A006512 -id:A006512 - cp's OEIS frontend

A182481 a(n) is the least k such that 6kn-1 and 6kn+1 are twin primes, and a(n)=0, if such k does not exist.

1, 1, 1, 3, 1, 2, 1, 4, 2, 1, 3, 1, 4, 5, 2, 2, 1, 1, 2, 2, 7, 5, 1, 3, 1, 2, 5, 16, 2, 1, 7, 1, 1, 5, 2, 2, 9, 1, 8, 1, 5, 9, 4, 5, 1, 3, 1, 4, 3, 2, 7, 1, 20, 5, 2, 8, 14, 1, 3, 21, 43, 4, 6, 3, 5, 8, 4, 9, 2, 1, 3, 1, 14, 15, 9, 30, 1, 4, 22, 7, 20, 21, 9

Offset: 1

Vladimir Shevelev, May 01 2012

nonn

Conjecture: a(n)>0; equivalently, for every n, the arithmetic progression {6*k*n-1} contains infinitely many lessers of twin primes (A001359).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001359, A006512, A014574, A001097, A077800, A002822.

Table[k = 0; While[! (PrimeQ[6*k*n - 1] && PrimeQ[6*k*n + 1]), k++]; k, {n, 100}] (* T. D. Noe, May 02 2012 *)

a(n)=my(k);n*=6;until(isprime(n*k++-1)&&isprime(n*k+1),);k \\ Charles R Greathouse IV, May 01 2012

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

Original entry on oeis.org

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.

A234200 a(n) = |{0 < k < n/2: kphi(n-k) - 1 and kphi(n-k) + 1 are both prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 3, 3, 2, 3, 2, 2, 3, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 2, 2, 4, 4, 3, 2, 4, 4, 3, 2, 3, 7, 2, 4, 4, 3, 7, 3, 6, 5, 3, 6, 5, 4, 3, 4, 3, 7, 4, 6, 3, 3, 4, 6, 7, 3, 7, 4, 6, 8, 2, 4, 6, 7, 8, 5, 2, 2, 10, 6, 3, 7, 7, 3, 7, 6, 2, 7, 4, 2, 6, 7, 9, 8, 4, 1, 3, 2, 4, 5, 8, 10, 4, 10, 7

Offset: 1

Zhi-Wei Sun, Dec 21 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 4.
(ii) If n > 3 is different from 9 and 29, then k*sigma(n-k) - 1 and k*sigma(n-k) + 1 are both prime for some 0 < k < n.
Obviously, either of the two parts implies the twin prime conjecture. We have verified part (i) for n up to 10^8.

			a(5) = 1 since 2*phi(3) - 1 = 3 and 2*phi(3) + 1 = 5 are both prime.
a(7) = 1 since 3*phi(4) - 1 = 5 and 3*phi(4) + 1 = 7 are both prime.
a(18) = 1 since 5*phi(13) - 1 = 59 and 5*phi(13) + 1 = 61 are both prime.
a(91) = 1 since 13*phi(78) - 1 = 311 and 13*phi(78) + 1 = 313 are both prime.
a(101) = 1 since 6*phi(95) - 1 = 431 and 6*phi(95) + 1 = 433 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, New representation problems involving Euler's totient function, a message to Number Theory List, Dec. 18, 2013.

Cf. A000010, A000040, A001359, A006512, A014574, A233542, A233544, A233547, A233566, A233567, A233867, A233918.

TQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
a[n_]:=Sum[If[TQ[k*EulerPhi[n-k]],1,0],{k,1,(n-1)/2}]
Table[a[n],{n,1,100}]

A237127 Number of ways to write n = k + m (0 < k < m) with k and m terms of A072281.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 3, 2, 3, 2, 3, 3, 3, 2, 2, 4, 3, 3, 2, 2, 4, 3, 4, 3, 4, 4, 3, 3, 4, 5, 4, 1, 3, 3, 5, 4, 4, 4, 4, 5, 3, 4, 2, 4, 4, 4, 5, 2, 4, 1, 4, 4, 4, 4, 1, 3, 4, 4, 5, 5

Offset: 1

Zhi-Wei Sun, Feb 04 2014

nonn

Conjecture: a(n) > 0 for all n > 11.

