A001359 -id:A001359 - cp's OEIS frontend

A276826 a(n) is the maximal difference between the corresponding terms of sequences defined in the same way as A159559, but with initial terms A001359(n-1)+2 and A001359(n-1) respectively.

4, 14, 6, 6, 6, 12, 6, 8, 14, 14, 18, 36, 24, 65, 18, 6, 10, 6, 84, 14, 162, 10, 54, 84, 179, 10, 23, 12, 18, 18, 24, 128, 18, 24, 28, 10, 10, 72, 34, 23, 12, 18, 6, 6, 12, 34, 8, 644, 12, 12, 6, 29, 24, 12, 18, 28, 28, 24, 22, 22, 10, 14, 12, 12, 16, 6, 58

Offset: 2

Vladimir Shevelev, Sep 19 2016

nonn

It seems likely that 6 occurs infinitely often.

			Since A276703(3)=4 (cf. example there), a(2)=4.

V. Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2009.
Vladimir Shevelev, Peter J. C. Moses, Constellations of primes generated by twin primes, arXiv:1610.03385 [math.NT], 2016.

Cf. A001359, A159559, A229019, A276676, A276703, A276767.

More terms from Peter J. C. Moses, Sep 19 2016

A277118 For a lesser p of twin primes, let B_k be A159559, but with initial term k; then a(n) is the smallest m such that B_(p+2)(m)-B_p(m)>6, where p = A001359(n-1), or a(n) = 0 if there is no such m.

Original entry on oeis.org

0, 13, 0, 0, 0, 9, 0, 11, 11, 5, 3, 15, 3, 7, 3, 0, 3, 0, 3, 5, 7, 3, 11, 5, 3, 5, 11, 3, 9, 3, 3, 7, 3, 5, 5, 3, 5, 3, 5, 11, 3, 5, 0, 0, 5, 5, 7, 5, 13, 7, 0, 5, 3, 3, 3, 3, 7, 3, 3, 3, 5, 3, 7, 3, 3, 0, 3, 5, 5, 3, 11, 11, 5, 3, 5, 7, 5, 3, 0, 3, 3, 3, 3, 3

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Sep 30 2016

nonn

Theorem: a(n) takes only the values 0, 3, 5, 7, 9, 11, 13, 15, and 17.

Peter J. C. Moses, Table of n, a(n) for n = 2..5001
Vladimir Shevelev, "Nearest" twin primes, Post to seqfan, Sep 21 2016.
Vladimir Shevelev, Peter J. C. Moses, Constellations of primes generated by twin primes, arXiv:1610.03385 [math.NT], 2016.

Cf. A022009, A159559, A229019, A276676, A276703, A276767, A276826, A276831, A276848.

nextcomposite(n)=if(n<4, return(4)); n=ceil(n); if(isprime(n), n+1, n)
do(p)=my(a=p,b=p+2,f); for(n=3,17, f=if(isprime(n), nextprime, nextcomposite); a=f(a+1); b=f(b+1); if(b-a > 6, return(n))); 0
p=2; forprime(q=3,1e3, if(q-p==2, print1(do(p)", ")); p=q) \\ Charles R Greathouse IV, Oct 17 2016

a(n) = 3 on a subsequence of measure 1. - Charles R Greathouse IV, Oct 17 2016

A356793 Decimal expansion of sum of squares of reciprocals of lesser twin primes, Sum_{j>=1} 1/(A001359(j))^2.

Original entry on oeis.org

1, 6, 5, 6, 1, 8, 4, 6, 5, 3, 9, 5

Offset: 0

Artur Jasinski, Sep 04 2022

Alternative definition: sum of squares of reciprocals of primes whose distance from the next prime is equal to 2.
Convergence table:
   k       A001359(k)      Sum_{j=1..k} 1/A001359(j)^2
10000000   3285916169 0.165618465394273171950874120818
20000000   7065898967 0.165618465394707600197099741096
30000000  11044807451 0.165618465394836120901019351544
40000000  15151463321 0.165618465394895965582366015390
50000000  19358093939 0.165618465394930089884704869090
60000000  23644223231 0.165618465394951950670948192842
Using the Hardy-Littlewood prediction of the density of twin primes (see A347278), the contribution to the sum after the last entry in the table above can be estimated as 9.056*10^(-14), making the infinite sum ~= 0.16561846539504... . - Hugo Pfoertner, Sep 28 2022

			0.165618465395...

