A071700 -id:A071700 - cp's OEIS frontend

A182221 Composite numbers in A182140 but not in A071700.

255, 385, 34561, 65535, 147455, 195841, 1034881, 4070401, 4746241, 5040001, 16675201, 22704001, 36067201, 47013121, 83623935, 136967041, 168720001, 271878145, 549141119, 613092481, 836567041, 1039779841, 1049759999, 1548072961, 2556902401, 2646067201

Offset: 1

José María Grau Ribas, Apr 19 2012

nonn

A182140 includes the prime numbers and A071700.

J. M. Grau, A. M. Oller-Marcen, M. Rodríguez, D. Sadornil, Fermat test with gaussian base and Gaussian pseudoprimes, arXiv preprint arXiv:1401.4708 [math.NT], 2014.

Cf. A182140, A071700, A060968, A201629.

fa=FactorInteger;isA071700[n_]:=Length@fa[n]==2&&fa[n][[1,2]]==fa[n][[2,2]]==1&&Mod[Sqrt[n+1],4]==0; A060968[p_, s_] := Which[Mod[p, 4] == 1, p^(s-1)*(p-1), Mod[p, 4]==3, p^(s-1)*(p+1), s==1, 2, True, 2^(s+1)]; A060968[1]=1; A060968[n_] := Product[A060968[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}]; A201629[n_]:=Which[Mod[n, 4]==1, (n-1), Mod[n, 4]==3, (n+1), True, n];
Select[Range[10000], !isA071700[#]&&!PrimeQ[#]&&A060968[#]==A201629[#]&]

isA071700(n)=my(k=sqrtint(n\16)); n==16*k^2+32*k+15 && isprime(4*k+3) && isprime(4*k+5)
is(n)=if(isprime(n),return(0)); my(f=factor(n)[, 1]); n*prod(i=if(n%2, 1, 2), #f, if(f[i]%4==1, 1-1/f[i], 1+1/f[i]))*if(n%4, 1, 2)==if(n%2,(n+1)\4*4,n) && !isA071700(n) \\ Charles R Greathouse IV, Jul 03 2013

a(15)-a(23) from Charles R Greathouse IV, Jul 03 2013

a(24)-a(26) from Charles R Greathouse IV, Jul 05 2013

A037074 Numbers that are the product of a pair of twin primes.

Original entry on oeis.org

15, 35, 143, 323, 899, 1763, 3599, 5183, 10403, 11663, 19043, 22499, 32399, 36863, 39203, 51983, 57599, 72899, 79523, 97343, 121103, 176399, 186623, 213443, 272483, 324899, 359999, 381923, 412163, 435599, 656099, 675683, 685583, 736163

Offset: 1

Felice Russo

Each entry is the product of p and p+2 where both p and p+2 are prime, i.e., the product of the lesser and greater of a twin prime pair.
Except for the first term, all entries have digital root 8. - Lekraj Beedassy, Jun 11 2004
The above statement follows from p > 3 => (p,p+2) = (6k-1,6k+1) => p*(p+2) = 36k^2 - 1 == 8 (mod 9), and A010888 === A010878 (mod 9). - M. F. Hasler, Jan 11 2013
Albert A. Mullin states that m is a product of twin primes iff phi(m)*sigma(m) = (m-3)*(m+1), where phi(m) = A000010(m) and sigma(m) = A000203(m). Of course, for a product of distinct primes p*q we know sigma(p*q) = (p+1)*(q+1) and if p, q, are twin primes, say q = p + 2, then sigma(p*q) = (p+1)*(q+1) = (p+1)*(p+3). - Jonathan Vos Post, Feb 21 2006
Also the area of twin prime rectangles. A twin prime rectangle is a rectangle whose sides are components of twin prime pairs. E.g., the twin prime pair (3,5) produces a 3 X 5 unit rectangle which has area 15 square units. - Cino Hilliard, Jul 28 2006
Except for 15, a product of twin primes is of the form 36k^2 - 1 (cf. A136017, A002822). - Artur Jasinski, Dec 12 2007
A072965(a(n)) = 1; A072965(m) mod A037074(n) > 0 for all m. - Reinhard Zumkeller, Jan 29 2008
The number of terms less than 10^(2n) is A007508(n). - Robert G. Wilson v, Feb 08 2012
If m is the product of twin primes, then sigma(m) = m + 1 + 2*sqrt(m + 1), phi(m) = m + 1 - 2*sqrt(m + 1). pmin(m) = sqrt(m + 1) - 1, pmax(m) = sqrt(m + 1) + 1. - Wesley Ivan Hurt, Jan 06 2013
Semiprimes of the form 4*k^2 - 1. - Vincenzo Librandi, Apr 13 2013

