, 9, 5, 1, 2, 2, 4, 0, 7, 9, 5, 9, 2, 7, 0, 1, 0, 8, 2, 2, 2, 9, 4, 1, 4, 4, 9, 9, 3, 8, 1, 9, 3, 0, 0, 7, 1, 8, 2, 1, 2, 7, 2, 3, 4, 9, 6, 3, 6, 0, 4, 8, 4, 2, 7, 2, 9, 7

Offset: 1

Vaclav Kotesovec, May 02 2020

Product_{k>=1} (1 - 1/A007528(k)^2) = 9*A175646/Pi^2 = 0.9429084997268899069451546585312672145658112624159...

			1.058760201782545491315895454572153336734712663249...

Cf. A007528, A175646, A334479, A334481.

A334481 * A334482 = 54/(5*Pi^2).

More digits from Vaclav Kotesovec, Jun 27 2020

A095016 Number of 6k+5 primes (A007528) in range [2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 1, 3, 4, 6, 12, 23, 38, 68, 129, 236, 438, 806, 1516, 2864, 5388, 10178, 19327, 36839, 70201, 134151, 256884, 492947, 947240, 1822831, 3513553, 6781479, 13103713, 25349311, 49091941, 95168089, 184662764, 358633903, 697097374, 1356055862, 2639879029, 5142830496

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A007528, A036378, A095015.

a(n) = A036378(n) - A095015(n) for n >= 2.

a(34)-a(38) from Amiram Eldar, Jun 12 2024

A354543 Convolution of A007528 and A002476.

Original entry on oeis.org

35, 142, 357, 746, 1351, 2250, 3533, 5248, 7467, 10232, 13675, 17910, 22979, 28972, 35931, 44192, 53677, 64392, 76727, 90640, 106209, 123614, 142849, 164232, 187841, 213802, 242181, 273080, 306733, 343266, 382745, 425218, 470685, 519740, 572275, 628302, 688277, 752440, 820557, 892634, 969475

Offset: 2

J. M. Bergot and Robert Israel, Aug 17 2022

nonn

Convolution of the primes == 1 (mod 6) and the primes == 5 (mod 6).

			a(4) = A007528(1)*A002476(3) + A007528(2)*A002476(2) + A007528(3)*A002476(1) = 7*17 + 13*11 + 19*5 = 357.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A002476, A007528, A354542.

P1:= select(isprime, [seq(i,i=1..10000,6)]):
P5:= select(isprime, [seq(i,i=5..10000,6)]):
seq(add(P1[i]*P5[n-i],i=1..n-1), n=1..min(nops(P1),nops(P5))+1);

a(n) = Sum_{j=1..n-1} A007528(j)*A002476(n-j).

A354510 Primes of the form p+q^2+r where p,q,r are three consecutive members of A007528.

Original entry on oeis.org

13007, 28211, 36857, 39227, 86441, 272507, 345731, 459671, 467867, 553529, 599087, 746507, 777911, 788561, 910127, 1354901, 1425653, 1512923, 1587587, 1710869, 2039171, 2509061, 2624411, 3196913, 3617597, 3896657, 4161611, 4260077, 4359749, 4460549, 4536893, 4639757, 5171093, 5280791, 5673911, 5963351

Offset: 1

J. M. Bergot and Robert Israel, Aug 16 2022

nonn

Primes of the form p+q^2+r where p, q and r are consecutive members of the sequence of primes of the form 6*k-1.

All terms == 5 (mod 6).

			a(3) = 36857 is in the sequence because 36857 = 179 + 191^2 + 197 and 179 = A007528(21), 191 = A007528(22) and 197 = A007528(23).

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

Cf. A007528.

q:= 5: r:= 11: count:= 0: R:= NULL:
while count < 40 do
  p:= q; q:= r;
  do r:= r+6 until isprime(r);
  if isprime(p+q^2+r) then count:= count+1; R:= R, p+q^2+r fi
od:
R;

Select[#[[1]] + #[[2]]^2 + #[[3]] & /@ Partition[Select[Prime[Range[400]], Mod[#1, 6] == 5 &], 3, 1], PrimeQ] (* Amiram Eldar, Aug 16 2022 *)

A144918 Duplicate of A007528.

Original entry on oeis.org

5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

dead

A144920 Duplicate of A007528.

Original entry on oeis.org

5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane, Jul 21 2020

dead

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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A334479 Decimal expansion of Product_{k>=1} (1 + 1/A007528(k)^3).

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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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A334480 Decimal expansion of Product_{k>=1} (1 - 1/A007528(k)^3).

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A334482 Decimal expansion of Product_{k>=1} (1 + 1/A007528(k)^2).

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A095016 Number of 6k+5 primes (A007528) in range [2^n,2^(n+1)].

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A354543 Convolution of A007528 and A002476.

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A354510 Primes of the form p+q^2+r where p,q,r are three consecutive members of A007528.

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