A001359 -id:A001359 - cp's OEIS frontend

A308344 a(n) = (A001359(n+1)^2 - 1)/24, where A001359 = lesser of twin primes; or: pentagonal numbers (A000326) whose indices are twin ranks (A002822).

1, 5, 12, 35, 70, 145, 210, 425, 477, 782, 925, 1335, 1520, 1617, 2147, 2380, 3015, 3290, 4030, 5017, 7315, 7740, 8855, 11310, 13490, 14950, 15862, 17120, 18095, 27270, 28085, 28497, 30602, 32340, 43265, 44290, 45850, 46905, 49595, 55200, 62935, 67947, 69230, 70525

Offset: 1

M. F. Hasler and A. Dinculescu, Jul 04 2019

nonn

Subsequence of A024702 which considers all primes rather than only twins.
This sequence seems to play an important role in studying the twin prime conjecture; see also A057767, A273257, and related.
Dinculescu calls the numbers M(j) = (prime(j)^2 - 1)/6 "basic numbers", and [M(j), M(j+1)] a "twin interval" when j is the index of a twin prime. He notes that the length of such an interval equals four times the corresponding twin rank k(j) = (prime(j) + prime(j+1))/6, see near eq.(3.3) in the 2018 paper.

			Sequence A001359 = {3, 5, 11, 17, 29, ...} lists the lesser members of pairs of twin primes, (3, 5), (5, 7), (11, 13), (17, 19), ...
We ignore the first and start with the second pair, (5, 7). We have (5^2 - 1)/24 = 1 = a(1).
Next comes the pair (11, 13), whence (11^2 - 1)/24 = 120/24 = 5 = a(2), etc.

M. F. Hasler, Table of n, a(n) for n = 1..10000
A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

Cf. A000326, A002822, A001359, A024702.

(#^2-1)/24&/@Rest[Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]] == 2&][[All,1]]] (* Harvey P. Dale, Sep 05 2020 *)

PARI
```
a(n)=A000326(A002822(n))
```

a(n)=(A001359(n+1)^2-1)/24 \\ or implemented as follows:
p=0;forprime(q=5,oo,p+2==q&&print1(p^2\24",");p=q)

a(n) = (A001359(n+1)^2 - 1)/24 = A000326(A002822(n)).

A339625 a(n) is the number of ways to write 6*n = p + q with p a lesser twin prime (A001359) and q a greater twin prime (A006512).

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 2, 4, 2, 3, 2, 4, 4, 3, 4, 0, 4, 2, 6, 5, 2, 4, 2, 5, 4, 4, 4, 6, 2, 6, 2, 4, 6, 5, 12, 3, 6, 2, 4, 8, 6, 8, 8, 2, 6, 3, 6, 10, 4, 13, 2, 6, 4, 4, 10, 4, 10, 4, 6, 3, 4, 6, 10, 5, 8, 1, 0, 6, 2, 12, 4, 6, 6, 2, 10, 3, 10, 6, 6, 7, 2, 8, 4, 6, 6, 0, 6, 6, 6, 9, 2, 6, 2, 5, 6, 4

Offset: 1

J. M. Bergot and Robert Israel, Dec 10 2020

nonn

If 6*n = p + q, then also 6*n = (p+2) + (q-2), with p+2 a greater and q-2 a lesser twin prime. Thus a(n) is odd if and only if n/2 is in A002822.

			a(4)=3 because 6*4 = 24 = 5 + 19 = 11 + 13 = 17 + 7 where (5,7), (11,13) and (17,19) are twin prime pairs.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001359, A006512, A002822, A339630.

a(n)=0 for n in A243956.

N:= 600: # for a(1)..a(floor(N/6)))
P:= select(isprime, {seq(i,i=3..N,2)}):
T1:= sort(convert(P intersect map(`-`,P,2),list)):
T2:= map(`+`,T1,2):
V:= Vector(N):
nT:= nops(T1):
for i from 1 to nT do
  for j from 1 to nT do
    v:= T1[i]+T2[j];
    if v > N then break fi;
    V[v]:= V[v]+1;
od od:
seq(V[6*i],i=1..N/6);

A339630 a(n) is the first number k such that there are exactly n ways to write 6*k = p + q with p a lesser twin prime (A001359) and q a greater twin prime (A006512), or 0 if there is no such k.