Clearly, this implies the twin prime conjecture.

			 a(13) = 1 since 13 = 5 + 8 with phi(5) - 1 = 3, phi(5) + 1 = 5, phi(8) - 1 = 3 and phi(8) + 1 = 5 all prime.
a(60) = 1 since 60 = 18 + 42 with phi(18) - 1 = 5, phi(18) + 1 = 7, phi(42) - 1 = 11 and phi(42) + 1 = 13 all prime.
a(84) = 1 since 84 = 7 + 77 with phi(7) - 1 = 5, phi(7) + 1 = 7, phi(77) - 1 = 59 and phi(77) + 1 = 61 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A001359, A006512, A014574, A072281.

PQ[n_]:=PrimeQ[EulerPhi[n]-1]&&PrimeQ[EulerPhi[n]+1]
a[n_]:=Sum[If[PQ[k]&&PQ[n-k],1,0],{k,1,(n-1)/2}]
Table[a[n],{n,1,70}]

A242490 Smallest even number k such that lpf(k-3) = prime(n) while lpf(k-1) > lpf(k-3), where lpf=least prime factor (A020639).

Original entry on oeis.org

6, 8, 80, 14, 224, 20, 440, 854, 32, 1460, 1742, 44, 2282, 3434, 4190, 62, 5432, 4760, 74, 12194, 8930, 8054, 12374, 13292, 104, 15350, 110, 14282, 31982, 17402, 18212, 140, 24050, 152, 25220, 29990, 28202, 32234, 33392, 182, 43262, 194, 44972, 200, 47564

Offset: 2

Vladimir Shevelev, May 16 2014

nonn

Note that the "small terms" {6,8,14,20,32,44,...} correspond to a(n) for which {a(n)-3, a(n)-1} is a twin pair such that the corresponding positions form sequence A029707.

If we change the definition to consider k for which {k-3, k-1} is not a twin pair, we obtain a closely related sequence 12,38,80,212,224,530,440,854,1250,1460,1742,... which shows a "model behavior" of A242490, if there are only a finite number of twin primes. - Vladimir Shevelev, May 19 2014

			Let n=2, prime(2)=3. Then lpf(6-3)=3, but lpf(6-1)=5>3. Since k=6 is the smallest such k, a(2)=6.

Peter J. C. Moses, Table of n, a(n) for n = 2..1001

Cf. A001359, A006512, A242489.

a(n)=my(p=prime(n),k=p+3); while(factor(k-3)[1,1]Charles R Greathouse IV, May 30 2014

Correction and more terms from Peter J. C. Moses, May 19 2014

A242767 Numbers of repetitions of terms in A242758.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, 2, 6, 9, 6, 5, 4, 3, 4, 20, 2, 2, 4, 4, 19, 2, 3, 2, 4, 8, 11, 5, 3, 3, 3, 10, 5, 4, 2, 17, 3, 6, 3, 3, 9, 9, 2, 6, 2, 6, 5, 6, 2, 3, 2, 3, 9, 4, 7, 3, 7, 20, 4, 7, 6, 5, 3, 7, 3, 20, 2, 14, 4

Offset: 2

Vladimir Shevelev, May 22 2014

nonn

If {pA242758. If this number occurs k times in A242758, then we say that k is the index of the pair of twin primes {p,q} with p in A001359.

Is this the same as A027833 shifted by two indices? - R. J. Mathar, May 23 2014

Peter J. C. Moses, Table of n, a(n) for n = 2..5001

Cf. A001359, A006512, A242489, A242490, A242720, A242758.

From the construction of A242758, in supposition of an infinity of twin primes, we have a(2)=1; for n>=3, a(n) = A027833(n-2). Otherwise, A027833 is finite, while A242758 will coincide with A242720 after the last pair of twin primes. - Vladimir Shevelev, May 26 2014

More terms from Peter J. C. Moses, May 22 2014

A282321 Lesser of twin primes congruent to 11 (mod 30).