			a(2)=35 because 5*7=35, that is (5,7) is the 2nd pair of twin primes.

Albert A. Mullin, "Bicomposites, twin primes and arithmetic progression", Abstract 04T-11-48, Abstracts of AMS, Vol. 25, No. 4, 2004, p. 795.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A001359, A006512, A014574, A136017, A074480 (multiplicative closure), A209328.
Cf. A071700 (subsequence).
Cf. A075369.

a037074 = subtract 1 . a075369  -- Reinhard Zumkeller, Feb 10 2015
-- Reinhard Zumkeller, Feb 10 2015, Aug 14 2011

[p*(p+2): p in PrimesUpTo(1000) | IsPrime(p+2)];  // Bruno Berselli, Jul 08 2011

IsSemiprime:=func; [s: n in [1..500] | IsSemiprime(s) where s is 4*n^2-1]; // Vincenzo Librandi, Apr 13 2013

ZL:=[]: for p from 1 to 863 do if (isprime(p) and isprime(p+2) ) then ZL:=[op(ZL),(p*(p+2))]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007
for i from 1 to 150 do if ithprime(i+1) = ithprime(i) + 2 then print({ithprime(i)*ithprime(i+1)}); fi; od; # Zerinvary Lajos, Mar 19 2007

s = Select[ Prime@ Range@170, PrimeQ[ # + 2] &]; s(s + 2) (* Robert G. Wilson v, Feb 21 2006 *)
(* For checking large numbers, the following code is better. For instance, we could use the fQ function to determine that 229031718473564142083 is in this sequence. *) fQ[n_] := Block[{fi = FactorInteger[n]}, Last@# & /@ fi == {1, 1} && Differences[ First@# & /@ fi] == {2}]; Select[ Range[750000], fQ] (* Robert G. Wilson v, Feb 08 2012 *)
Times@@@Select[Partition[Prime[Range[500]],2,1],Last[#]-First[#]==2&] (* Harvey P. Dale, Oct 16 2012 *)

g(n) = for(x=1,n,if(prime(x+1)-prime(x)==2,print1(prime(x)*prime(x+1)","))) \\ Cino Hilliard, Jul 28 2006

a(n) = A001359(n)*A006512(n). A000010(a(n))*A000203(a(n)) = (a(n)-3)*(a(n)+1). - Jonathan Vos Post, Feb 21 2006
a(n) = (A014574(n))^2 - 1. a(n+1) = (6*A002822(n))^2 - 1. - Lekraj Beedassy, Sep 02 2006
a(n) = A075369(n) - 1. - Reinhard Zumkeller, Feb 10 2015
Sum_{n>=1} 1/a(n) = A209328. - Amiram Eldar, Nov 20 2020
A000010(a(n)) == 0 (mod 8). - Darío Clavijo, Oct 26 2022

More terms from Erich Friedman

A071698 Lesser members of twin prime pairs of form (4k+3, 4k+5), k >= 0.

