A040040 -id:A040040 - cp's OEIS frontend

A223138 Numbers n such that sigma(n+1) - sigma(n-1) = n / k for some integer, where sigma(n) = A000203 (sum of divisors of n).

4, 5, 6, 9, 10, 12, 18, 30, 32, 42, 54, 56, 60, 72, 101, 102, 108, 129, 138, 144, 150, 172, 176, 180, 192, 198, 204, 216, 220, 228, 240, 252, 270, 282, 312, 348, 384, 420, 432, 462, 522, 544, 570, 600, 618, 642, 648, 660, 792, 810, 822, 828, 858, 882, 900, 1020

Offset: 1

Jaroslav Krizek, May 01 2013

nonn

Supersequence of A014574 for k = n/2 (average of twin prime pairs).

Corresponding values of integers k: 2, 1, 3, 3, -10, 6, 9, 15, 2, 21, 3, 7, 30, 36, -101, 51, 54, -43, 69, 12, 75, -2, -22, 90, 96, 99, 17, -27, -5, 114, 120, 7, 135, 141, 156, 174, 2, 210, 216, 231, 261, -8, 285, 300, 309, 321, 9, 330, -18, 405, 411, 414, 429, 441, 75, 510, ... (supersequence of A040040).

			Number 5 is in sequence because sigma(6) - sigma(4) = 12 - 7 = 5; k=1.

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

Cf. A000203, A014574, A040040.

Select[Range[1000], DivisorSigma[1, # + 1] - DivisorSigma[1, # - 1] != 0 && IntegerQ[#/(DivisorSigma[1, # + 1] - DivisorSigma[1, # - 1])] &] (* T. D. Noe, May 02 2013 *)

A243811 Numbers k such that 2k+3 and 2k+5 are both prime.

Original entry on oeis.org

0, 1, 4, 7, 13, 19, 28, 34, 49, 52, 67, 73, 88, 94, 97, 112, 118, 133, 139, 154, 172, 208, 214, 229, 259, 283, 298, 307, 319, 328, 403, 409, 412, 427, 439, 508, 514, 523, 529, 544, 574, 613, 637, 643, 649, 658, 712, 724, 739, 742, 802, 808, 832, 847, 859, 892, 934

Offset: 1

Vincenzo Librandi, Jun 11 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A040040, A067076, A089038.

[n: n in [0..1000] | IsPrime(2*n+3) and IsPrime(2*n+5)];

Select[Range[0, 1000], PrimeQ[2 # + 3] && PrimeQ[2 # + 5] &]

a(n) = A040040(n)-2.

A264263 The number of distinct nontrivial integral cevians of an isosceles triangle, with base of length 1 and legs of length n, that divide the base into two integral parts.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 3, 3, 1, 3, 3, 2, 5, 3, 1, 3, 7, 3, 3, 3, 1, 5, 5, 2, 5, 3, 3, 7, 3, 1, 5, 11, 3, 3, 3, 1, 5, 11, 3, 4, 4, 3, 7, 3, 3, 7, 7, 3, 5, 5, 1, 7, 7, 1, 3, 3, 3, 11, 11, 5, 5, 7, 3, 3, 3, 3, 15, 7, 1, 3, 7, 7, 11, 5, 1, 5, 11, 3, 3, 7, 3, 7, 7, 2

Offset: 1

Colin Barker, Nov 10 2015

A cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension).
A nontrivial cevian is one that does not coincide with a side of the triangle.
If a(n) = 1 then the length of the unique cevian is n^2.
It seems that a(n) = 1 if and only if n is the average of twin prime pairs divided by 2 (A040040).

			a(4) = 2 because for legs of length 4 there are two cevians, of length 6 and 16, that divide the base into two integral parts.