Jeffrey P.S. Lay, Sign changes in Mertens' first and second theorems, arXiv:1505.03589 [math.NT], 2015.
Mark B. Villarino, Mertens' Proof of Mertens' Theorem, arXiv:math/0504289 [math.HO], 2005.
Marek Wolf, Generalized Brun's constants, IFTUWr 910/97 (1998), 1-15.
Marek Wolf, Some heuristics on the gaps between consecutive primes, arXiv:1102.0481 [math.NT]. 2011.

Cf. A006512, A065421, A077800, A078437, A085548, A096247, A160910, A194098, A209328, A209329, A242301, A242302, A242303, A242304, A306539, A342714.

Cf. A347278.

Data extended to ...3, 9, 5 by Hugo Pfoertner, Sep 28 2022

A359637 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 prime factors, counted with multiplicity.

Original entry on oeis.org

7, 97, 349, 13309, 33613, 5594749, 84477247, 1524981247, 60924074749

Offset: 2

Hugo Pfoertner, Jan 16 2023

			a(2) = 7: 8 = 2^3, 9 = 3^2, 10 = 2*5 all have at least the minimum number of 2 prime factors;
a(3) = 97: 98 = 2*7^2, 99 = 3^2*11, 100 = 2^2*5^2 have a minimum of 3 prime factors;
a(4) = 349: 350 = 2*5^2*7, 351 = 3^3*13, 352 = 2^5*11 have a minimum of 4 prime factors.

Cf. A001222, A001359, A062502, A359636.

a359637(maxp) = {my (k=2, pp=3); forprime (p=5, maxp, my(mi=oo); if (p-pp>2, for (j=pp+1, p-1, my(mo=bigomega(j)); if (mo=k, print1(pp,", "); k++)); pp=p)};
a359637(10^8)

A073831 Maximum of A073830(k) for k between A001359(n) and A001359(n+1).

Original entry on oeis.org

8, 56, 182, 552, 1406, 2862, 4556, 9506, 10712, 17292, 19460, 30102, 32942, 37442, 49952, 54522, 69432, 77006, 94556, 113906, 167690, 177662, 209306, 259590, 317532, 352242, 376382, 398792, 427062, 636006, 658532, 678152

Offset: 1

Reinhard Zumkeller, Aug 12 2002

nonn

a(n) = A073830(A073832(n)); a(n) < A037074(n+2).

A073831 := proc(n)
    A073830(A073832(n)) ;
end proc:
seq(A073831(n),n=1..50) ; # R. J. Mathar, Feb 21 2017

f[n_] := Mod[4*((n - 1)! + 1) + n, n*(n + 2)];
pp = Select[Prime[Range[200]], PrimeQ[# + 2] & ];
a[n_] := Max[f /@ Range[pp[[n]], pp[[n + 1]]]];
Array[a, Length[pp] - 1] (* Jean-François Alcover, Feb 22 2018 *)

A073832 k between A001359(n) and A001359(n+1) such that A073830(k) is maximal.

Original entry on oeis.org

4, 7, 13, 23, 37, 53, 67, 97, 103, 131, 139, 173, 181, 193, 223, 233, 263, 277, 307, 337, 409, 421, 457, 509, 563, 593, 613, 631, 653, 797, 811, 823, 853, 877, 1013, 1021, 1039, 1051, 1087, 1129, 1223, 1259, 1283, 1297, 1307, 1423, 1447, 1471, 1483, 1601

Offset: 1

Reinhard Zumkeller, Aug 12 2002

nonn

A073830(a(n)) = A073831(n).

