A006512 -id:A006512 - cp's OEIS frontend

A373405 Sum of the n-th maximal antirun of odd primes differing by more than two.

3, 5, 18, 30, 71, 109, 202, 199, 522, 210, 617, 288, 990, 372, 390, 860, 701, 1281, 829, 1194, 1645, 4578, 852, 2682, 4419, 3300, 2927, 2438, 1891, 2602, 14660, 1632, 1650, 3378, 3480, 18141, 2052, 3121, 2112, 4310, 8922, 13131, 6253, 3851, 3889, 3929, 13788

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The length of this run is given by A027833 (except initial term).

An antirun of a sequence (in this case A000040\{2}) is an interval of positions at which consecutive terms differ by more than one.

			Row-sums of:
   3
   5
   7  11
  13  17
  19  23  29
  31  37  41
  43  47  53  59
  61  67  71
  73  79  83  89  97 101

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

The partial sums are a subset of A071148 (partial sums of odd primes).
Functional neighbors: A001359, A006512, A027833 (partial sums A029707), A373404, A373406, A373411, A373412.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
Cf. A025157, A038664, A046933, A293697, A350842 (strict A350844), A371201.

Total/@Split[Select[Range[3,1000],PrimeQ],#1+2!=#2&]//Most

A029908 Starting with n, repeatedly sum prime factors (with multiplicity) until reaching 0 or a fixed point. Then a(n) is the fixed point (or 0).

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 5, 5, 7, 11, 7, 13, 5, 5, 5, 17, 5, 19, 5, 7, 13, 23, 5, 7, 5, 5, 11, 29, 7, 31, 7, 5, 19, 7, 7, 37, 7, 5, 11, 41, 7, 43, 5, 11, 7, 47, 11, 5, 7, 5, 17, 53, 11, 5, 13, 13, 31, 59, 7, 61, 5, 13, 7, 5, 5, 67, 7, 5, 5, 71, 7, 73, 5, 13, 23, 5, 5, 79, 13, 7, 43, 83, 5, 13

Offset: 1

Dann Toliver

nonn

That is, the sopfr function (see A001414) applied repeatedly until reaching 0 or a fixed point.
For n > 1, the sequence reaches a fixed point which is either 4 or a prime.
A002217(n) is number of terms in sequence from n to a(n). - Reinhard Zumkeller, Apr 08 2003
Because sopfr(n) <= n (with equality at 4 and the primes), the first appearance of all primes is in the natural order: 2,3,5,7,11,... . - Zak Seidov, Mar 14 2011
The terms 0, 2, 3 and 4 occur exactly once, because no number > 5 can have factors that sum to be < 5, and therefore can never enter a trajectory that will drop below 5. - Christian N. K. Anderson, May 19 2013
For all primes p, where p is contained in A001359, then a(p^2) = p + 2. (A006512). Proof: p^2 has prime factors (p, p). This sums to 2p. 2p has factors (2, p). This sums to p + 2. Since p was the lesser of a twin prime, then p + 2 is the greater of a twin prime. - Ryan Bresler, Nov 04 2021

			20 -> 2+2+5 = 9 -> 3+3 = 6 -> 2+3 = 5, so a(20)=5.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Sum of Prime Factors.

Cf. A001414 (sum of prime factors of n).
Cf. A081758, A002217, A075860.
Cf. A001414, A056239, A008475, A082081, A082083.

f:= proc(n) option remember;
if isprime(n) then n
else `procname`(add(x[1]*x[2], x = ifactors(n)[2]))
fi
end proc:
f(1):= 0: f(4):= 4:
map(f, [$1..100]); # Robert Israel, Apr 27 2015

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] slog[x_] := slog[x_] := Apply[Plus, ba[x]*ep[x]] Table[FixedPoint[slog, w], {w, 1, 128}]
f[n_] := Plus @@ Flatten[ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n]; Array[ FixedPoint[f, # ] &, 87] (* Robert G. Wilson v, Jan 18 2006 *)
fz[n_]:=Plus@@(#[[1]]*#[[2]]&/@FactorInteger@n); Array[FixedPoint[fz,#]&,1000] (* Zak Seidov, Mar 14 2011 *)

from sympy import factorint
def a(n, pn):
    if n == pn:
        return n
    else:
        return a(sum(p*e for p, e in factorint(n).items()), n)
print([a(i, None) for i in range(1, 100)]) # Gleb Ivanov, Nov 05 2021

A066388 Numbers j such that j and 2j are both between a pair of twin primes.

