A006512 -id:A006512 - cp's OEIS frontend

A242720 Smallest even k such that the pair {k-3,k-1} is not a twin prime pair and lpf(k-1) > lpf(k-3) >= prime(n), where lpf = least prime factor (A020639).

12, 38, 80, 212, 224, 440, 440, 854, 1250, 1460, 1742, 2282, 2282, 3434, 4190, 4664, 4760, 4760, 6890, 8054, 8054, 8054, 12374, 12830, 12830, 13592, 13592, 14282, 17402, 17402, 18212, 22502, 22502, 22502, 25220, 28202, 28202, 32234, 32402, 32402, 38012

Offset: 2

Vladimir Shevelev, May 21 2014

nonn

The sequence is nondecreasing. See comment in A242758.
a(n) >= prime(n)^2+3. Conjecture: a(n) <= prime(n)^4. - Vladimir Shevelev, Jun 01 2014
Conjecture. There are only a finite number of composite numbers of the form a(n)-1. Peter J. C. Moses found only two: a(16)-1 = 4189 = 59*71 and a(20)-1 = 6889 = 83^2 and no others up to a(2501). Most likely, there are no others. - Vladimir Shevelev, Jun 09 2014

Peter J. C. Moses, Table of n, a(n) for n = 2..2501
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014, (Section 10).

Cf. A020639, A001359, A006512, A070155, A242489, A242490, A242758, A242847, A246748, A246821, A246824.

lpf[n_] := FactorInteger[n][[1, 1]];
Clear[a]; a[n_] := a[n] = For[k = If[n <= 2, 2, a[n-1]], True, k = k+2, If[Not[PrimeQ[k-3] && PrimeQ[k-1]] && lpf[k-1] > lpf[k-3] >= Prime[n], Return[k]]];
Table[a[n], {n, 2, 50}] (* Jean-François Alcover, Nov 02 2018 *)

lpf(k) = factorint(k)[1,1];
vector(60, n, k=6; while((isprime(k-3) && isprime(k-1)) || lpf(k-1)<=lpf(k-3) || lpf(k-3)Colin Barker, Jun 01 2014

Conjecturally, a(n) ~ (prime(n))^2, as n goes to infinity (cf. A246748, A246821). - Vladimir Shevelev, Sep 02 2014

For n>=3, a(n) >= (prime(n)+1)^2 + 2. Equality holds for terms of A246824. - Vladimir Shevelev, Sep 04 2014

A373673 First element of each maximal run of powers of primes (including 1).

Original entry on oeis.org

1, 7, 11, 13, 16, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Gus Wiseman, Jun 15 2024

nonn

A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.
The last element of the same run is A373674.
Consists of all powers of primes k such that k-1 is not a power of primes.

			The maximal runs of powers of primes begin:
   1   2   3   4   5
   7   8   9
  11
  13
  16  17
  19
  23
  25
  27
  29
  31  32
  37
  41
  43
  47
  49

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For composite antiruns we have A005381, max A068780, length A373403.
For prime antiruns we have A006512, max A001359, length A027833.
For composite runs we have A008864, max A006093, length A176246.
For prime runs we have A025584, max A067774, length A251092 or A175632.
For runs of prime-powers:
- length A174965
- min A373673 (this sequence)
- max A373674
- sum A373675
For runs of non-prime-powers:
- length A110969 (firsts A373669, sorted A373670)
- min A373676
- max A373677
- sum A373678
For antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
- sum A373576
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
- sum A373679
A000961 lists all powers of primes (A246655 if not including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A000040, A007674, A014963, A053806, A054265, A068781, A072284, A073051, A120992, A373400.

pripow[n_]:=n==1||PrimePowerQ[n];
Min/@Split[Select[Range[100],pripow],#1+1==#2&]//Most

A373674 Last element of each maximal run of powers of primes (including 1).

Original entry on oeis.org

5, 9, 11, 13, 17, 19, 23, 25, 27, 29, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Gus Wiseman, Jun 16 2024

nonn

A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.
The first element of the same run is A373673.
Consists of all powers of primes k such that k+1 is not a power of primes.