Colin Barker, Table of n, a(n) for n = 1..1000
Wikipedia, Cevian
Wikipedia, Isosceles triangle

Cf. A040040, A264264.

ceviso(n) = {
  my(d, L=List());
  for(k=1, n^2,
    if(issquare(n^2+k^2-k, &d) && d!=n,
      listput(L, d)
    )
  );
  Vec(L)
}
vector(100, n, #ceviso(n))

A306247 Numbers k such that 2k - p is not a prime where p is any prime divisor of 4k^2 - 1.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 15, 19, 21, 26, 29, 30, 34, 36, 40, 48, 49, 51, 54, 61, 63, 64, 69, 74, 75, 79, 82, 84, 86, 89, 90, 95, 96, 99, 103, 106, 110, 111, 112, 114, 119, 120, 135, 139, 141, 146, 147, 149, 151, 152, 153, 154, 156, 161, 166, 169, 173, 174, 179, 180, 184, 186, 187, 190, 194

Offset: 1

Juri-Stepan Gerasimov, Jan 31 2019

Primes in a(n): 2, 3, 19, 29, 61, 79, 89, 103, 139, 149, 151, 173, 179, ...

			1 is a term because 4*1^2 - 1 = 3 and 2*1 - 3 = -1 (nonprime);
2 is a term because 4*2^2 - 1 = 15 and 2*2 - 15 = -11 (nonprime);
3 is a term because 4*3^2 - 1 = 35 and 2*3 - 35 = -29 (nonprime);
6 is a term because 4*6^2 - 1 = 143 = 11*13 and 2*6 - 11 = 1 (nonprime), 2*6 - 13 = -1 (nonprime);
9 is a term because 4*9^2 - 1 = 323 = 17*19 and 2*9 - 17 = 1 (nonprime), 2*9 - 19 = -1 (nonprime).

Includes A040040.

Cf. A306261.

filter:= proc(n) andmap(`not` @ isprime, map(p -> 2*n-p, numtheory:-factorset(4*n^2-1))) end proc:
select(filter, [$1..300]); # Robert Israel, Jan 31 2019

Select[Range@ 200, AllTrue[2 # - FactorInteger[4 #^2 - 1][[All, 1]], ! PrimeQ@ # &] &] (* Michael De Vlieger, Feb 03 2019 *)

isok(k) = {my(pf = factor(4*k^2-1)[,1]); #select(x->isprime(2*k-x), pf) == 0;} \\ Michel Marcus, Mar 02 2019

A306261(a(n)) > 1 for n >= 4.

A365416 Numbers k such that 2k-1 and 2k+1 are both prime powers (A246655).

Original entry on oeis.org

2, 3, 4, 5, 6, 9, 12, 13, 14, 15, 21, 24, 30, 36, 40, 41, 51, 54, 63, 69, 75, 84, 90, 96, 99, 114, 120, 121, 135, 141, 156, 174, 180, 210, 216, 231, 261, 285, 300, 309, 321, 330, 364, 405, 411, 414, 420, 429, 441, 510, 516, 525, 531, 546, 576, 615, 639, 645, 651, 660, 684

Offset: 1

Jianing Song, Oct 22 2023

According to Pillai's conjecture, k = 13 is the only term such that 2*k-1 and 2*k+1 both have exponent greater than 1.

			41 is a term since 2*41-1 = 81 is a prime power, and 2*41+1 = 83 is a prime.

Jianing Song, Table of n, a(n) for n = 1..10000
Wikipedia, Catalan's conjecture. Pillai's conjecture.

Cf. A246655. Supersequence of A040040 and 2*A365411.

Cf. A088071, A175593.

isA365416(n) = isprimepower(2*n-1) && isprimepower(2*n+1)

A094949 Phi(m)*sigma(m), where m is the product of exactly two primes that differ by 2, where phi=A000010, sigma=A000203.

Original entry on oeis.org

192, 1152, 20160, 103680, 806400, 3104640, 12945600, 26853120, 108201600, 136002240, 362597760, 506160000, 1049630400, 1358807040, 1536796800, 2702128320, 3317529600, 5314118400, 6323748480, 9475464960, 14665694400

Offset: 1

Lekraj Beedassy, Jun 19 2004

nonn

If m=p*q for the twin prime pair (p, q), then the relation p^2 + q^2 = 2*(m+2) is evident from equations p*(p+2)=m=q*(q-2). Now phi(m)=(p-1)*(q-1)=p^2 - 1 and sigma(m)=(p+1)*(q+1)=q^2 - 1, so that phi(m)*sigma(m)=(p*q)^2 -(p^2 + q^2)+1=m^2-2*(m+2)+1=(m-3)*(m+1).