Original entry on oeis.org

11, 41, 71, 101, 191, 281, 311, 431, 461, 521, 641, 821, 881, 1031, 1061, 1091, 1151, 1301, 1451, 1481, 1721, 1871, 1931, 2081, 2111, 2141, 2381, 2591, 2711, 2801, 3251, 3371, 3461, 3581, 3671, 3821, 3851, 4001, 4091, 4241, 4271, 4421, 4481, 4721, 4931, 5021

Offset: 1

Martin Renner, Feb 11 2017

nonn

The union of [this sequence and A282322] is A132241.
The union of [{3, 5}, this sequence, A282323 and A060229] is the lesser of twin primes sequence A001359.
The union of [{3, 5, 7}, A282321 to A282326] is the twin primes sequence A001097.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Subset of A001097, A001359, A132232, A132241 and A132247.

Cf. A006512, A232880, A232881, A232882, A282322, A282323, A060229, A060229, A282326.

[p: p in PrimesUpTo(5000) | IsPrime(p+2) and p mod 30 eq 11 ]; // Vincenzo Librandi, Feb 12 2017

a:={}:
for i from 1 to 1229 do
  if isprime(ithprime(i)+2) and ithprime(i) mod 30 = 11 then
    a:={op(a),ithprime(i)}:
  fi:
od:
a;

list(lim)=my(v=List(), p=2); forprime(q=3, lim+2, if(q-p==2 && q%30==13, listput(v, p)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A321597 Number of permutations tau of {1,...,n} such that k*tau(k) + 1 is prime for every k = 1,...,n.

Original entry on oeis.org

1, 2, 1, 6, 1, 24, 9, 38, 36, 702, 196, 7386, 3364, 69582, 45369, 885360, 110224, 14335236, 640000, 19867008, 11009124, 1288115340, 188485441, 17909627257, 4553145529, 363106696516, 149376066064, 11141446425852, 990882875761, 371060259505399, 16516486146304, 1479426535706319, 497227517362801, 102319410607145600, 32589727661167504, 12597253470226980096

Offset: 1

Zhi-Wei Sun, Nov 14 2018

Conjecture: (i) a(n) > 0 for all n > 0. Similarly, for any integer n > 2, there is a permutation tau of {1,...,n} such that k*tau(k) - 1 is prime for every k = 1,...,n.
(ii) For any integer n > 2, there is a permutation tau of {1,...,n} such that k + tau(k) - 1 and k + tau(k) + 1 are twin prime for every k = 1,...,n.
Obviously, part (ii) of this conjecture implies the twin prime conjecture. P. Bradley proved in arXiv:1809.01012 that for any positive integer n there is a permutation tau of {1,...,n} such that k + tau(k) is prime for every k = 1,...,n.

			a(3) = 1, and (1,3,2) is a permutation of {1,2,3} with 1*1 + 1 = 2, 2*3 + 1 = 7 and 3*2 + 1 = 7 all prime.
a(5) = 1, and (1,5,4,3,2) is a permutation of {1,2,3,4,5} with 1*1 + 1 = 2, 2*5 + 1 = 11, 3*4 + 1 = 13, 4*3 + 1 = 13 and 5*2 + 1 = 11 all prime.

Paul Bradley, Prime number sums, arXiv:1809.01012 [math.GR], 2018.
Zhi-Wei Sun, Primes arising from permutations, Question 315259 on Mathoverflow, Nov. 14, 2018.
Zhi-Wei Sun, On permutations of {1, ..., n} and related topics, arXiv:1811.10503 [math.CO], 2018.

Cf. A000040, A001359, A006512, A014574.

V[n_]:=V[n]=Permutations[Table[i,{i,1,n}]]
tab={};Do[r=0;Do[Do[If[PrimeQ[i*Part[V[n],k][[i]]+1]==False,Goto[aa]],{i,1,n}];r=r+1;Label[aa],{k,1,n!}];tab=Append[tab,r],{n,1,11}]

a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(i*j + 1))); \\ Jinyuan Wang, Jun 13 2020

a(12)-a(26) from Alois P. Heinz, Nov 17 2018
a(27)-a(30) from Jinyuan Wang, Jun 13 2020
a(31)-a(36) from Vaclav Kotesovec, Aug 19 2021

A365795 Numbers k such that omega(k) = 3 and its prime factors satisfy the equation p_1 + p_2 = p_3.