Original entry on oeis.org

1, 2, 3, 4, 8, 20, 19, 80, 40, 90, 48, 270, 35, 50, 117, 140, 110, 644, 215, 714, 222, 430, 345, 350, 315, 850, 390, 930, 620, 1110, 602, 1040, 385, 2290, 590, 780, 798, 910, 735, 990, 1020, 1700, 700, 770, 595, 1760, 950, 3380, 875, 5660, 1330, 1120, 975, 5970, 1085, 2990, 1400, 3980, 1815, 4570

Offset: 0

J. M. Bergot and Robert Israel, Dec 10 2020

nonn

If n is odd, a(n)/2 (if nonzero) is in A002822.

			a(4) = 8 because 6*8 = 48 can be written as p+q in exactly 4 ways: 48 = 5 + 43 = 17 + 31 = 29 + 19 = 41 + 7, and no smaller number has this property.

Robert Israel, Table of n, a(n) for n = 0..1330

Cf. A001359, A002822, A006512, A339625.

# given list A339625
T:= Array(0..max(A339625)):
for n from 1 to nops(A339625) do
  if T[A339625[n]] = 0 then T[A339625[n]]:= n fi
od:
for k from 1 while T[k] <> 0 do od:
seq(T[i],i=0..k-1);

A339625(a(n)) = n if a(n) > 0.

A357934 Products of two distinct lesser twin primes A001359.

Original entry on oeis.org

15, 33, 51, 55, 85, 87, 123, 145, 177, 187, 205, 213, 295, 303, 319, 321, 355, 411, 447, 451, 493, 505, 535, 537, 573, 591, 649, 681, 685, 697, 717, 745, 781, 807, 843, 895, 933, 955, 985, 1003, 1041, 1111, 1135, 1177, 1189, 1195, 1207, 1257, 1293, 1345, 1383, 1405, 1507, 1555, 1563

Offset: 1

Artur Jasinski, Oct 21 2022

nonn

R. J. Mathar, Table of n, a(n) for n = 1..10000

Cf. A001359. Subsequence of A006881.

omega := proc(n)
    nops(numtheory[factorset](n)) ;
end proc:
isA357934 := proc(n)
    local pe,p,q;
    if numtheory[bigomega](n)= 2 and omega(n) =2 then
        pe := ifactors(n)[2] ;
        p := op(1,op(1,pe)) ;
        q := op(1,op(2,pe)) ;
        if isprime(p+2) and isprime(q+2) then
            true;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
for n from 10 to 2000 do
    if isA357934(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Feb 13 2025

d = {};less = Select[Range[1607], PrimeQ[#] && PrimeQ[# + 2] &];Do[Do[AppendTo[d, less[[m]] less[[n]]], {m, n + 1, Length[less]}], {n,
  1, Length[less] - 1}]; Take[Sort[d], 55]

list(lim)=my(v=List(),p=5); forprime(q=7,lim\3+2, if(q-p==2, my(r=3); forprime(s=5,min(lim\p+2,p), if(s-r==2, listput(v, p*r)); r=s)); p=q); Set(v) \\ Charles R Greathouse IV, Oct 21 2022

A359639 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have at least n odd prime factors, counted with multiplicity.

Original entry on oeis.org

97, 1999, 101527, 6666547, 272572999, 3819770107, 410274361249

Offset: 2

Hugo Pfoertner, Jan 16 2023

			a(2) = 97: 98 = 2*7^2, 99 = 3^2*11, 100 = 2^2*5^2 have 2 or 3 odd prime factors, so the minimum 2 is achieved.
a(3) = 1999: 2000 has the 3 odd prime factors 5^3, 2001 = 3*23*29, 2002 = 2*7*11*13.