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

Cf. A001359, A006512, A037074.

EulerPhi[#]DivisorSigma[1,#]&/@Times@@@Select[Partition[Prime[ Range[ 200]],2,1],#[[2]]-#[[1]]==2&] (* Harvey P. Dale, Apr 13 2017 *)

PARI
```
{m=400;p=1;while(p
				
```

a(n)=(m-3)*(m+1), where m=A037074(n).
a(n)=192*A002415(k), where k=A040040(n-1).
a(n) = (A120875(n))^2 - 4 = 4*((A120876(n))^2 - 1). - Lekraj Beedassy, Jul 09 2006

Corrected and extended by Jason Earls, Rick L. Shepherd, Vladeta Jovovic and Klaus Brockhaus, Jun 20 2004

A113710 a(n) = A113709(n)/(prime(n+1) - prime(n)).

Original entry on oeis.org

2, 3, 2, 6, 4, 9, 5, 4, 15, 6, 10, 21, 11, 8, 9, 30, 11, 17, 36, 13, 20, 14, 12, 25, 51, 26, 54, 28, 9, 32, 22, 69, 14, 75, 26, 27, 41, 28, 29, 90, 19, 96, 49, 99, 17, 18, 56, 114, 58, 39, 120, 25, 42, 43, 44, 135, 46, 70, 141, 29, 21, 77, 156, 79, 23, 56, 34, 174, 88, 59, 45, 62

Offset: 2

Leroy Quet, Nov 06 2005

nonn

Records are in A040040. - Andres Cicuttin, Nov 26 2016

			Between the primes 67 and 71 is the composite 68 and 68 is divisible by (71-67)=4. So 68/(71-67)= 17 is in the sequence.

Cf. A113709.

a(n) = floor(p(n+1)/(p(n+1)-p(n))) = ceiling(p(n)/(p(n+1)-p(n))), where p(n) is the n-th prime. - Leroy Quet, Feb 03 2007

a(n) = A113709(n)/A001223(n). - Omar E. Pol, Nov 26 2016

More terms from R. J. Mathar, Aug 31 2007

A123200 Numbers k such that 1000000k-1 and 1000000k+1 are twin primes.

Original entry on oeis.org

24, 30, 198, 345, 348, 432, 438, 471, 492, 609, 669, 774, 777, 858, 864, 1032, 1083, 1125, 1218, 1395, 1536, 1824, 1914, 1929, 2088, 2139, 2301, 2334, 2376, 2418, 2448, 2460, 2544, 2763, 2832, 2970, 3021, 3297, 3369, 3384, 3495, 3528, 3540, 3633, 3777

Offset: 1

Jonathan Vos Post, Nov 05 2006

			a(6) = 432 because 431999999 and 432000001 are primes.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A000040, A040040, A045753, A002822, A124065.

a:=proc(n) if isprime(10^6*n-1)=true and isprime(10^6*n+1)=true then n else fi end: seq(a(n),n=1..4500); # Emeric Deutsch, Nov 16 2006

Select[Range[3800],AllTrue[#*10^6+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 13 2017 *)

More terms from Emeric Deutsch, Nov 16 2006

A160919 Averages of twin prime pairs that are sums of 5 consecutive averages of twin prime pairs.

Original entry on oeis.org

108, 570, 858, 1452, 3330, 6792, 7458, 9420, 9630, 10710, 10890, 13722, 17388, 18120, 25032, 27582, 27792, 34032, 68712, 68898, 72270, 76830, 78978, 81372, 89820, 90402, 95232, 99708, 104472, 119772, 122868, 125790, 138078, 165312

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 30 2009

			Averages of twin prime pairs: 4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, ...
108 = 6 + 12 + 18 + 30 + 42, 570 = 72 + 102 + 108 + 138 + 150, ...

