A037074 -id:A037074 - cp's OEIS frontend

A071139 Numbers k such that the sum of distinct primes dividing k is divisible by the largest prime dividing k.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 70, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 107, 109, 113, 120, 121, 125, 127, 128, 131, 137, 139, 140, 149, 150, 151, 157, 163, 167, 169, 173, 179, 180

Offset: 1

Labos Elemer, May 13 2002

nonn

All primes and prime powers are terms, as are certain other composites (see Example section).
If k is a term then every multiple of k having no prime factors other than those of k are also terms. E.g., since 286 = 2*11*13 is a term, so are 572 = 286*2 and 3146 = 286*11.
If k = 2*p*q where p and q are twin primes, then sum = 2+p+q = 2q is divisible by q, the largest prime factor, so 2*A037074 is a subsequence.

			30 = 2*3*5; sum of distinct prime factors is 2+3+5 = 10, which is divisible by 5, so 30 is a term;
2181270 = 2*3*5*7*13*17*47; sum of distinct prime factors is 2+3+5+7+13+17+47 = 94, which is divisible by 47, so 2181270 is a term.

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

Cf. A008472, A006530, A000961, A025475, A037074.

a071139 n = a071139_list !! (n-1)
a071139_list = filter (\x -> a008472 x `mod` a006530 x == 0) [2..]
-- Reinhard Zumkeller, Apr 18 2013

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] Do[s=sb[n]/ma[n]; If[IntegerQ[s], Print[{n, ba[n]}]], {n, 2, 1000000}]

isok(n) = if (n != 1, my(f=factor(n)[,1]); (sum(k=1, #f~, f[k]) % vecmax(f)) == 0); \\ Michel Marcus, Jul 09 2018

A008472(k)/A006530(k) is an integer.

Edited by Jon E. Schoenfield, Jul 08 2018

A138389 Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 20, 21, 23, 24, 25, 29, 31, 33, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199

Offset: 1

Vladimir Shevelev, May 08 2008

nonn

Note that every i not exceeding n/2 for which (n,i)=1 divides binomial(n-i-1,i-1). For n>33, a(n) is either prime or square of a prime or a product of twin primes. For a proof, see link of V. Shevelev.

Numbers n such that A178105(n) = 0. - Michel Marcus, Feb 07 2016

Peter J. C. Moses, Table of n, a(n) for n = 1..10000
V. Shevelev, On divisibility of binomial(n-i-1,i-1) by i, Intl. J. of Number Theory, 3, no.1 (2007), 119-139.

Cf. A000040, A001248, A077800, A037074, A178105.

Select[Range@ 200, Function[n, NoneTrue[Select[Range@ Floor[n/2], ! CoprimeQ[#, n] &], Divisible[Binomial[n - # - 1, # - 1], #] &]]] (* Michael De Vlieger, Feb 07 2016, Version 10 *)

isok(n) = {my(md = -1); for (d=2, n\2, if (((binomial(n-d-1,d-1) % d) == 0) && (gcd(n, d) > 1), if (md == -1, md = d, md = min(d, md)));); (md == -1);} \\ Michel Marcus, Feb 07 2016

A051779 Primes of form pq + 2 where p and q are twin primes.

Original entry on oeis.org

17, 37, 22501, 32401, 57601, 72901, 176401, 324901, 1664101, 1742401, 5336101, 6502501, 7452901, 11289601, 11492101, 18147601, 21622501, 34222501, 34574401, 40449601, 45968401, 81000001, 85377601, 92736901, 110880901, 118592101

Offset: 1

Joe DeMaio (jdemaio(AT)kennesaw.edu), Dec 09 1999

Starting with 3rd term, 22501, all terms are of the form 900n^2+1 with n=5, 6, 8, 9, 14, 19, 43, 44, 77, 85 (A125251). - Zak Seidov, Dec 07 2008

Primes of the form (p^2 + q^2)/2, where p and q are twin primes. - Thomas Ordowski and Altug Alkan, Mar 19 2017

			The third term 22501 is a member of the sequence because 22501=149*151+2, 22501 is prime and {149,151} is a twin prime pair.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A048880, A051779, A000040, A001359, A005384, A006512, A037074, A054735.