Original entry on oeis.org

6, 30, 660, 810, 2130, 2550, 3330, 3390, 5850, 6270, 10530, 33180, 41610, 44130, 53550, 55440, 57330, 63840, 65100, 70380, 70980, 72270, 74100, 74760, 78780, 80670, 81930, 87540, 93240, 102300, 115470, 124770, 133980, 136950, 156420

Offset: 1

Jud McCranie, Dec 23 2001

nonn

Also terms of A014574 such that twice the term is also in A014574. Related to a problem of anti-divisors.

All a(n) > 6 must be a multiple of 30: As for elements of A014574, we must have a(n) = 6k, and k = 5m+-1 would lead to a(n)-+1 divisible by 5, while k = 5m+-2 would lead to 2*a(n)+-1 divisible by 5, so only k=5m is possible. - M. F. Hasler, Nov 27 2010

			j = 30 is a term since 29 and 31 are prime, as are 59 and 61.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Eric Weisstein's World of Mathematics, Bitwin Chain.

Subsequence of A014574.
Subsequences: A118859, A118860, A349321.
Cf. A001359, A006512, A117499.
Cf. A118859, A118860, A349321, A348348.

lst={}; Do[p1=Prime[n]; p2=Prime[n+1]; d=2; If[p2-p1==d, w=p1+1; If[PrimeQ[2*w-1]&&PrimeQ[2*w+1], AppendTo[lst, w]]], {n, 1, 10^4}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)

{ n=0; forstep (m=2, 10^9, 2, if (isprime(m - 1) && isprime(m + 1) && isprime(2*m - 1) && isprime(2*m + 1), write("b066388.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 13 2010

A117499(a(n)) = 4. - Reinhard Zumkeller, Mar 23 2006

{k in A014574: 2*k in A014574}. - R. J. Mathar, Jan 20 2025

A373410 Minimum of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

Original entry on oeis.org

4, 9, 25, 28, 45, 49, 50, 64, 76, 81, 99, 100, 117, 121, 125, 126, 136, 148, 153, 169, 172, 176, 189, 208, 225, 243, 244, 245, 261, 276, 280, 289, 297, 316, 325, 333, 343, 344, 351, 352, 361, 364, 369, 376, 388, 405, 424, 425, 441, 460, 476, 477, 496, 508, 513

Offset: 1

Gus Wiseman, Jun 06 2024

nonn

The maximum is given by A068781.
An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.
Consists of 4 and all nonsquarefree numbers n such that n - 1 is also nonsquarefree.

			Row-minima of:
   4   8
   9  12  16  18  20  24
  25  27
  28  32  36  40  44
  45  48
  49
  50  52  54  56  60  63
  64  68  72  75
  76  80
  81  84  88  90  92  96  98
  99

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

Functional neighbors: A005381, A006512, A053806, A068781, A373408, A373409, A373412.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A049094, A061399, A077641, A077643, A112926, A143658, A294242, A350842, A372683, A372684, A373125.

First/@Split[Select[Range[100],!SquareFreeQ[#]&],#1+1!=#2&]

a(1) = 4; a(n>1) = A068781(n-1) + 1.

A164292 Binary sequence identifying the twin primes (characteristic function of twin primes: 1 if n is a twin prime else 0).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Carlos Alves, Aug 12 2009

nonn

Similar to prime binary digit sequence A010051.
In decimal notation A164292=0.1646823906345389353962381...
See also A164293 (similar to prime decimal sequence A051006).
a(A001097(n))=1; a(A001359(n))=1; a(A006512(n))=1. - Reinhard Zumkeller, Mar 29 2010
Characteristic function of A001097. - Georg Fischer, Aug 04 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions [From _Reinhard Zumkeller_, Mar 29 2010]

Cf. A001097, A010051, A051006, A057427, A129950, A164293.

a164292 1 = 0
a164292 2 = 0
a164292 n = signum (a010051' n * (a010051' (n - 2) + a010051' (n + 2)))
-- Reinhard Zumkeller, Feb 03 2014

Table[(PrimePi[n] - PrimePi[n - 1]) * Ceiling[(PrimePi[n + 2] - PrimePi[n + 1] + PrimePi[n - 2] - PrimePi[n - 3])/2], {n, 100}] (* Wesley Ivan Hurt, Jan 31 2014 *)
Table[If[PrimeQ[n]&&AnyTrue[n+{2,-2},PrimeQ],1,0],{n,120}] (* Harvey P. Dale, Jan 04 2025 *)

a(n) = A057427(A010051(n)*(A010051(n-2)+A010051(n+2))), for n>2. - Reinhard Zumkeller, Mar 29 2010

a(n) = c(n) * ceiling(( c(n+2) + c(n-2) )/2), where c is the prime characteristic. - Wesley Ivan Hurt, Jan 31 2014

A166944 a(1)=2; a(n) = a(n-1) + gcd(n, a(n-1)) if n is even, a(n) = a(n-1) + gcd(n-2, a(n-1)) if n is odd.