Original entry on oeis.org

3, 11, 59, 71, 107, 179, 191, 227, 239, 311, 347, 419, 431, 599, 659, 827, 1019, 1031, 1091, 1151, 1319, 1427, 1451, 1487, 1607, 1619, 1667, 1787, 1871, 1931, 2027, 2087, 2111, 2267, 2339, 2591, 2687, 2711, 2999, 3119, 3167, 3251, 3299

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

Corresponding greater members: A071699(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A001359, A071695, A071700.

Cf. A010051, subsequence of A004767.

a071698 n = a071698_list !! (n-1)
a071698_list = [x | x <- [3, 7 ..], a010051' x == 1, a010051' (x+2) == 1]
-- Reinhard Zumkeller, Aug 05 2014

[4*k+3:k in [0..1000]|IsPrime(4*k+3) and IsPrime(4*k+5)]; // Marius A. Burtea, Nov 06 2019

Transpose[Select[Table[4n+{3,5},{n,0,1000}],AllTrue[#,PrimeQ]&]][[1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 16 2015 *)

a(n) = 2*A241557(n+1)-1. - Hilko Koning, Nov 06 2019

A071697 Product of twin primes of form (4k+1,4k+3), k>0.

Original entry on oeis.org

35, 323, 899, 1763, 10403, 19043, 22499, 39203, 72899, 79523, 213443, 272483, 324899, 381923, 412163, 656099, 675683, 736163, 777923, 1102499, 1127843, 1512899, 1633283, 1664099, 1695203, 2196323, 2883203, 2965283, 3526883, 3802499, 3992003, 4334723, 4536899

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

Cf. A037074, A071700.

Cf. A071695, A071696.

[16*n^2+16*n+3: n in [1..700]| IsPrime(4*n+1) and IsPrime(4*n+3)]; // Vincenzo Librandi, Feb 24 2015

lista(nn) = {forprime(p=3, nn, if ((((p-1) % 4) == 0) && isprime(p+2), print1(p*(p+2), ", ")););} \\ Michel Marcus, Feb 24 2015

a(n) = A071695(n)*A071696(n).

More terms from Michel Marcus, Feb 24 2015

A071699 Greater members of twin prime pairs of form (4k+3, 4k+5), k >= 0.

Original entry on oeis.org

5, 13, 61, 73, 109, 181, 193, 229, 241, 313, 349, 421, 433, 601, 661, 829, 1021, 1033, 1093, 1153, 1321, 1429, 1453, 1489, 1609, 1621, 1669, 1789, 1873, 1933, 2029, 2089, 2113, 2269, 2341, 2593, 2689, 2713, 3001, 3121, 3169, 3253, 3301

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

Corresponding lesser members: A071698(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A071696, A071698, A071700.
Cf. A010051, subsequence of A016813.
Intersection of A006512 and A002144.

a071699 n = a071699_list !! (n-1)
a071699_list = [x | x <- [5, 9 ..], a010051' x == 1, a010051' (x-2) == 1]
-- Reinhard Zumkeller, Aug 05 2014

[4*(k+1)+1:k in [0..1000]|IsPrime(4*k+3) and IsPrime(4*k+5)]; // Marius A. Burtea, Nov 06 2019

#+2&/@Select[4Range[0,850]+3,PrimeQ[#]&&PrimeQ[#+2]&] (* Harvey P. Dale, Aug 24 2011 *)

a(n) = 2*A241557(n+1) + 1. - Hilko Koning, Nov 06 2019

A074480 Multiplicative closure of twin prime pair products (A037074).

Original entry on oeis.org

1, 15, 35, 143, 225, 323, 525, 899, 1225, 1763, 2145, 3375, 3599, 4845, 5005, 5183, 7875, 10403, 11305, 11663, 13485, 18375, 19043, 20449, 22499, 26445, 31465, 32175, 32399, 36863, 39203, 42875, 46189, 50625, 51983, 53985, 57599, 61705

Offset: 1

Reinhard Zumkeller, Aug 23 2002

			a(99) = 1040399 = 1019*1021.
a(101) = 1090125 = (3*5)*(3*5)*(3*5)*(17*19).
a(103) = 1101275 = (5*7)*(5*7)*(29*31).
a(105) = 1126125 = (3*5)*(3*5)*(5*7)*(11*13).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A037074, A001359, A006512, A062505, A072965.