Cf. A125251. - Zak Seidov, Dec 07 2008

with (numtheory): for n from 1 to 2000 do if (ithprime(n+1)-ithprime(n)=2) then if (tau(ithprime(n)*ithprime(n+1)+2)=2) then print(ithprime(n),ithprime(n+1), ithprime(n)*ithprime(n+1)+2); fi; fi; od;

lst={};Do[p=Prime[n];If[Length[Divisors[p-2]]==4&&(Divisors[p-2][[3]]-Divisors[p-2][[2]])==2, AppendTo[lst, p]], {n, 6*10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 08 2008 *)
Select[(First[#]Last[#]+2)&/@Select[Partition[Prime[Range[2700]], 2,1], Last[#]-First[#]==2&],PrimeQ]  (* Harvey P. Dale, Mar 11 2011 *)
Select[2 + Times @@@ Select[ Partition[ Prime@ Range@ 1350, 2, 1], First[#] + 2 == Last[#] &], PrimeQ] (* Robert G. Wilson v, Mar 12 2001 *)

{A037074(k) + 2} INTERSECT {A000040}. {A001359(k) * A006512(k) + 2} INTERSECT {A000040}. {A054735(k)^2 + 2*A054735(k) + 2} INTERSECT {A000040}. - Jonathan Vos Post, May 11 2006

Edited by R. J. Mathar, Aug 08 2008

A111192 Product of the n-th sexy prime pair.

Original entry on oeis.org

55, 91, 187, 247, 391, 667, 1147, 1591, 1927, 2491, 3127, 4087, 4891, 5767, 7387, 9991, 10807, 11227, 12091, 17947, 23707, 25591, 28891, 30967, 37627, 38407, 51067, 52891, 55687, 64507, 67591, 70747, 75067, 78391, 96091, 98587, 111547, 122491, 126727, 136891

Offset: 1

Shawn M Moore (sartak(AT)gmail.com), Oct 23 2005

nonn

Semiprime of the form 4*m^2-9 = (2*m-3)*(2*m+3). - Vincenzo Librandi, Jan 26 2016

			a(2)=91 because the second sexy prime pair is (7, 13) and 7*13=91.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes. - _N. J. A. Sloane_, Mar 07 2021].

Cf. A023201, A104229.

Cf. A037074, A143206, A195118; intersection of A143205 and A001358.

a111192 n = a111192_list !! (n-1)
a111192_list = f a000040_list where
   f (p:ps@(q:r:_)) | q - p == 6 = (p*q) : f ps
                    | r - p == 6 = (p*r) : f ps
                    | otherwise  = f ps
-- Reinhard Zumkeller, Sep 13 2011

IsSemiprime:=func; [s: n in [1..300] | IsSemiprime(s) where s is 4*n^2-9]; // Vincenzo Librandi, Jan 26 2016

#(#+6)&/@Select[Prime[Range[100]], PrimeQ[#+6]&] (* Harvey P. Dale, Dec 17 2010 *)
(* For checking large numbers, the following code is better. For instance, we could use the fQ function to determine that 229031718473564142083 is not in this sequence. *) fQ[n_] := Block[{fi = FactorInteger[n]}, Last@# & /@ fi == {1, 1} && Differences[ First@# & /@ fi] == {6}]; Select[ Range[125000], fQ] (* Robert G. Wilson v, Feb 08 2012 *)
Select[Table[4 n^2 - 9, {n, 300}], PrimeOmega[#] == 2 &] (* Vincenzo Librandi, Jan 26 2016 *)

a(n) = A023201(n) * A046117(n). - Reinhard Zumkeller, Sep 13 2011

A143206 Product of the n-th cousin prime pair.

Original entry on oeis.org

21, 77, 221, 437, 1517, 2021, 4757, 6557, 9797, 11021, 12317, 16637, 27221, 38021, 50621, 53357, 77837, 95477, 99221, 123197, 145157, 159197, 194477, 210677, 216221, 239117, 250997, 378221, 416021, 455621, 549077, 576077, 594437, 680621

Offset: 1

Reinhard Zumkeller, Aug 12 2008

nonn

Intersection of A143203 and A001358.