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, 2288, 2400, 2430, 2450

Offset: 1

Stefano Spezia, Sep 19 2023

nonn

The lower prime factor p_1 is equal to 2 and the other two are twin primes: p_3 - p_2 = 2.

			60 is a term since 60 = 2^2*3*5 and 2 + 3 = 5.
286 is a term since 286 = 2*11*13 and 2 + 11 = 13.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A001221, A001359, A006512, A071142.

Subsequence of A033992 and of A071140.

Select[Range[2500],PrimeNu[#]==3&&Part[First/@FactorInteger[#],1]+Part[First/@FactorInteger[#],2]==Part[First/@FactorInteger[#],3]&]

isok(k) = if (omega(k)==3, my(f=factor(k)[,1]); f[1] + f[2] == f[3]); \\ Michel Marcus, Sep 19 2023

A373825 Position of first appearance of n in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

Original entry on oeis.org

1, 2, 13, 11, 105, 57, 33, 69, 59, 29, 227, 129, 211, 341, 75, 321, 51, 45, 407, 313, 459, 301, 767, 1829, 413, 537, 447, 1113, 1301, 1411, 1405, 2865, 1709, 1429, 3471, 709, 2543, 5231, 1923, 679, 3301, 2791, 6555, 5181, 6345, 11475, 2491, 10633

Offset: 1

Gus Wiseman, Jun 21 2024

nonn

Positions of first appearances in A373819.

			The runs of odd primes differing by 2 begin:
   3   5   7
  11  13
  17  19
  23
  29  31
  37
  41  43
  47
  53
  59  61
  67
  71  73
  79
with lengths:
3, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, ...
which have runs beginning:
  3
  2 2
  1
  2
  1
  2
  1 1
  2
  1
  2
  1 1 1 1
  2 2
  1 1 1
with lengths:
1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 3, 2, 4, 3, ...
with positions of first appearances a(n).

Firsts of A373819 (run-lengths of A251092).
For antiruns we have A373827 (sorted A373826), firsts of A373820, run-lengths of A027833 (partial sums A029707, firsts A373401, sorted A373402).
The sorted version is A373824.
A000040 lists the primes.
A001223 gives differences of consecutive primes (firsts A073051), run-lengths A333254 (firsts A335406), run-lengths of run-lengths A373821.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
For composite runs: A005381, A054265, A068780, A176246, A373403, A373404.
Cf. A006560, A006562, A037201, A038664, A069010, A122535.

t=Length/@Split[Length/@Split[Select[Range[3,10000], PrimeQ],#1+2==#2&]//Most]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#1]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]

cp's OEIS Frontend

A182481 a(n) is the least k such that 6*k*n-1 and 6*k*n+1 are twin primes, and a(n)=0, if such k does not exist.

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

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A234200 a(n) = |{0 < k < n/2: k*phi(n-k) - 1 and k*phi(n-k) + 1 are both prime}|, where phi(.) is Euler's totient function.

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A237127 Number of ways to write n = k + m (0 < k < m) with k and m terms of A072281.

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A242490 Smallest even number k such that lpf(k-3) = prime(n) while lpf(k-1) > lpf(k-3), where lpf=least prime factor (A020639).

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A242767 Numbers of repetitions of terms in A242758.

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A282321 Lesser of twin primes congruent to 11 (mod 30).

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A321597 Number of permutations tau of {1,...,n} such that k*tau(k) + 1 is prime for every k = 1,...,n.

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A182481 a(n) is the least k such that 6kn-1 and 6kn+1 are twin primes, and a(n)=0, if such k does not exist.

A234200 a(n) = |{0 < k < n/2: kphi(n-k) - 1 and kphi(n-k) + 1 are both prime}|, where phi(.) is Euler's totient function.