Cf. A071700 (subsequence).

import Data.Set (Set, singleton, delete, findMin, deleteFindMin, insert)
a074480 n = a074480_list !! (n-1)
a074480_list = multClosure a037074_list where
  multClosure []     = [1]
  multClosure (b:bs) = 1:h [b] (singleton b) bs where
   h cs s []    = m:h (m:cs) (foldl (flip insert) s' $ map (*m) cs) []
    where (m, s') = deleteFindMin s
   h cs s xs'@(x:xs)
    | m < x     = m:h (m:cs) (foldl (flip insert) s' $ map (*m) cs) xs'
    | otherwise = x:h (x:cs) (foldl (flip insert) s  $ map (*x) (x:cs)) xs
    where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 14 2011

max = 70000; t1 = Select[Prime /@ Range[PrimePi[Sqrt[max]]], PrimeQ[# + 2] &]; pairs = Join[{1}, t1*(t1 + 2)]; f[pairs_] := Outer[Times, pairs, pairs] // Flatten // Union // Select[#, # <= max &] &; FixedPoint[f, pairs] (* Jean-François Alcover, Dec 11 2012 *)

A072965(a(n)) = 1.

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/A037074(k)) = 1.117122860973... . - Amiram Eldar, Apr 13 2025

A182140 Numbers n such that A060968(n) = A201629(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 255, 257

Offset: 1

José María Grau Ribas, Apr 14 2012

nonn

Includes prime numbers and the sequence A071700.

a(n) = A240960(n) for n <= 35. - Reinhard Zumkeller, Aug 05 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A060968, A201629, A071700, A240960.

a182140 n = a182140_list !! (n-1)
a182140_list = [x | x <- [1..], a060968 x == a201629 x]
-- Reinhard Zumkeller, Aug 05 2014

fa = FactorInteger; A060968[p_, s_] := Which[Mod[p, 4] == 1, p^( s - 1)*(p - 1), Mod[p, 4] == 3, p^(s - 1)*(p + 1), s == 1,2, True, 2^(s + 1)]; A060968[1] = 1; A060968[n_] := Product[A060968[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}]; A201629[n_] := Which[Mod[n, 4] == 1, (n - 1), Mod[n, 4] == 3, (n + 1), True, n]; Select[Range[1000], A060968[#] == A201629[#] &]

is(n)=my(f=factor(n)[, 1]); n*prod(i=if(n%2, 1, 2), #f, if(f[i]%4==1, 1-1/f[i], 1+1/f[i]))*if(n%4, 1, 2)==if(n%2,(n+1)\4*4,n) \\ Charles R Greathouse IV, Jul 03 2013

cp's OEIS Frontend

A182221 Composite numbers in A182140 but not in A071700.

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A037074 Numbers that are the product of a pair of twin primes.

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A071698 Lesser members of twin prime pairs of form (4*k+3, 4*k+5), k >= 0.

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A071697 Product of twin primes of form (4*k+1,4*k+3), k>0.

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A071699 Greater members of twin prime pairs of form (4*k+3, 4*k+5), k >= 0.

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Haskell

Magma

Mathematica

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A074480 Multiplicative closure of twin prime pair products (A037074).

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A182140 Numbers n such that A060968(n) = A201629(n).

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A071698 Lesser members of twin prime pairs of form (4k+3, 4k+5), k >= 0.

A071697 Product of twin primes of form (4k+1,4k+3), k>0.

A071699 Greater members of twin prime pairs of form (4k+3, 4k+5), k >= 0.