			The maximal runs of powers of primes begin:
   1   2   3   4   5
   7   8   9
  11
  13
  16  17
  19
  23
  25
  27
  29
  31  32
  37
  41
  43
  47
  49

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For prime antiruns we have A001359, min A006512, length A027833.
For composite runs we have A006093, min A008864, length A176246.
For prime runs we have A067774, min A025584, length A251092 or A175632.
For squarefree runs we have A373415, min A072284, length A120992.
For nonsquarefree runs we have min A053806, length A053797.
For runs of prime-powers:
- length A174965
- min A373673
- max A373674 (this sequence)
- sum A373675
For runs of non-prime-powers:
- length A110969 (firsts A373669, sorted A373670)
- min A373676
- max A373677
- sum A373678
For antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
- sum A373576
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
- sum A373679
A000961 lists all powers of primes (A246655 if not including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A000040, A005381, A007674, A014963, A068780, A068781, A073051, A373400.

pripow[n_]:=n==1||PrimePowerQ[n];
Max/@Split[Select[Range[nn],pripow],#1+1==#2&]//Most

A071538 Number of twin prime pairs (p, p+2) with p <= n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 30 2002

nonn

The convention is followed that a twin prime is <= n if its smaller member is <= n.
Except for (3, 5), every pair of twin primes is congruent (-1, +1) (mod 6). - Daniel Forgues, Aug 05 2009
This function is sometimes known as pi_2(n). If this name is used, there is no obvious generalization for pi_k(n) for k > 2. - Franklin T. Adams-Watters, Jun 01 2014

			a(30) = 5, since (29,31) is included along with (3,5), (5,7), (11,13) and (17,19).

S. Lang, The Beauty of Doing Mathematics, pp. 12-15; 21-22, Springer-Verlag NY 1985.

Daniel Forgues, Table of n, a(n) for n = 1..99998
Thomas R. Nicely, Some Results of Computational Research in Prime Numbers.
Eric Weisstein's World of Mathematics, Twin Primes.

Cf. A007508, A033843, A001359, A006512.

primePi2[1] = 0; primePi2[n_] := primePi2[n] = primePi2[n - 1] + Boole[PrimeQ[n] && PrimeQ[n + 2]]; Table[primePi2[n], {n, 100}] (* T. D. Noe, May 23 2013 *)

A071538(n) = local(s=0,L=0); forprime(p=3,n+2,L==p-2 & s++; L=p); s
/* For n > primelimit, one may use: */ A071538(n) = { local(s=isprime(2+n=precprime(n))&n,L); while( n=precprime(L=n-2),L==n & s++); s }
/* The following gives a reasonably good estimate for small and for large values of n (cf. A007508): */
A071538est(n) = 1.320323631693739*intnum(t=2,n+1/n,1/log(t)^2)-log(n) /* (The constant 1.320... is A114907.) */ \\ M. F. Hasler, Dec 10 2008

Definition edited by Daniel Forgues, Jul 29 2009

A205302 Greater of Hamming's twin primes.

Original entry on oeis.org

3, 7, 19, 23, 31, 43, 47, 71, 101, 103, 139, 151, 167, 197, 199, 271, 283, 311, 317, 367, 383, 397, 409, 457, 461, 463, 503, 523, 571, 619, 643, 647, 677, 743, 751, 773, 811, 823, 859, 863, 883, 887, 911, 937, 941, 991, 1013, 1051, 1063, 1117, 1231, 1279, 1291, 1301, 1303, 1483, 1487

Offset: 1

Vladimir Shevelev, Jan 28 2012

The Hamming distance between a(n) and the previous prime is 1 (cf. A205510).

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A205510, A205511, A205533.

Cf. A001097, A001359, A006512.

Select[Partition[Prime[Range[250]], 2, 1], DigitCount[BitXor[First[#], Last[#]], 2, 1] == 1 &] [[;; , 2]] (* Amiram Eldar, Aug 06 2023 *)

n=0;for(i=1,80,until(A205510(n++)==1,);print1(prime(n+1)","))  \\ M. F. Hasler, Jan 29 2012

a(10)-a(57) from Peter J. C. Moses, Jan 28 2012

Values verified by M. F. Hasler, Jan 29 2012

A205511 Lesser of the n-th pair of Hamming twin primes.

Original entry on oeis.org

2, 5, 17, 19, 29, 41, 43, 67, 97, 101, 137, 149, 163, 193, 197, 269, 281, 307, 313, 359, 379, 389, 401, 449, 457, 461, 499, 521, 569, 617, 641, 643, 673, 739, 743, 769, 809, 821, 857, 859, 881, 883, 907, 929, 937, 983, 1009, 1049, 1061, 1109, 1229, 1277, 1289, 1297

Offset: 1

Vladimir Shevelev, Jan 28 2012

The Hamming distance between a(n) and the next prime is 1, cf. A205510.