Cf. A001222, A001359, A087436, A359637, A359640.

a087436(n) = bigomega (n >> valuation (n, 2));
a359639(maxp) = {my(k=2,pp=5); forprime (p=7, maxp, my(mi=oo); if (p-pp>2, for (j=pp+1, p-1, my(mo=a087436(j)); if (mo=k, print1(pp,", "); k++)); pp=p)};
a359639(3*10^8)

A079328 Let f(n)=A001359(n) be the smaller member of the n-th pair of twin primes. Then a(n) is the average of f(n) and f(n+1).

Original entry on oeis.org

4, 8, 14, 23, 35, 50, 65, 86, 104, 122, 143, 164, 185, 194, 212, 233, 254, 275, 296, 329, 383, 425, 446, 491, 545, 584, 608, 629, 650, 734, 815, 824, 842, 869, 950, 1025, 1040, 1055, 1076, 1121, 1190, 1253, 1283, 1295, 1310, 1373, 1439, 1466, 1484, 1547, 1613

Offset: 1

Vincenzo Origlio (vincenzo.origlio(AT)itc.cnr.it), Feb 13 2003

nonn

Average of two consecutive lesser of twin primes. - Juri-Stepan Gerasimov, Aug 26 2011.

Eric Weisstein's World of Mathematics, Twin Primes

Cf. A001359, A079329.

s=Select[Range[2000], PrimeQ[ # ]&&PrimeQ[ #+2]&]; (Drop[s, 1]+Drop[s, -1])/2

Edited by Dean Hickerson, Feb 14 2003

A097973 Least m>p such that p|m, p+1|m+1 and p+2|m+2, for twin prime pairs (p, p+2), p in A001359.

Original entry on oeis.org

63, 215, 1727, 5831, 26999, 74087, 215999, 373247, 1061207, 1259711, 2628071, 3374999, 5831999, 7077887, 7762391, 11852351, 13823999, 19682999, 22425767, 30371327, 42144191, 74087999, 80621567, 98611127, 142236647, 185192999

Offset: 1

Lekraj Beedassy, Sep 07 2004

nonn

			The triple {a(6), a(6)+1, a(6)+2}, for instance, i.e., (74087=41*1807, 74088=42*1764, 74089=43*1723) is the smallest one whose elements are respectively divisible by those of (41, 42, 43), (41, 43) being the 6th twin prime pair.

Robert Israel, Table of n, a(n) for n = 1..10000

map(t -> (t+1)^3-1, select(t -> isprime(t) and isprime(t+2), [3,seq(i,i=5..10^4,6)])); # Robert Israel, May 16 2018

a(n) = p*(p+1)*(p+2) + p = (p+1)^3 - 1 (p=A001359, p+1=A014574).

A174046 Places n for which A001359(n) and A023200(n) is a twin prime pair.

Original entry on oeis.org

2, 3, 4, 6, 14, 16, 29, 356, 358, 359, 403, 446, 464, 485, 652, 655, 764, 861, 866, 1123, 1301, 1304, 1324, 1328, 1358, 1486, 1610, 2631, 2632, 3735, 3931, 3953, 3956, 3957, 4679, 4855, 4931, 5222, 5226, 5269, 5283, 5292, 5403, 5427, 5445

Offset: 1

Vladimir Shevelev, Mar 06 2010

nonn

			2 is in the sequence because A001359(2)=5 and A023200(2)=7 are twin primes.

Michel Marcus, Table of n, a(n) for n = 1..149

Cf. A023200, A001359, A046132, A006512, A174042

lista(nn)  = {vp = primes(nn); va = select(x->isprime(x+2), vp); vb = select(x->isprime(x+4), vp); for (n=1, min(#va, #vb), if (vb[n] == va[n]+2, print1(n, ", ")););} \\ Michel Marcus, Jul 22 2017

Terms beyond 29 from R. J. Mathar, Nov 03 2011

Edited by Michel Marcus, Jul 22 2017

A229500 Smaller of Fermi-Dirac twin primes (A229064) which are not the smaller of twin primes (A001359).

