A037074 -id:A037074 - cp's OEIS frontend

A209328 Decimal expansion of the sum of the inverse twin prime products.

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

Offset: 0

R. J. Mathar, Jan 19 2013

Summation up to the lesser prime(100000) gives 0.1079839703839.., summation up to the lesser prime(5000000) gives 0.10798397490956.. and summation up to the lesser prime(100000000) gives 0.107983974949..

The constant splits Brun's constant B=A065421 into two portions: define L=Sum_{n>=1} 1/A001359(n) and U=Sum_{n>=1} 1/A006512(n). Then B=U+L and this constant here = (L-U)/2. This leads to the estimates L=1.059064 and U=0.843096. - R. J. Mathar, Feb 05 2013

			0.10798397495... = 1/(3*5) + 1/(5*7) + 1/(11*13) + .. = Sum_{n>=1} 1/A037074(n).

Cf. A001359, A006512, A037074, A065421.

A071142 Numbers of the form 2pq where (p,q) is a twin prime pair.

Original entry on oeis.org

30, 70, 286, 646, 1798, 3526, 7198, 10366, 20806, 23326, 38086, 44998, 64798, 73726, 78406, 103966, 115198, 145798, 159046, 194686, 242206, 352798, 373246, 426886, 544966, 649798, 719998, 763846, 824326, 871198, 1312198, 1351366, 1371166, 1472326, 1555846

Offset: 1

Labos Elemer, May 13 2002

nonn

For each term k, A008472(k)/A006530(k) = (2+p+q)/q = (q+q)/q = 2.

			a(1) = 2 * (product of 1st twin prime pair) = 2*3*5 = 30.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Equal[lf[n], 3]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]

a(n) = 2*A037074(n).

Edited by Jon E. Schoenfield, Sep 30 2023

A136016 a(n) = 9*n^2-1.

Original entry on oeis.org

8, 35, 80, 143, 224, 323, 440, 575, 728, 899, 1088, 1295, 1520, 1763, 2024, 2303, 2600, 2915, 3248, 3599, 3968, 4355, 4760, 5183, 5624, 6083, 6560, 7055, 7568, 8099, 8648, 9215, 9800, 10403, 11024, 11663, 12320, 12995, 13688, 14399, 15128, 15875, 16640

Offset: 1

Artur Jasinski, Dec 10 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..3000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001359, A006512, A037074, A248897.

Cf. A005563, A016777, A016789, A073010, A136017.

[9*n^2-1: n in [1..50]]; // Vincenzo Librandi, May 09 2011

Table[9n^2 - 1, {n, 1, 100}]
LinearRecurrence[{3,-3,1},{8,35,80},50] (* Harvey P. Dale, Oct 09 2012 *)

a(n)=9*n^2-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A005563(3*n-1). - Paul Curtz, Oct 28 2008
a(2*n) = A136017(n). - Paul Curtz, Sep 30 2008
a(n) = A016777(n)*A016789(n-1). - Reinhard Zumkeller, Feb 15 2009
G.f.: x*(-8-11*x+x^2) / ( x-1 )^3. - R. J. Mathar, Jul 01 2011
From Amiram Eldar, Jul 31 2020: (Start)
Sum_{n>=1} 1/a(n) = 1/2 - sqrt(3)*Pi/18.
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/9 - 1/2. (End)
From Amiram Eldar, Feb 04 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = 2*Pi/(3*sqrt(3)) (A248897).
Product_{n>=1} (1 - 1/a(n)) = sqrt(2/3)*sin(sqrt(2)*Pi/3). (End)
a(n) = a(-n) for all n in Z. Sum_{n in Z} 1/a(n) = -Pi/3^(3/2) = -A073010. - Michael Somos, May 21 2023
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Jun 19 2025

A210503 Numbers k that form a primitive Pythagorean triple with k' and sqrt(k^2 + k'^2), where k' is the arithmetic derivative of k.

