A037074 -id:A037074 - cp's OEIS frontend

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

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

A073831 Maximum of A073830(k) for k between A001359(n) and A001359(n+1).

Original entry on oeis.org

8, 56, 182, 552, 1406, 2862, 4556, 9506, 10712, 17292, 19460, 30102, 32942, 37442, 49952, 54522, 69432, 77006, 94556, 113906, 167690, 177662, 209306, 259590, 317532, 352242, 376382, 398792, 427062, 636006, 658532, 678152

Offset: 1

Reinhard Zumkeller, Aug 12 2002

nonn

a(n) = A073830(A073832(n)); a(n) < A037074(n+2).

A073831 := proc(n)
    A073830(A073832(n)) ;
end proc:
seq(A073831(n),n=1..50) ; # R. J. Mathar, Feb 21 2017

f[n_] := Mod[4*((n - 1)! + 1) + n, n*(n + 2)];
pp = Select[Prime[Range[200]], PrimeQ[# + 2] & ];
a[n_] := Max[f /@ Range[pp[[n]], pp[[n + 1]]]];
Array[a, Length[pp] - 1] (* Jean-François Alcover, Feb 22 2018 *)

A171727 The number of twin prime pairs in the interval (p^2,p*q), where (p,q) runs over the twin prime pairs (A001359(n),A006512(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 1, 3, 2, 2, 4, 7, 3, 3, 5, 7, 4, 4, 7, 6, 11, 9, 5, 11, 9, 9, 11, 10, 11, 9, 11, 11, 12, 11, 12, 18, 12, 12, 16, 11, 16, 20, 14, 16, 15, 20, 16, 22, 13, 22, 16, 17, 21, 20, 20, 23, 22, 23, 20, 21, 21, 26, 20, 28, 24, 24, 23, 24, 25, 21, 24, 37, 27, 21, 28, 24, 31

Offset: 1

Jaspal Singh Cheema, Dec 16 2009

nonn

If you graph the order of the twin primes along the x-axis (i.e., first twin, second, third, ...) and the number of twins in the sequence given above along the y-axis, a clear pattern emerges. As you go farther along the x-axis, the number of twin primes, on average, within the interval increases. The pattern appears to be nonlinear. If one could prove that there's at least one twin prime within each interval, the twin prime conjecture would be proved since the n-th twin produces larger intervals with more twin primes. The evidence seems overwhelming.

			The first twin prime pair (3,5) corresponds to the interval (9,15), which contains one twin prime pair (11,13), so a(1) = 1.
The fifth twin prime pair (29,31) corresponds to the interval (841,899), which contains the twin prime pairs (857,859) and (881,883), so a(5) = 2.

C. C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Perseus Books, 1999.
J. Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Penguin Books Canada Ltd., 2004.
M. du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins Publishers Inc., 2004.

J. S. Cheema, Table of n, a(n) for n = 1..1044

Cf. A001359, A006512, A108570, A037074. - Charlie Neder, Feb 12 2019

{for(k=1, 300, if(prime(k+1)-prime(k)==2, my(c=0); forprime(m=prime(k)^2, prime(k)*prime(k+1), c+=isprime(m+2)); print1(c, ", ")))} \\ Zhandos Mambetaliyev, Mar 28 2021

Partially edited by Michel Marcus, Mar 19 2013

Edited by Charlie Neder, Feb 12 2019

A386991 Numbers k such that k^2 + sopfr(k)^2 is a square, where sopfr = A001414.

Original entry on oeis.org

1, 8, 15, 35, 112, 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

Robert Israel, Aug 12 2025

nonn

Includes A037074 because if k = p*(p+2) where p and p+2 are primes, k^2 + sopfr(k)^2 = p^2*(p+2)^2 + (2*p+2)^2 = (p^2 + 2*p + 2)^2.

Are 1, 8 and 112 the only terms not in A037074?

			a(3) = 15 is a term because the sum of prime factors of 15 is 3+5 = 8 and 15^2 + 8^2 = 289 = 17^2.

Robert Israel, Table of n, a(n) for n = 1..300

Cf. A001414, A386246. Includes A037074.

sopfr:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
filter:= t -> issqr(t^2 + sopfr(t)^2):
select(filter, [$1..10^5]);

Sopfr[1]=0;Sopfr[n_]:= Plus @@ Times @@@ FactorInteger@ n;Select[Range[500000],IntegerQ[Sqrt[#^2+Sopfr[#]^2]]&] (* James C. McMahon, Aug 14 2025 *)

A063531 Numbers k such that sigma(k) + 1 is a square.