Original entry on oeis.org

7, 9, 23, 47, 79, 81, 167, 359, 839, 1367, 1847, 2207, 2399, 3719, 5039, 6561, 7919, 10607, 11447, 14639, 16127, 17159, 19319, 28559, 29927, 36479, 44519, 49727, 54287, 57119, 66047, 85847, 97967, 113567, 128879, 177239, 196247, 201599, 218087, 241079, 273527

Offset: 1

Vladimir Shevelev, Sep 25 2013

nonn

The sequence conjecturally infinite. All squares of the sequence are 9, 81, 6561,... and have the form 3^(2^k), k>=1. However, we conjecture that there is only a finite number of them, or, the same, there is only a finite number of primes of the form 3^(2^k) + 2.

Peter J. C. Moses, Table of n, a(n) for n = 1..1200

Cf. A229064, A001359.

More terms from Peter J. C. Moses, Sep 25 2013

A270541 a(n) = A001359(n) - A001359(n+1) - A001359(n+2) + A001359(n+3).

Original entry on oeis.org

4, 6, 6, 6, 0, 12, -6, 0, 6, 0, 0, -24, 18, 6, 0, 0, 0, 24, 42, -24, -42, 48, 18, -30, -30, -6, 0, 126, -6, -144, 18, 18, 108, -12, -120, 0, 12, 48, 48, -12, -66, -36, 6, 96, 6, -78, -18, 90, 6, -72, 18, -24, 36, 60, -60, -30, 12, -6, 12, 6, -24, -30, 12, -12, 78, 18, -54, 0, 0, 138, 0, -102, -12, -42

Offset: 1

Altug Alkan, Mar 18 2016

sign

6*k appears for the form of a(n) for n > 1.
What is the most repeated value of a(n)?
See A270535 for the position of 0's in this sequence.

			a(1) = 4 because a(1) = A001359(1) - A001359(2) - A001359(3) + A001359(4) = 3 - 5 - 11 + 17 = 4.

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

Cf. A001359, A006512, A014574, A053319.

s = Select[Prime@Range[10^6], PrimeQ[# + 2] &]; Table[s[[n]] - s[[n + 1]] - s[[n + 2]] + s[[n + 3]], {n, 74}] (* Michael De Vlieger, Mar 19 2016, after Robert G. Wilson v at A001359 *)
#[[1]]-#[[2]]-#[[3]]+#[[4]]&/@Partition[Select[Partition[Prime[Range[400]],2,1],#[[2]]-#[[1]]==2&][[;;,1]],4,1] (* Harvey P. Dale, Jun 14 2025 *)

t(n, p=3) = { while( p+2 < (p=nextprime( p+1 )) || n-->0, ); p-2}
a(n) = t(n) + t(n+3) - t(n+1) - t(n+2);
for(n=1, 200, print1(a(n), ", "));

a(n) = A053319(n+2) - A053319(n).

cp's OEIS Frontend

A308344 a(n) = (A001359(n+1)^2 - 1)/24, where A001359 = lesser of twin primes; or: pentagonal numbers (A000326) whose indices are twin ranks (A002822).

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A339625 a(n) is the number of ways to write 6*n = p + q with p a lesser twin prime (A001359) and q a greater twin prime (A006512).

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A339630 a(n) is the first number k such that there are exactly n ways to write 6*k = p + q with p a lesser twin prime (A001359) and q a greater twin prime (A006512), or 0 if there is no such k.

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A357934 Products of two distinct lesser twin primes A001359.

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A359639 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have at least n odd prime factors, counted with multiplicity.

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A079328 Let f(n)=A001359(n) be the smaller member of the n-th pair of twin primes. Then a(n) is the average of f(n) and f(n+1).

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A097973 Least m>p such that p|m, p+1|m+1 and p+2|m+2, for twin prime pairs (p, p+2), p in A001359.

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A174046 Places n for which A001359(n) and A023200(n) is a twin prime pair.

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