Original entry on oeis.org

15, 35, 143, 323, 899, 1763, 3599, 4641, 5183, 10403, 11663, 13585, 19043, 22499, 32399, 35581, 36863, 39203, 51983, 57599, 72899, 79523, 97343, 121103, 176399, 186623, 213443, 272483, 324899, 359999, 381923, 412163, 435599, 446641, 622081, 656099, 675683

Offset: 1

Paolo P. Lava, Jan 25 2013

nonn

A037074 is a subsequence of this sequence.
If k is the product of a pair of twin primes we have k=p(p+2), k'=2(p+1) and sqrt(k^2+k'^2)=(p+1)^2+1=p(p+2)+2=k+2. These numbers are relatively prime and therefore they form a primitive Pythagorean triple.
Also in the sequence are the following numbers with four distinct prime factors:
        4641 = 3*7*13*17       [form p(p+4)*q(q+4)],
       13585 = 5*11*13*19      [form p(p+6)*q(q+6)],
       35581 = 7*13*17*23      [form p(p+6)*q(q+6)],
      446641 = 13*17*43*47     [form p(p+4)*q(q+4)],
      622081 = 17*23*37*43     [form p(p+6)*q(q+6)],
      700321 = 19*29*31*41     [form p(p+10)*q(q+10)],
From Ray Chandler, Jan 25 2017: (Start)
    24887581 = 47*53*97*103    [form p(p+6)*q(q+6)],
    43518577 = 59*67*101*109   [form p(p+8)*q(q+8)],
   115539901 = 83*97*113*127   [form p(p+14)*q(q+14)],
   158682817 = 89*101*127*139  [form p(p+12)*q(q+12)],
   305162941 = 103*113*157*167 [form p(p+10)*q(q+10)],
  1093514641 = 103*107*313*317 [form p(p+4)*q(q+4)],
  1415940061 = 167*193*197*223 [form p(p+26)*q(q+26)].
And one term with six distinct prime factors:
   650344079 = 7*11*37*53*59*73. (End)

			m=57599, m'=480, sqrt(57599^2 + 480^2) = 57601.

Ray Chandler, Table of n, a(n) for n = 1..500 (terms 1..100 from Paolo P. Lava)

Cf. A003415, A009003, A009004, A037074.

with(numtheory);
A210503:= proc(q)
local a,n,p;
for n from 1 to q do
  a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
  if trunc(sqrt(n^2+a^2))=sqrt(n^2+a^2) and gcd(n,gcd(a,n^2+a^2))=1 then print(n); fi;
od; end:
A210503(100000);

from math import sqrt
from sympy import factorint
from gmpy2 import mpz, is_square, gcd
A210503 = []
for n in range(2, 10**5):
    nd = sum([mpz(n*e/p) for p, e in factorint(n).items()])
    if is_square(nd**2+n**2) and gcd(gcd(n, nd), mpz(sqrt(nd**2+n**2))) == 1:
        A210503.append(n) # Chai Wah Wu, Aug 21 2014

A322362 a(n) = gcd(n, A166590(n)), where A166590 is completely multiplicative with a(p) = p+2 for prime p.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 5, 16, 1, 2, 1, 4, 3, 2, 1, 8, 1, 2, 1, 4, 1, 10, 1, 32, 1, 2, 7, 4, 1, 2, 3, 8, 1, 6, 1, 4, 5, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 20, 1, 2, 9, 64, 5, 2, 1, 4, 1, 14, 1, 8, 1, 2, 5, 4, 1, 6, 1, 16, 1, 2, 1, 12, 1, 2, 1, 8, 1, 10, 1, 4, 3, 2, 1, 32, 1, 2, 1, 4, 1, 2, 1, 8, 105

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A001359, A006512, A037074, A066086, A166590, A322361.

a[n_] := If[n == 1, 1, GCD[n, Times@@ ((First[#]+2)^Last[#] &/@FactorInteger[n])]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018~ *)

A166590(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] += 2); factorback(f); };
A322362(n) = gcd(n, A166590(n));

a(n) = gcd(n, A166590(n)).

a(A037074(n)) = A006512(n).

A059267 Numbers n with 2 divisors d1 and d2 having difference 2: d2 - d1 = 2; equivalently, numbers that are 0 (mod 4) or have a divisor d of the form d = m^2 - 1.