Original entry on oeis.org

2, 4, 5, 6, 9, 12, 13, 14, 21, 22, 23, 24, 25, 26, 39, 40, 45, 54, 55, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 129, 130, 135, 138, 139, 140, 147, 148, 149, 150, 151, 152, 153, 154, 155, 160, 161, 162, 163

Offset: 1

Vladimir Shevelev, Oct 24 2009

nonn

Conjecture: Every record of differences a(n)-a(n-1) more than 5 is the greater of twin primes (A006512).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol.11(2008), Article 08.2.8. arXiv:0710.3217 [math.NT]
V. Shevelev, An infinite set of generators of primes based on the Rowland idea and conjectures concerning twin primes, arXiv:0910.4676 [math.NT], 2009. - _Vladimir Shevelev_, Oct 27 2009
V. Shevelev, Three theorems on twin primes, arXiv:0911.5478 [math.NT], 2009-2010. - _Vladimir Shevelev_, Dec 03 2009

Cf. A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293.

A166944 := proc(n) option remember; if n = 1 then 2; else p := procname(n-1) ; if type(n,'even') then p+igcd(n,p) ; else p+igcd(n-2,p) ; end if; end if; end proc: # R. J. Mathar, Sep 03 2011

nxt[{n_,a_}]:={n+1,If[OddQ[n],a+GCD[n+1,a],a+GCD[n-1,a]]}; Transpose[ NestList[ nxt,{1,2},70]][[2]] (* Harvey P. Dale, Feb 10 2015 *)

print1(a=2); for(n=2, 100, d=gcd(a, if(n%2, n-2, n)); print1(", "a+=d)) \\ Charles R Greathouse IV, Oct 13 2017

Terms beginning with a(18) corrected by Vladimir Shevelev, Nov 10 2009

A181490 Numbers k such that 32^k-1 and 32^k+1 are twin primes (A001097).

Original entry on oeis.org

1, 2, 6, 18

Offset: 1

M. F. Hasler, Oct 30 2010

Sequences A181491 and A181492 list the corresponding primes.
No more terms below three million. - Charles R Greathouse IV, Mar 14 2011
Intersection of A002235 and A002253. - Jeppe Stig Nielsen, Mar 05 2018

Cf. A001097, A001359, A002235, A002253, A006512, A014574, A294730.

Filtered([1..300],k->IsPrime(3*2^k-1) and IsPrime(3*2^k+1)); # Muniru A Asiru, Mar 11 2018

a:=k->`if`(isprime(3*2^k-1) and isprime(3*2^k+1),k,NULL); seq(a(k),k=1..1000); # Muniru A Asiru, Mar 11 2018

fQ[n_] := PrimeQ[3*2^n - 1] && PrimeQ[3*2^n + 1]; k = 1; lst= {}; While[k < 15001, If[fQ@k, AppendTo[lst, k]; Print@k]; k++ ] (* Robert G. Wilson v, Nov 05 2010 *)
Select[Range[20],AllTrue[3*2^#+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 24 2014 *)

PARI
```
for( k=1,999, ispseudoprime(3<
				
```

Equals { k | A007283(k) in A014574 } = { k | A153893(k) in A001359 }.

Pari program repaired by Charles R Greathouse IV, Mar 14 2011

A236457 Primes p with q = p + 2 and prime(q) + 2 both prime.

Original entry on oeis.org

3, 5, 11, 41, 107, 311, 461, 599, 641, 1277, 1619, 1997, 2309, 2381, 2789, 3671, 4787, 5099, 6659, 6701, 6827, 7457, 7487, 8219, 8537, 8597, 9929, 10709, 11117, 12071, 12107, 12251, 13709, 17747, 18047, 18251, 18521, 22091, 22637, 23027

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

According to the conjecture in A236456, this sequence should have infinitely many terms.