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A205510, A205302, A205533.

Cf. A001097, A001359, A006512.

Select[Prime[Range[211]],Count[IntegerDigits[BitXor[#,NextPrime[#]],2],1]==1 &] (* Jayanta Basu, May 26 2013 *)

n=0;for(i=1,10^4,until(A205510(n++)==1,);write("c:/temp/b205511.txt",i" "prime(n))) \\ M. F. Hasler, Jan 29 2012

Terms from a(11) were added by Peter J. C. Moses

A205533 Intersection of A205302 and A205511.

Original entry on oeis.org

19, 43, 101, 197, 457, 461, 643, 743, 859, 883, 937, 1301, 1483, 1579, 1877, 1949, 1997, 2083, 2129, 2141, 2221, 2237, 2251, 2381, 2539, 2609, 2617, 2659, 2663, 3019, 3221, 3389, 3461, 3701, 4003, 4157, 4517, 4637, 5573, 5741, 5783, 6763, 6899, 6907, 7349, 7757, 7877

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Jan 28 2012

Both Hamming's distances between a(n) and the previous prime and between a(n) and the next prime equal 1.

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

Cf. A205302, A205511, A205510, A205509, A001097, A001359, A006512.

Select[Partition[Prime[Range[1000]], 3, 1], DigitCount[BitXor[#[[1]], #[[2]]], 2, 1] == DigitCount[BitXor[#[[2]], #[[3]]], 2, 1] == 1 &] [[;; , 2]] (* Amiram Eldar, Aug 06 2023 *)

More terms from Amiram Eldar, Aug 06 2023

A373400 Numbers k such that the k-th maximal run of composite numbers has length different from all prior maximal runs. Sorted positions of first appearances in A176246 (or A046933 shifted).

Original entry on oeis.org

1, 3, 8, 23, 29, 33, 45, 98, 153, 188, 216, 262, 281, 366, 428, 589, 737, 1182, 1830, 1878, 2190, 2224, 3076, 3301, 3384, 3426, 3643, 3792, 4521, 4611, 7969, 8027, 8687, 12541, 14356, 14861, 15782, 17005, 19025, 23282, 30801, 31544, 33607, 34201, 34214, 38589

Offset: 1

Gus Wiseman, Jun 10 2024

nonn

The unsorted version is A073051.

A run of a sequence (in this case A002808) is an interval of positions at which consecutive terms differ by one.

			The maximal runs of composite numbers begin:
   4
   6
   8   9  10
  12
  14  15  16
  18
  20  21  22
  24  25  26  27  28
  30
  32  33  34  35  36
  38  39  40
  42
  44  45  46
  48  49  50  51  52
  54  55  56  57  58
  60
  62  63  64  65  66
  68  69  70
  72
  74  75  76  77  78
  80  81  82
  84  85  86  87  88
  90  91  92  93  94  95  96
  98  99 100
The a(n)-th rows are:
   4
   8   9  10
  24  25  26  27  28
  90  91  92  93  94  95  96
 114 115 116 117 118 119 120 121 122 123 124 125 126
 140 141 142 143 144 145 146 147 148
 200 201 202 203 204 205 206 207 208 209 210

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

The unsorted version is A073051, firsts of A176246.
For squarefree runs we have the triple (1,3,5), firsts of A120992.
For prime runs we have the triple (1,2,3), firsts of A175632.
For squarefree antiruns we have A373128, firsts of A373127.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For prime antiruns we have A373402, unsorted A373401, firsts of A027833.
For composite runs we have the triple (1,2,7), firsts of A373403.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A006512, A007674, A049093, A068781, A072284, A077641, A174965, A251092, A373198, A373408, A373411.

t=Length/@Split[Select[Range[10000],CompositeQ],#1+1==#2&]//Most;
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A143201 Product of distances between prime factors in factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 6, 3, 1, 1, 2, 1, 4, 5, 10, 1, 2, 1, 12, 1, 6, 1, 6, 1, 1, 9, 16, 3, 2, 1, 18, 11, 4, 1, 10, 1, 10, 3, 22, 1, 2, 1, 4, 15, 12, 1, 2, 7, 6, 17, 28, 1, 6, 1, 30, 5, 1, 9, 18, 1, 16, 21, 12, 1, 2, 1, 36, 3, 18, 5, 22, 1, 4, 1, 40, 1, 10, 13, 42, 27, 10, 1, 6, 7, 22