Sum_{n>=2} 1/a(n) > 0.02187310784. - R. J. Mathar, Jan 23 2013

			a(1) = 3*7 = 3*(3+4) = 21;
a(2) = 7*11 = 7*(7+4) = 77;
a(3) = 13*17 = 13*(13+4) = 221;
a(4) = 19*23 = 19*(19+4) = 437.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cousin Primes

Cf. A037074, A111192.

a143206 n = a143206_list !! (n-1)
a143206_list = (3*7) : f a000040_list where
   f (p:ps@(p':_)) | p'-p == 4 = (p*p') : f ps
                   | otherwise = f ps
-- Reinhard Zumkeller, Sep 13 2011

[(p*(p+4)): p in PrimesUpTo(1000)| IsPrime(p+4)]; // Vincenzo Librandi, Jan 04 2018

fQ[n_] := Block[{fi = FactorInteger@ n}, Last@# & /@ fi == {1, 1} && Differences[ First@# & /@ fi] == {4}]; Select[ Range@ 700000, fQ] (* Robert G. Wilson v, Feb 08 2012 *)

lista(nn) = forprime(p=2, nn, if (isprime(q=p+4), print1(p*q, ", "))); \\ Michel Marcus, Jan 04 2018

a(n) = A023200(n)*A046132(n).

A167054 Values of A167053(k)-A167053(k-1)-1 not equal to 1.

Original entry on oeis.org

15, 19, 41, 83, 167, 337, 673, 1361, 2729, 5471, 10949, 21911, 43853, 87719, 175447, 350899, 701819, 1403641, 2807303, 5614657, 11229331, 22458671, 44917381, 89834777, 179669557, 359339171, 718678369

Offset: 1

Vladimir Shevelev, Oct 27 2009

All terms of the sequence are primes or products of twin primes (A037074).

Cf. A167053, A166944, A166945, A116533, A163961, A163963, A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293

Values from a(3) on replaced by R. J. Mathar, Dec 17 2009

More terms from Amiram Eldar, Sep 13 2019

A071140 Numbers n such that sum of distinct primes dividing n is divisible by largest prime dividing n; n is neither a prime, nor a true power of prime.

Original entry on oeis.org

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, 2160, 2240, 2250

Offset: 1

Labos Elemer, May 13 2002

nonn

a(n) are the numbers such that the difference between the largest and the smallest prime divisor equals the sum of the other distinct prime divisors. - Michel Lagneau, Nov 13 2011
The statement above is only true for 966 of the first 1000 terms. The first counterexample is a(140) = 15015. - Donovan Johnson, Apr 10 2013
Lagneau's definition can be simplified to the largest prime divisor equals the sum of the other distinct prime divisors. - Christian N. K. Anderson, Apr 15 2013

			n = 70 = 2*5*7 has a form of 2pq, where p and q are twin primes; n = 3135 = 3*5*11*19, sum = 3+5+11+19 = 38 = 2*19, divisible by 19.

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

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

Subsequence of A024619.

a071140 n = a071140_list !! (n-1)
a071140_list = filter (\x -> a008472 x `mod` a006530 x == 0) a024619_list
-- Reinhard Zumkeller, Apr 18 2013

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] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Greater[s, 1], Print[{n, ba[n]}]], {n, 2, 1000000}]
(* Second program: *)
Select[Range@ 2250, And[Length@ # > 1, Divisible[Total@ #, Last@ #]] &[FactorInteger[#][[All, 1]] ] &] (* Michael De Vlieger, Jul 18 2017 *)

A008472(n)/A006530(n) is an integer and n has at least 3 distinct prime factors.

A008472(a(n)) mod A006530(a(n)) = 0 and A010055(a(n)) = 0. - Reinhard Zumkeller, Apr 18 2013

A071147 Smallest squarefree number k with exactly n prime factors such that the sum of the prime factors is divisible by the largest prime dividing k, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 0, 30, 3135, 3570, 72930, 1231230, 14804790, 497668710, 14908423530, 278196808890, 12192694624110, 550939666387110, 21275256232500270, 1458502323630662310, 87988283090327810190, 3254611619240885033130, 261462818462495728868790, 9965666894849284108299810, 557940830126698960967415390, 90544636506979071680577724410

Offset: 0

Labos Elemer, May 13 2002

nonn

No solution exists for n=2, so a(2)=0.

			a(0) =       1 = 1;
a(1) =       2 = 2;
a(3) =      30 = 2 *  3 *  5;
a(4) =    3135 = 3 *  5 * 11 * 19;
a(5) =    3570 = 2 *  3 *  5 *  7 * 17;
a(6) =   72930 = 2 *  3 *  5 * 11 * 13 * 17;
a(7) = 1231230 = 2 *  3 *  5 *  7 * 11 * 13 * 41.

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071146.