See A236458 for a similar sequence.

			a(1) = 3 since 3 + 2 = 5 and prime(5) + 2 = 13 are both prime, but 2 + 2 = 4 is not.

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

Cf. A000040, A001359, A006512, A236119, A236456, A236458.

p[n_]:=PrimeQ[n+2]&&PrimeQ[Prime[n+2]+2]
In[2]:= n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10000}]
Select[Prime[Range[2600]],AllTrue[{#+2,Prime[#+2]+2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 21 2021 *)

s=[]; forprime(p=2, 24000, q=p+2; if(isprime(q) && isprime(prime(q)+2), s=concat(s, p))); s \\ Colin Barker, Jan 26 2014

A373820 Run-lengths (differing by 0) of antirun-lengths (differing by > 2) of odd primes.

Original entry on oeis.org

2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Jun 22 2024

nonn

Run-lengths of the version of A027833 with 1 prepended.

			The antiruns 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  83  89  97 101
 103 107
 109 113 127 131 137
 139 149
 151 157 163 167 173 179
 181 191
 193 197
 199 211 223 227
 229 233 239
 241 251 257 263 269
 271 277 281
with lengths:
1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, ...
with runs:
  1  1
  2  2
  3  3
  4
  3
  6
  2
  5
  2
  6
  2  2
  4
  3
  5
  3
  4
with lengths a(n).

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

Run-lengths of A027833 (if we prepend 1), partial sums A029707.
For runs we have A373819, run-lengths of A251092.
Positions of first appearances are A373827, sorted A373826.
A000040 lists the primes.
A001223 gives differences of consecutive primes, run-lengths A333254, 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, A373403, A373404.
Cf. A005117, A006560, A006562, A037201, A038664, A049579, A073051, A076259, A122535, A176246.

Length/@Split[Length/@Split[Select[Range[3,1000],PrimeQ],#2-#1>2&]//Most]//Most

A053319 Distance between the smaller members of successive twin prime pairs.

Original entry on oeis.org

2, 6, 6, 12, 12, 18, 12, 30, 6, 30, 12, 30, 12, 6, 30, 12, 30, 12, 30, 36, 72, 12, 30, 60, 48, 30, 18, 24, 18, 150, 12, 6, 30, 24, 138, 12, 18, 12, 30, 60, 78, 48, 12, 12, 18, 108, 24, 30, 6, 120, 12, 48, 30, 24, 66, 84, 6, 54, 18, 48, 30, 54, 6, 24, 18, 12, 96, 30, 42, 30, 42

Offset: 1

Labos Elemer, Mar 06 2000

nonn

Conjecture: a(n) < log(A014574(n))^3 for n > 2. - Thomas Ordowski, Jul 21 2012

This is of course also the distance between the larger members of successive twin primes. - Franklin T. Adams-Watters, Jun 03 2014

M. F. Hasler, Table of n, a(n) for n = 1..107406

First differences of A001359, A006512, and of A014574.

Differences[Transpose[Select[Partition[Prime[Range[450]],2,1],Last[#]-First[#]==2&]][[1]]]  (* Harvey P. Dale, Feb 08 2011 *)

{cnt=0; lp= /*last*/ ltp=/*last twin(upper)*/ 5;
forprime(p=lp+1,default(primelimit), if(p-lp != 2,lp=p;next);
write("b053319.txt",cnt++" "p-ltp);/* print1(p-ltp", ");*/ ltp=lp=p)} \\ M. F. Hasler, May 26 2007

Definition clarified by Harvey P. Dale, Feb 08 2011

cp's OEIS Frontend

A373405 Sum of the n-th maximal antirun of odd primes differing by more than two.

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A029908 Starting with n, repeatedly sum prime factors (with multiplicity) until reaching 0 or a fixed point. Then a(n) is the fixed point (or 0).

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A066388 Numbers j such that j and 2j are both between a pair of twin primes.

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A373410 Minimum of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

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A164292 Binary sequence identifying the twin primes (characteristic function of twin primes: 1 if n is a twin prime else 0).

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A166944 a(1)=2; a(n) = a(n-1) + gcd(n, a(n-1)) if n is even, a(n) = a(n-1) + gcd(n-2, a(n-1)) if n is odd.

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A181490 Numbers k such that 3*2^k-1 and 3*2^k+1 are twin primes (A001097).

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A181490 Numbers k such that 32^k-1 and 32^k+1 are twin primes (A001097).