Offset: 1

Reinhard Zumkeller, Aug 12 2008

nonn

a(n) is the product of the sum of 1 and first differences of prime factors of n with multiplicity, with a(n) = 1 for n = 1 or prime n. - Michael De Vlieger, Nov 12 2023.
a(A007947(n)) = a(n);
A006093 and A001747 give record values and where they occur:
A006093(n)=a(A001747(n+1)) for n>1.
a(n) = 1 iff n is a prime power: a(A000961(n))=1;
a(n) = 2 iff n has exactly 2 and 3 as prime factors:
a(A033845(n))=2;
a(n) = 3 iff n is in A143202;
a(n) = 4 iff n has exactly 2 and 5 as prime factors:
a(A033846(n))=4;
a(n) = 5 iff n is in A143203;
a(n) = 6 iff n is in A143204;
a(n) = 7 iff n is in A143205;
a(n) <> A006512(k)+1 for k>1.
a(A033849(n))=3; a(A033851(n))=3; a(A033850(n))=5; a(A033847(n))=6; a(A033848(n))=10. [Reinhard Zumkeller, Sep 19 2011]

			a(86) = a(43*2) = 43-2+1 = 42;
a(138) = a(23*3*2) = (23-3+1)*(3-2+1) = 42;
a(172) = a(43*2*2) = (43-2+1)*(2-2+1) = 42;
a(182) = a(13*7*2) = (13-7+1)*(7-2+1) = 42;
a(276) = a(23*3*2*2) = (23-3+1)*(3-2+1)*(2-2+1) = 42;
a(330) = a(11*5*3*2) = (11-5+1)*(5-3+1)*(3-2+1) = 42.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A000961, A001747, A006093, A020639, A033845, A033846.
Cf. A033847, A033848, A033849, A033850.
Cf. A143203, A143204, A143205.
Cf. A027746.

a143201 1 = 1
a143201 n = product $ map (+ 1) $ zipWith (-) (tail pfs) pfs
   where pfs = a027748_row n
-- Reinhard Zumkeller, Sep 13 2011

Table[Times@@(Differences[Flatten[Table[First[#],{Last[#]}]&/@ FactorInteger[ n]]]+1),{n,100}] (* Harvey P. Dale, Dec 07 2011 *)

a(n) = f(n,1,1) where f(n,q,y) = if n=1 then y else if q=1 then f(n/p,p,1)) else f(n/p,p,y*(p-q+1)) with p = A020639(n) = smallest prime factor of n.

A236458 Primes p with p + 2 and prime(p) + 2 both prime.

Original entry on oeis.org

3, 5, 17, 41, 1949, 2309, 2711, 2789, 2801, 3299, 3329, 3359, 3917, 4157, 4217, 4259, 4637, 5009, 5021, 5231, 6449, 7757, 8087, 8219, 8627, 9419, 9929, 10007, 10937, 11777, 12071, 14321, 15647, 15971, 16061, 16901, 18131, 18251, 18287, 18539

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

According to the conjecture in A236470, the sequence should have infinitely many terms. This is stronger than the twin prime conjecture.

See A236457 and A236467 for similar sequences.

			a(1) = 3 since 3 + 2 = 5 and prime(3) + 2 = 7 are both prime, but 2 + 2 = 4 is composite.

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

Cf. A000040, A001359, A006512, A236119, A236456, A236457, A236467, A236470.

p[n_]:=PrimeQ[n+2]&&PrimeQ[Prime[n]+2]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]

s=[]; forprime(p=2, 20000, if(isprime(p+2) && isprime(prime(p)+2), s=concat(s, p))); s \\ Colin Barker, Jan 26 2014

cp's OEIS Frontend

A242720 Smallest even k such that the pair {k-3,k-1} is not a twin prime pair and lpf(k-1) > lpf(k-3) >= prime(n), where lpf = least prime factor (A020639).

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A373673 First element of each maximal run of powers of primes (including 1).

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A373674 Last element of each maximal run of powers of primes (including 1).

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A071538 Number of twin prime pairs (p, p+2) with p <= n.

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A205302 Greater of Hamming's twin primes.

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A205511 Lesser of the n-th pair of Hamming twin primes.

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A205533 Intersection of A205302 and A205511.

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A373400 Numbers k such that the k-th maximal run of composite numbers has length different from all prior maximal runs. Sorted positions of first appearances in A176246 (or A046933 shifted).

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