Original entry on oeis.org

3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 66, 68, 69, 70, 72, 75, 76, 78, 80, 81, 84, 87, 88, 90, 92, 93, 96, 99, 100, 102, 104, 105, 108, 111, 112, 114, 116, 117, 120, 123, 124, 126, 128

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Jan 23 2001

nonn

Complement of A099477; A008586, A008585 and A037074 are subsequences - Reinhard Zumkeller, Oct 18 2004
These numbers have an asymptotic density of ~ 0.522. This corresponds to all numbers which are multiples of 4 (25%), or of 3 (having 1 & 3 as divisors: + (1-1/4)*1/3 = 1/4), or of 5*7, or of 11*13, etc. (Generally, multiples of lcm(k,k+2), but multiples of 3 and 4 are already taken into account in the 50% covered by the first 2 terms.) - M. F. Hasler, Jun 02 2012
By considering divisors of the form m^2-1 with m <= 200 it is possible to prove that the density of this sequence is in the interval (0.5218, 0.5226). The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 52, 521, 5219, 52206, 522146, 5221524, 52215473, 522155386, 5221555813, ..., so the asymptotic density of this sequence can be estimated empirically by 0.522155... . - Amiram Eldar, Sep 25 2022

			a(18) = 35 because 5 and 7 divide 35 and 7 - 5 = 2.

M. F. Hasler, Table of n, a(n) for n = 1..3131

Cf. A099475, A099477, A008585, A008586, A037074.

isA059267 := proc(n)
    local m ;
    if modp(n,4)=0 then
        true;
    else
        for m from 2 to ceil(sqrt(n)) do
            if modp(n,m^2-1) = 0 then
                return true ;
            end if;
        end do;
        false ;
    end if;
end proc:
for n from 1 to 130 do
    if isA059267(n) then
        printf("%d,",n) ;
    end if;
end do:

d1d2Q[n_]:=Mod[n,4]==0||AnyTrue[Sqrt[#+1]&/@Divisors[n],IntegerQ]; Select[ Range[ 200],d1d2Q] (* Harvey P. Dale, May 31 2020 *)

isA059267(n)={ n%4==0 || fordiv( n,d, issquare(d+1) && return(1))} \\ M. F. Hasler, Aug 29 2008

is_A059267(n) = fordiv( n,d, n%(d+2)||return(1)) \\ M. F. Hasler, Jun 02 2012

A099475(a(n)) > 0. - Reinhard Zumkeller, Oct 18 2004

More terms from James Sellers, Jan 24 2001

Removed comments linking to A143714, which seem wrong, as observed by Ignat Soroko, M. F. Hasler, Jun 02 2012

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

A072026 Swap twin prime pairs >(3,5) in prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 9, 14, 13, 12, 11, 10, 21, 16, 19, 18, 17, 28, 15, 26, 23, 24, 49, 22, 27, 20, 31, 42, 29, 32, 39, 38, 35, 36, 37, 34, 33, 56, 43, 30, 41, 52, 63, 46, 47, 48, 25, 98, 57, 44, 53, 54, 91, 40, 51, 62, 61, 84, 59, 58, 45, 64, 77, 78

Offset: 1

Reinhard Zumkeller, Jun 07 2002

			a(143)=a(11*13)=a(11)*a(13)=13*11=143; a(77)=a(7*11)=a(7)*a(11)=5*13=65.

Cf. A007510, A037074, A072027, A001359, A006512, A072028, A072029, A061898, A064505.

a[n_] := Product[{p, e} = pe; If[p <= 3, p, If[PrimeQ[p+2], p+2, If[PrimeQ[p-2], p-2, p]]]^e, {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Nov 20 2021 *)

a(a(n)) = n, a self-inverse permutation of natural numbers.
a(n) = n for single primes (A007510) and products of twin prime pairs (A037074).
Multiplicative with a(p) = (if p<=3 then p else (if p+2 is prime then p+2 else (if p-2 is prime then p-2 else p))), p prime.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p, q primes > 3, p = q+2} ((p^2-p)*(q^2-q)/((p^2-q)*(q^2-p))) = 0.53439004468579249988... . - Amiram Eldar, Dec 24 2022

A074040 Product of first n twin prime pair products.

Original entry on oeis.org

15, 525, 75075, 24249225, 21800053275, 38433493923825, 138322144631846175, 716923675626858725025, 7458156997546211316435075, 86984485062381462583582279725, 1656445549042930191979157352803175

Offset: 1

Reinhard Zumkeller, Aug 13 2002

nonn

			The first two twin prime pairs are (3,5) and (5,7), their products: 15 and 35, therefore a(2)=15*35=525.