Michael S. Branicky, Table of n, a(n) for n = 1..2600

A073832 := proc(n)
    local k,kmx,a ;
    kmx := 0 ;
    a := A001359(n)+1 ;
    for k from A001359(n)+1 to A001359(n+1)-1 do
        if A073830(k) > kmx then
            a := k ;
            kmx := A073830(k) ;
        end if;
    end do:
    a ;
end proc:
seq(A073832(n),n=1..50) ; # R. J. Mathar, Feb 21 2017

f[n_] := Mod[4*((n - 1)! + 1) + n, n*(n + 2)];
pp = Select[Prime[Range[300]], PrimeQ[# + 2] & ];
a[n_] := MaximalBy[Range[pp[[n]], pp[[n + 1]]], f];
Array[a, Length[pp] - 1] // Flatten (* Jean-François Alcover, Feb 22 2018 *)

from math import factorial
from itertools import islice, pairwise
from sympy import isprime, nextprime, primerange
def f(n): return (4*(factorial(n-1) + 1) + n)%(n*(n + 2))
def bgen(): # generator of A001359
    p, q = 2, 3
    while True:
        if q - p == 2: yield p
        p, q = q, nextprime(q)
def agen(): # generator of terms
    for p, q in pairwise(bgen()):
        yield max((f(k), k) for k in range(p+1, q))[1]
print(list(islice(agen(), 80))) # Michael S. Branicky, Aug 13 2024

A078864 Smallest primes from A001359, each belonging to those different residue class of mod 210 which are listed in A078859. Arranged according to possible least positive residues mod 210.

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 419, 1427

Offset: 1

Labos Elemer, Dec 13 2002

			Several terms are equal to corresponding ones in A078859, while others are larger like: 1427=210.6+167 where r=167 is in A078859.

Cf. A001359, A008364, A078859-A078864.

f[x_] := Mod[Prime[x], 210] d[x_] := Prime[x+1]-Prime[x] t=Table[0, {210}]; Do[s=f[n]; If[Equal[d[n], 2]&&s<211&&t[[s]]==0, t[[s]]=Prime[n]], {n, 1, 10000}]; t

A095017 Number of lesser twin primes (A001359) in range ]2^n, 2^(n+1)].

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 7, 7, 12, 26, 45, 70, 113, 215, 355, 666, 1153, 2071, 3785, 6965, 12495, 22643, 41608, 76371, 140944, 261752, 484968, 904799, 1689477, 3160113, 5928904, 11139071, 20970782, 39535081, 74697745, 141342490, 267812262, 508194094, 965623233, 1837147717

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

Conjecture: a(n) > 0 for all n. This holds for all n <= 100. - Charles R Greathouse IV, May 14 2012

It appears that a(n+1) is approximately 2 * a(n) * (n/(n+1))^2. The 2 accounts for each segment of numbers being twice as large as the previous segment. The first n/(n+1) accounts for primes being less common as numbers increase in size. The second n/(n+1) accounts for twin primes being a less common gap size as numbers increase in size. This formula has increasing accuracy as the numbers increase and is better than 0.1% by the end of the known sequence. - Jerry M Lagrou, Jan 05 2025

Jerry M Lagrou, Table of n, a(n) for n = 1..52
Igoris Belovas and Martynas Sabaliauskas, On integer sequences associated with admissible prime k-tuples, Ann. Univ. Craiova-Math. Comp. Sci. Series (2025) Vol. 52, No. 1.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Cf. A033843, A095016, A036378.

Table[ps = Prime[Range[PrimePi[2^n] + 1, PrimePi[2^(n+1) + 1]]]; Count[Differences[ps], 2], {n, 25}] (* T. D. Noe, May 08 2012 *)

a095017(maxex2)={my (L=List([1]), p2=8, n2=0, pp=5); forprime (p=7, 2^maxex2, if (p>p2, p2*=2; listput(L,n2); n2=0); if (p-pp==2, n2++); pp=p); Vec(L)};
a095017(30) \\ Hugo Pfoertner, Feb 05 2024

a(34) and beyond from Jerry M Lagrou, Dec 02 2023

A096474 Difference prime(q+2) - prime(q) as q runs through the lesser members of twin primes (A001359).