A008472(k)/A006530(k) is an integer; k is squarefree and has exactly n prime factors.

Corrected and extended by Donovan Johnson, Apr 22 2008

Name corrected by Jon E. Schoenfield, Jul 08 2018

A071700 Product of twin primes of form (4k+3,4(k+1)+1), k>=0.

Original entry on oeis.org

15, 143, 3599, 5183, 11663, 32399, 36863, 51983, 57599, 97343, 121103, 176399, 186623, 359999, 435599, 685583, 1040399, 1065023, 1192463, 1327103, 1742399, 2039183, 2108303, 2214143, 2585663, 2624399, 2782223

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

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

Subsequence of A037074, A071700 and of A182140.

Cf. A071697, a(n) = A071698(n)*A071699(n).

a071700 n = a071700_list !! (n-1)
a071700_list = [x * y | x <- [3, 7 ..], a010051' x == 1,
                        let y = x + 2, a010051' y == 1]
-- Reinhard Zumkeller, Aug 05 2014

for(k=0,1e3,if(isprime(4*k+3)&&isprime(4*k+5),print1(16*k^2+32*k +15", "))) \\ Charles R Greathouse IV, Jul 03 2013

is(n)=my(k=sqrtint(n\16)); n==16*k^2+32*k+15 && isprime(4*k+3) && isprime(4*k+5) \\ Charles R Greathouse IV, Jul 03 2013

is(n)=my(t); n%16==15 && issquare(n+1,&t) && isprime(t-1) && isprime(t+1) \\ Charles R Greathouse IV, Dec 12 2016

list(lim)=my(v=List(),p=3); forprime(q=5,sqrtint(1+lim\1)+1, if(q-p==2 && p%4==3, listput(v,p*q)); p=q); Vec(v) \\ Charles R Greathouse IV, Dec 12 2016

a(n) >> n^2 log^4 n. - Charles R Greathouse IV, Jul 03 2013

A075369 Square associated with twin primes (p,p+2): p(p+2) + 1. Square of the average of twin primes.

Original entry on oeis.org

16, 36, 144, 324, 900, 1764, 3600, 5184, 10404, 11664, 19044, 22500, 32400, 36864, 39204, 51984, 57600, 72900, 79524, 97344, 121104, 176400, 186624, 213444, 272484, 324900, 360000, 381924, 412164, 435600, 656100, 675684, 685584, 736164

Offset: 1

Amarnath Murthy, Sep 20 2002

nonn

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

Cf. A014574, A037074, A120875.

a075369 = (^ 2) . a014574  -- Reinhard Zumkeller, Feb 10 2015

[a: n in [1..300] | IsSquare(a) where a is NthPrime(n)*NthPrime(n+1)+1]; // Vincenzo Librandi, Nov 19 2015

P:= select(isprime,{seq(2*i+1,i=1..1000)}):
T:= P intersect map(`+`,P,2):
sort(convert(map(t -> (t-1)^2, T), list)); # Robert Israel, Nov 18 2015

f[n_]:=Prime[n]*Prime[n+1]+1; lst={}; Do[If[IntegerQ[Sqrt[f[n]]],AppendTo[lst,f[n]]],{n,4*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 10 2010 *)

p=2; forprime(b=3, 1e3, if(b-p==2, print1(b*p+1", ")); p=b) \\ Altug Alkan, Nov 10 2015

a(n) = A037074(n) + 1. - Jon E. Schoenfield, Jan 13 2015
a(n) = A014574(n)^2. - Jon E. Schoenfield, Jan 14 2015
a(n) = A120875(n) + 2. - Jason Kimberley, Oct 22 2015

cp's OEIS Frontend

A071139 Numbers k such that the sum of distinct primes dividing k is divisible by the largest prime dividing k.

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A138389 Binomial primes: positive integers n such that every i not coprime to n and not exceeding n/2 does not divide binomial(n-i-1,i-1).

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A051779 Primes of form pq + 2 where p and q are twin primes.

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Maple

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A111192 Product of the n-th sexy prime pair.

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Magma

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A143206 Product of the n-th cousin prime pair.

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Magma

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A167054 Values of A167053(k)-A167053(k-1)-1 not equal to 1.

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A071140 Numbers n such that sum of distinct primes dividing n is divisible by largest prime dividing n; n is neither a prime, nor a true power of prime.

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A071700 Product of twin primes of form (4k+3,4(k+1)+1), k>=0.