Original entry on oeis.org

2, 7, 8, 14, 15, 23, 32, 33, 35, 47, 54, 56, 57, 60, 72, 78, 79, 84, 87, 92, 95, 120, 123, 124, 128, 138, 143, 154, 165, 167, 174, 184, 190, 196, 213, 223, 235, 242, 252, 253, 258, 267, 295, 312, 315, 319, 323, 327, 348, 359, 375, 378, 380, 393, 412, 423, 439

Offset: 1

Labos Elemer, Aug 02 2001

nonn

Numbers k such that A000203(k) = -1 + m^2 for some m.

			If k = p(p+2) is a product of twin primes (from A037074), then sigma(k) + 1 = 1 + (p+1)(p+3) = (p+2)^2, square of the larger twin. Other solutions can be either special primes = m^2 - 2 or composites like 120: sigma(120) = 120 + 60 + ... + 1 = 360 = 19^2 - 1. Square number solution is, e.g., 196: sigma(196) = 399 = 20^2 - 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000203, A028871, A037074, A063530, A088580.

Select[Range[500],IntegerQ[Sqrt[DivisorSigma[1,#]+1]]&] (* Harvey P. Dale, Jul 02 2021 *)

{ n=0; for (a=1, 10^9, if (issquare(sigma(a) + 1), write("b063531.txt", n++, " ", a); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 25 2009

Minor edits from Franklin T. Adams-Watters, Aug 29 2009

A071146 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n has exactly 7 distinct prime factors and n is squarefree.

Original entry on oeis.org

1231230, 2062830, 2181270, 3327870, 3594990, 4224990, 4320030, 4671030, 5162430, 5411406, 5414430, 6767670, 7052430, 7432230, 7870830, 7947030, 8150142, 8273265, 8287230, 8569470, 8804334, 9378390, 10630830, 10705695, 10757838, 10776990, 10900230

Offset: 1

Labos Elemer, May 13 2002

nonn

			n = pqrstu, p<q<r<s<t<u, primes, p+q+r+s+t+u = ku; n = 9378390 = 2*3*5*7*17*37*71; sum = 2+3+5+7+17+37+71 = 142 = 2*71

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], 7]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]

A008472(n)/A006530(n) is an integer; A001221(n) = 7, n is squarefree.

A126249 p(p+1)(p+2)/6 where (p,p+2) are twin primes.

Original entry on oeis.org

10, 35, 286, 969, 4495, 12341, 35990, 62196, 176851, 209934, 437989, 562475, 971970, 1179616, 1293699, 1975354, 2303960, 3280455, 3737581, 5061836, 7023974, 12347930, 13436856, 16435111, 23706021, 30865405, 35999900, 39338069

Offset: 1

Lekraj Beedassy, Dec 21 2006

nonn

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

Cf. A126248.

ZL:=[]:for p from 1 to 617 do if (isprime(p) and isprime(p+2) ) then ZL:=[op(ZL),(binomial((p+2),p-1))]; fi; od; print(ZL); # Zerinvary Lajos, Mar 08 2007

Times@@# Mean[#]/6&/@Select[Partition[Prime[Range[250]],2,1],#[[2]]- #[[1]] == 2&] (* Harvey P. Dale, Sep 20 2014 *)

a(n) = A001359(n)*A014574(n)*A006512(n)/6;
a(n) = A037074(n)*A014574(n)/6;
a(n) = A007531(A006512(n))/6.
a(n) = A037074(n)*A002822(n-1), for n > 1.
a(n) = A126248(n)/6.

A138637 Products of prime quadruples.

Original entry on oeis.org

5005, 46189, 121330189, 1445140189, 463236778189, 4862973196189, 12359548828189, 18898278256189, 112254342850189, 144149198626189, 1022657400370189, 7924420639216189, 28604961973900189, 59910402098980189

Offset: 1

Jonathan Vos Post, May 14 2008

Product of numbers n, n+2, n+6 and n+8 when are all prime. Quadruplet analog of A037074. Subset of A014613.

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

Cf. A007530, A014613, A037074.

isA007530 := proc(n) local q; if isprime(n) and n>=5 then q := nextprime(n) ; if q-n = 2 then q := nextprime(q) ; if q -n = 6 then q := nextprime(q) ; RETURN( q-n = 8 ) ; else RETURN(false) ; fi ; else RETURN(false) ; ; fi ; else RETURN(false) ; ; fi ; end: A007530 := proc(n) option remember ; local a; if n = 1 then 5 ; else a := nextprime(A007530(n-1)) ; while not isA007530(a) do a := nextprime(a) ; od: RETURN(a) ; fi ; end: A138637 := proc(n) local p ; p := A007530(n) ; p*(p+2)*(p+6)*(p+8) ; end: seq(A138637(n),n=1..20) ; # R. J. Mathar, May 18 2008

a = {}; For[n = 1, n < 5000, n++, If[{Prime[n+1]-Prime[n], Prime[n+2]-Prime[n+1], Prime[n+3]-Prime[n+2]} == {2, 4, 2}, AppendTo[a, Prime[n]*Prime[n+1]*Prime[n+2]* Prime[n+3]]]]; a (* Stefan Steinerberger, May 18 2008 *)
Times@@@Select[Partition[Prime[Range[2500]],4,1],Differences[#]=={2,4,2}&] (* Harvey P. Dale, Sep 10 2018 *)

a(n) = A007530(n)*A007530(n+2)*A007530(n+6)*A007530(n+8).