Original entry on oeis.org

6, 6, 10, 8, 18, 12, 6, 14, 16, 12, 24, 18, 24, 18, 16, 14, 24, 18, 24, 18, 10, 12, 18, 40, 28, 20, 24, 18, 28, 10, 12, 12, 8, 8, 22, 16, 12, 12, 14, 14, 26, 36, 24, 30, 24, 8, 18, 30, 12, 22, 22, 16, 18, 24, 10, 14, 18, 14, 10, 20, 10, 32, 32, 12, 10, 44, 30, 18, 16, 36, 14, 12

Offset: 1

Labos Elemer, Jun 23 2004

nonn

			{q, q+2} = {17, 19} is the 4th twin-pair and prime(19) - prime(17) = 67 - 59 = 8, so a(4) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1270 from Vincenzo Librandi)

Cf. A001359, A006512, A057470, A057473.

{ta=Table[0, {1300}], tb=Table[0, {1300}], tc=Table[0, {1300}], u=1}; Do[s=Prime[n+1]-Prime[n];If[Equal[s, 2], ta[[u]]=Prime[Prime[n+1]]-Prime[Prime[n]];tb[[u]]=n; tc[[u]]=Prime[n];u=u+1], {n, 1, 10000}];ta
Prime[#[[2]]]-Prime[#[[1]]]&/@Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&] (* Harvey P. Dale, Dec 26 2023 *)

lista(nn) = {forprime(q=2, nn, if (isprime(q+2), print1(prime(q+2)-prime(q), ", ")););} \\ Michel Marcus, Jul 27 2017

a(n) = prime(A006512(n)) - prime(A001359(n)).

a(n) = A057473(n) - A057470(n). - Michel Marcus, Jul 27 2017

A143738 Number of twin primes between n and n^2. Only smaller of twins (terms of A001359) are counted.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 8, 9, 11, 12, 14, 16, 16, 17, 17, 19, 20, 21, 22, 24, 26, 26, 26, 29, 30, 30, 31, 34, 36, 36, 39, 41, 42, 45, 45, 48, 49, 50, 53, 55, 59, 61, 63, 66, 66, 68, 70, 74, 74, 76, 77, 78, 83, 87, 90, 91, 93, 96, 100

Offset: 1

Alen Skugor (askugor(AT)gmail.com), Aug 30 2008

nonn

			For n = 6, between 6 and 36 the smaller of twin pairs are {11, 17, 29}, so a(6) = 3.

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

Cf. A001359, A080356.

[#[p:p in PrimesInInterval(n,n^2)|IsPrime(p+2)]:n in [1..80]]; // Marius A. Burtea, Dec 20 2019

Table[Count[Table[PrimeQ[j] && PrimeQ[j + 2], {j, n, n*n}], True], {n, 1, 100}]

cp's OEIS Frontend

A276826 a(n) is the maximal difference between the corresponding terms of sequences defined in the same way as A159559, but with initial terms A001359(n-1)+2 and A001359(n-1) respectively.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A277118 For a lesser p of twin primes, let B_k be A159559, but with initial term k; then a(n) is the smallest m such that B_(p+2)(m)-B_p(m)>6, where p = A001359(n-1), or a(n) = 0 if there is no such m.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A356793 Decimal expansion of sum of squares of reciprocals of lesser twin primes, Sum_{j>=1} 1/(A001359(j))^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A359637 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 prime factors, counted with multiplicity.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A073831 Maximum of A073830(k) for k between A001359(n) and A001359(n+1).

Views

Author

Keywords

Comments

Programs

Maple

Mathematica

A073832 k between A001359(n) and A001359(n+1) such that A073830(k) is maximal.

Views

Author

Keywords

Comments

Links

Programs

Maple

Mathematica

Python

A078864 Smallest primes from A001359, each belonging to those different residue class of mod 210 which are listed in A078859. Arranged according to possible least positive residues mod 210.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A095017 Number of lesser twin primes (A001359) in range ]2^n, 2^(n+1)].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A096474 Difference prime(q+2) - prime(q) as q runs through the lesser members of twin primes (A001359).

Views

Author

Keywords

Examples

Links