Cf. A074041, A001359, A006512 & A014574.

Cf. A037074, A077800, A079164.

a = {4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150} (* A014574 *); Table[ Product[a[[k]]^2 - 1, {k, 1, n}], {n, 1, 12}]
Rest[FoldList[Times,1,Times@@@Select[Partition[Prime[Range[50]],2,1],#[[2]]-#[[1]]==2&]]] (* Harvey P. Dale, Jan 19 2015 *)
step[{list_, q_}] := Module[{p=NextPrime[q]}, {Join[list, If[PrimeQ[p+2], {{p,p+2}}, {}]], p}]
pairList[n_] := First[NestWhile[step, {{{3, 5}}, 3}, Length[First[step[#]]]<=n&]]
a037074[n_] := Map[Apply[Times, #]&, pairList[n]]
a074040[n_] := Rest[FoldList[Times, 1, a037074[n]]]
a074040[11] (* Hartmut F. W. Hoft, Apr 27 2021 *)

a(1) = A037074(1) and a(n) = a(n-1)*A037074(n) for n>1.

a(n) = A079164(2*n).

Edited by Robert G. Wilson v, Aug 17 2002

Corrections in Comment and Example, and added references. Hartmut F. W. Hoft, Apr 27 2021

A221054 Numbers whose distinct prime factors can be partitioned into two equal sums.

Original entry on oeis.org

1, 30, 60, 70, 90, 120, 140, 150, 180, 240, 270, 280, 286, 300, 350, 360, 450, 480, 490, 540, 560, 572, 600, 646, 700, 720, 750, 810, 900, 960, 980, 1080, 1120, 1144, 1200, 1292, 1350, 1400, 1440, 1500, 1620, 1750, 1798, 1800, 1920, 1960, 2145, 2160, 2240, 2250, 2288, 2310, 2400, 2430, 2450, 2584, 2700, 2730, 2800, 2880, 3000, 3135

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 15 2013

nonn

This is a superset of 2*product of twin primes, A071142.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Table of n, a(n), and equal sums of factors for n=1..10000

Cf. A175592 (multiplicity of prime factors allowed).
Cf. A071139-A071147, especially A071140.
Cf. A008472, A037074, A071142.
Cf. A027748, A005843, A083207.

a221054 n = a221054_list !! (n-1)
a221054_list = filter (z 0 0 . a027748_row) $ tail a005843_list where
   z u v []     = u == v
   z u v (p:ps) = z (u + p) v ps || z u (v + p) ps
-- Reinhard Zumkeller, Apr 18 2013

q[n_] := Module[{p = FactorInteger[n][[;; , 1]], sum, x}, sum = Total[p]; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, p}], x][[1 + sum/2]] > 0]; Select[Range[3200], q] (* Amiram Eldar, May 31 2025 *)

isok(k) = my(f=factor(k), nb=#f~); for (i=0,2^nb-1, my(v=Vec(Vecrev(binary(i)), nb)); if (sum(k=1, nb, if (v[k], f[k,1])) == sum(k=1, nb, if (!v[k], f[k,1])), return(1));); \\ Michel Marcus, May 31 2025

Missing terms inserted by Michel Marcus, May 31 2025

cp's OEIS Frontend

A209328 Decimal expansion of the sum of the inverse twin prime products.

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A071142 Numbers of the form 2*p*q where (p,q) is a twin prime pair.

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A136016 a(n) = 9*n^2-1.

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A210503 Numbers k that form a primitive Pythagorean triple with k' and sqrt(k^2 + k'^2), where k' is the arithmetic derivative of k.

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A322362 a(n) = gcd(n, A166590(n)), where A166590 is completely multiplicative with a(p) = p+2 for prime p.

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A059267 Numbers n with 2 divisors d1 and d2 having difference 2: d2 - d1 = 2; equivalently, numbers that are 0 (mod 4) or have a divisor d of the form d = m^2 - 1.

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

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Magma

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A072026 Swap twin prime pairs >(3,5) in prime factorization of n.

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A071142 Numbers of the form 2pq where (p,q) is a twin prime pair.

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