More terms from Stefan Steinerberger and R. J. Mathar, May 18 2008

A143957 An integer >= 2 is included if {the difference between the largest and smallest primes dividing n} divides n+1.

Original entry on oeis.org

6, 12, 14, 15, 18, 20, 24, 35, 36, 39, 44, 45, 48, 50, 54, 63, 72, 75, 80, 84, 96, 108, 119, 135, 143, 144, 147, 152, 153, 155, 162, 175, 192, 200, 208, 216, 224, 225, 230, 231, 242, 245, 275, 279, 288, 294, 299, 315, 320, 323, 324, 374, 375, 384, 399, 405, 429

Offset: 1

Leroy Quet, Sep 05 2008

nonn

			The largest prime dividing 14 is 7. The smallest prime dividing 14 is 2. 7-2=5 divides 14+1=15. So 14 is in the sequence.

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

Cf. A143958. Includes A037074.

filter:= proc(n) local P;
  P:= numtheory:-factorset(n);
  if nops(P) = 1 then return false fi;
  n+1 mod (max(P)-min(P)) = 0
end proc:
select(filter, [$2..1000]); # Robert Israel, Nov 04 2020

Select[Range[2, 429], If[Or[PrimeQ[#], PrimePowerQ[#]], False, Mod[#1, Last[#2] - First[#2]] == 0 & @@ {# + 1, FactorInteger[#][[All, 1]]}] &] (* Michael De Vlieger, Nov 04 2020 *)

Extended by Ray Chandler, Nov 07 2008

A153196 Numbers n such that 6n+5 and 6n+7 are twin primes.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 11, 16, 17, 22, 24, 29, 31, 32, 37, 39, 44, 46, 51, 57, 69, 71, 76, 86, 94, 99, 102, 106, 109, 134, 136, 137, 142, 146, 169, 171, 174, 176, 181, 191, 204, 212, 214, 216, 219, 237, 241, 246, 247, 267, 269, 277, 282, 286, 297, 311, 312, 321, 324, 332

Offset: 1

Vincenzo Librandi, Dec 20 2008

Appears to be the partial sums of A160273 which are the successive differences (divided by 3) of the average of twin prime pairs divided by 2 (A040040). - Stephen Crowley, May 24 2009

			For n = 0, 6*n+5 = 5 and 6*n+7 = 7 are twin primes;
for n = 99, 6*n+5 = 599 and 6*n+7 = 601 are twin primes.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A001359 (lesser of twin primes), A002822 (6n-1, 6n+1 are twin primes).

Cf. A037074. - Vincenzo Librandi, Dec 26 2008

[ n: n in [0..335] | IsPrime(6*n+5) and IsPrime(6*n+7) ];

ZL := []; for p to 1000000 do if `and`(isprime(p), isprime(p+2)) then ZL := [op(ZL), ((p+2)^2-p^2)*(1/8)] end if end do; A160273 := [seq((ZL[i+1]-ZL[i])*(1/3), i = 2 .. nops(ZL)-1)]: ListTools[PartialSums]( A160273 ); # Stephen Crowley, May 24 2009

Select[Range[0, 350], PrimeQ[6 # + 5]&&PrimeQ[6 # + 7]&] (* Vincenzo Librandi, Apr 04 2013 *)

a(j) = (A001359(j+1)-5)/6.

a(j) = A002822(j)-1.

Edited and extended by Klaus Brockhaus, Dec 26 2008

cp's OEIS Frontend

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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A073831 Maximum of A073830(k) for k between A001359(n) and A001359(n+1).

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A171727 The number of twin prime pairs in the interval (p^2,p*q), where (p,q) runs over the twin prime pairs (A001359(n),A006512(n)).

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A386991 Numbers k such that k^2 + sopfr(k)^2 is a square, where sopfr = A001414.

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A063531 Numbers k such that sigma(k) + 1 is a square.

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A071146 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n has exactly 7 distinct prime factors and n is squarefree.

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A126249 p*(p+1)*(p+2)/6 where (p,p+2) are twin primes.

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A138637 Products of prime quadruples.

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A126249 p(p+1)(p+2)/6 where (p,p+2) are twin primes.

A153196 Numbers n such that 6n+5 and 6n+7 are twin primes.