A074480 -id:A074480 - cp's OEIS frontend

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

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

A062505 Numbers k such that if p is a prime that divides k, then either p + 2 or p - 2 is also prime.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 31, 33, 35, 39, 41, 43, 45, 49, 51, 55, 57, 59, 61, 63, 65, 71, 73, 75, 77, 81, 85, 87, 91, 93, 95, 99, 101, 103, 105, 107, 109, 117, 119, 121, 123, 125, 129, 133, 135, 137, 139, 143, 145, 147, 149, 151, 153, 155, 165

Offset: 1

Leroy Quet, Jul 09 2001

nonn

Multiplicative closure of twin primes (A001097).

			35 is included because 35 = 5*7 and both (5+2) and (7-2) are primes.
65 = 5*13 where the factors are members of twin prime pairs: (3,5) and (11,13), therefore a(29) = 65 is a term; but 69 is not because 69 = 3*23 and 23 = A007510(2) is a single prime.

Stephan Ramon Garcia and Steven J. Miller, 100 Years of Math Milestones: The Pi Mu Epsilon Centennial Collection, American Mathematical Society, 2019, pp. 35-37.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Range of A072963.

Cf. A001097, A074480.

[k:k in [1..170] | forall{p:p in PrimeDivisors(k)| IsPrime(p+2) or IsPrime(p-2)}]; // Marius A. Burtea, Dec 30 2019

nmax = 15 (* corresponding to last twin prime pair (197,199) *); tp[1] = 3; tp[n_] := tp[n] = (p = NextPrime[tp[n-1]]; While[ !PrimeQ[p+2], p = NextPrime[p]]; p); twins = Flatten[ Table[ {tp[n], tp[n]+2}, {n, 1, nmax}]]; max = Last[twins]; mult[twins_] := Select[ Union[ twins, Apply[ Times, Tuples[twins, {2}], {1}]], # <= max & ]; A062505 = Join[{1}, FixedPoint[mult, twins] ] (* Jean-François Alcover, Feb 23 2012 *)

A072965 In prime factorization of n replace all matching twin prime pairs with 1, where (3,5)-matches are replaced before (5,7).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2, 31, 32, 33, 34, 1, 36, 37, 38, 39, 40, 41, 42, 43, 44, 3, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 4, 61, 62, 63, 64, 65, 66, 67, 68, 69, 2, 71, 72, 73, 74

Offset: 1

Reinhard Zumkeller, Aug 20 2002

a(a(n)) = a(n); a(A037074(n)) = 1.
a(n) = 1 iff n = A074480(k) for some k.
a(n) mod A037074(k) > 0 for all k. - Reinhard Zumkeller, Jan 29 2008

			a(30)=a(2*3*5)=2*1=2; a(105)=a(3*5*7)=1*7=7; a(143)=a(11*13)=1; a(225)=a(3*3*5*5)=a((3*5)*(3*5))=1*1=1; a(525)=a(3*5*5*7)=a((3*5)*(5*7))=1*1=1.

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

Cf. A001359, A006512.

Cf. A027746.

a072965 n = f 1 (a027746_row n) where
   f y []      = y
   f y [p]     = p * y
   f y (2:ps)  = f (2 * y) ps
   f y (3:5:_) = a072965 (n `div` 15)
   f y (p:qs@(q:ps)) | q == p + 2 = f y ps
                     | otherwise  = f (p * y) qs
-- Reinhard Zumkeller, Oct 31 2012

a[n_] := Times @@ (Flatten[ (Table[#[[1]], {#[[2]]}] & ) /@ FactorInteger[n]] //. {p1___, p2_, p3_, p4___} /; p3 == p2 + 2 -> {p1, p4}); Table[a[n], {n, 1, 74}](* Jean-François Alcover, Nov 04 2011 *)

a(n)=my(f=factor(n),t);for(i=2,#f[,1],if(f[i-1,1]+2==f[i,1],t=min(f[i-1,2],f[i,2]);f[i-1,2]-=t;f[i,2]-=t));factorback(f) \\ Charles R Greathouse IV, Nov 04 2011

A079877 Numbers that are divisible by at least one pair of twin primes (A001097).

Original entry on oeis.org

15, 30, 35, 45, 60, 70, 75, 90, 105, 120, 135, 140, 143, 150, 165, 175, 180, 195, 210, 225, 240, 245, 255, 270, 280, 285, 286, 300, 315, 323, 330, 345, 350, 360, 375, 385, 390, 405, 420, 429, 435, 450, 455, 465, 480, 490, 495, 510, 525, 540, 555, 560, 570, 572

Offset: 1

Reinhard Zumkeller, Feb 20 2003

Numbers that are divisible by both primes of some twin-prime pair. Harvey P. Dale, Aug 18 2017

By definition, if k is in the sequence, then so is every positive multiple of k. - Richard Locke Peterson, Aug 17 2017

			429 = 3*11*13 = 3*A001359(3)*A006512(3), therefore 429 is a term.

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

Cf. A001097, A001359, A006512, A037074, A062505, A074480, A289484.

Select[Range[600],MemberQ[Differences[Transpose[FactorInteger[#]][[1]]], 2]&] (* Harvey P. Dale, Sep 19 2011 *)

Definition clarified by N. J. A. Sloane, Aug 18 2017

Definition further clarified by Sean A. Irvine, Aug 29 2025

A334141 Numbers that are the product of distinct twin primes.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 29, 31, 33, 35, 39, 41, 43, 51, 55, 57, 59, 61, 65, 71, 73, 77, 85, 87, 91, 93, 95, 101, 103, 105, 107, 109, 119, 123, 129, 133, 137, 139, 143, 145, 149, 151, 155, 165, 177, 179, 181, 183, 187, 191, 193, 195, 197, 199

Offset: 1

Ilya Gutkovskiy, Apr 15 2020

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Subsequence of A005117.

Cf. A001097, A037074, A048599, A074480, A073485.

N:= 1000: # for terms <= N
P:= select(isprime, {seq(i,i=3..N+2,2)}):
TP:= P intersect map(`+`,P,2):
TP:= map(t -> (t-2,t), TP):
TP:= sort(convert(TP,list)):
S:= {1}:
for i from 1 to nops(TP) do
  S0:= S;
  S:= S union map(`*`, select(`<=`,S,N/TP[i]),TP[i]);
od:
sort(convert(S,list)); # Robert Israel, Oct 28 2020

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

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A062505 Numbers k such that if p is a prime that divides k, then either p + 2 or p - 2 is also prime.

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A072965 In prime factorization of n replace all matching twin prime pairs with 1, where (3,5)-matches are replaced before (5,7).

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A079877 Numbers that are divisible by at least one pair of twin primes (A001097).

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A334141 Numbers that are the product of distinct twin primes.

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