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

Cf. A040040, A014574, A160916, A160917, A160918

PrimeNextTwinAverage[n_]:=Module[{k},k=n+1; While[ !PrimeQ[k-1]||!PrimeQ[k+1],k++ ];k];lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],a=n;b=PrimeNextTwinAverage[a]; c=PrimeNextTwinAverage[b]; d=PrimeNextTwinAverage[c];e=PrimeNextTwinAverage[d]; a=a+b+c+d+e; If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,a]]],{n,3*8!}];lst
Select[Total/@(Partition[Mean/@Select[Partition[Prime[Range[10000]],2,1],#[[2]]-#[[1]]==2&],5,1]),AllTrue[#+{1,-1},PrimeQ]&] (* Harvey P. Dale, Sep 26 2024 *)

A160920 Primes which are at the same time balanced primes of order 2, 3 and 4.

Original entry on oeis.org

236429, 1108477, 1829801, 2073263, 2191513, 2192789, 3236267, 3990031, 4248947, 4485683, 4986061, 6869969, 7711079, 8473811, 8480911, 9282173, 9327277, 9350123, 9547303, 9730649, 12077909, 12993917, 13165441, 13398611, 14129761, 14785907

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 30 2009

nonn

The intersection of A082077, A082078 and A082079.

Muniru A Asiru, Table of n, a(n) for n = 1..200
Wikipedia, Balanced prime

Cf. A040040, A051795, A082077, A082078, A082079.

P:=Filtered([1,3..2*10^7+1],IsPrime);;
a:=Intersection(List([2,3,4],b->List(Filtered(List([0..Length(P)-(2*b+1)],k->List([1..2*b+1],j->P[j+k])),i->Sum(i)/(2*b+1)=i[b+1]),m->m[b+1]))); # Muniru A Asiru, Apr 08 2018

isBalPr := proc(p,o) local r,s,i ; r := p ; if isprime(p) then s := p ; for i from 1 to o do r := nextprime(r) ; s := s+r ; end do: r := p ; for i from 1 to o do r := prevprime(r) ; s := s+r ; end do: s := s/(2*o+1) ; if s = p then true; else false; end if; else false; end if; end proc:
isA160920 := proc(p) isBalPr(p,2) and isBalPr(p,3) and isBalPr(p,4) ; end proc:
for i from 10 do p := ithprime(i) ; if isA160920(p) then printf("%d,\n",p); end if; end do: # R. J. Mathar, Dec 15 2010

PrimeNext[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k],k++ ];k];PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k];lst={};Do[p=Prime[n];a1=PrimePrev[p];a2=PrimePrev[a1];a3=PrimePrev[a2];a4=PrimePrev[a3];a5=PrimePrev[a4];b1=PrimeNext[p];b2=PrimeNext[b1];b3=PrimeNext[b2];b4=PrimeNext[b3];b5=PrimeNext[b4];If[(a1+a2+a3+a4+b1+b2+b3+b4)/8==p&&(a1+a2+a3+b1+b2+b3)/6==p&&(a1+a2+b1+b2)/4==p,AppendTo[lst,p]],{n,2*9!}];lst

cp's OEIS Frontend

A223138 Numbers n such that sigma(n+1) - sigma(n-1) = n / k for some integer, where sigma(n) = A000203 (sum of divisors of n).

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A243811 Numbers k such that 2*k+3 and 2*k+5 are both prime.

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Magma

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A264263 The number of distinct nontrivial integral cevians of an isosceles triangle, with base of length 1 and legs of length n, that divide the base into two integral parts.

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A306247 Numbers k such that 2k - p is not a prime where p is any prime divisor of 4k^2 - 1.

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Maple

Mathematica

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A365416 Numbers k such that 2*k-1 and 2*k+1 are both prime powers (A246655).

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PARI

A094949 Phi(m)*sigma(m), where m is the product of exactly two primes that differ by 2, where phi=A000010, sigma=A000203.

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Mathematica

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A113710 a(n) = A113709(n)/(prime(n+1) - prime(n)).

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A123200 Numbers k such that 1000000*k-1 and 1000000*k+1 are twin primes.

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A243811 Numbers k such that 2k+3 and 2k+5 are both prime.

A365416 Numbers k such that 2k-1 and 2k+1 are both prime powers (A246655).

A123200 Numbers k such that 1000000k-1 and 1000000k+1 are twin primes.