A054735 -id:A054735 - cp's OEIS frontend

A335303 Numbers k such that k divides sum of k-th twin prime pair.

1, 2, 3, 4, 5, 6, 8, 12, 18, 30, 60, 70, 78, 150, 220, 240, 340, 360, 396, 672, 840, 1188, 1365, 1665, 3080, 3770, 6420, 7317, 9952, 10356, 12972, 18318, 19218, 20544, 21996, 24750, 28656, 34543, 37449, 41910, 180622, 201570, 245115, 728028, 856420, 897022, 986794

Offset: 1

Metin Sariyar, May 31 2020

nonn

			8 is a term because sum of 8th twin prime pair is 71 + 73 = 8*18.

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curio for 34543

Cf. A001097, A001359, A006512, A054735, A077800, A335304.

l={};c=0;Do[If[PrimeQ[Prime[n]+2],c=c+1;If[IntegerQ[(2*Prime[n]+2)/c], AppendTo[l,c]]] ,{n,2,10000}];l

is(n) = {!bittest(n, 0)&&isprime(n\2-1)&&isprime(n\2+1)}; \\ A054735
lista(nn) = {my(nb=0); for (n=1, nn, if (is(n), nb++; if ((n % nb) == 0, print1(nb, ", "));););} \\ Michel Marcus, Jun 01 2020

A335304 Primes k such that k divides sum of k-th twin prime pair.

Original entry on oeis.org

2, 3, 5, 34543

Offset: 1

Metin Sariyar, May 31 2020

Primes in A335303.

a(5) > 2*10^11, if it exists. - Giovanni Resta, Jun 01 2020

			34543 is a term because sum of 34543th twin prime pair is 5388707 + 5388709 = 8*3*13*34543.

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curio for 34543

Cf. A001097, A001359, A006512, A054735, A077800, A335303.

l={};c=0;Do[If[PrimeQ[Prime[n]+2],c=c+1;If[IntegerQ[(2*Prime[n]+2)/c]&&PrimeQ[c], AppendTo[l,c]]],{n,2,10^6}];l

is(n) = {!bittest(n, 0)&&isprime(n\2-1)&&isprime(n\2+1)}; \\ A054735
lista(nn) = {my(nb=0); for (n=1, nn, if (is(n), nb++; if (isprime(nb) && ((n % nb) == 0), print1(nb, ", "));););} \\ Michel Marcus, Jun 01 2020

A349757 Even numbers that are both the sum of a twin prime pair and the sum of 1 and a prime.

Original entry on oeis.org

8, 12, 24, 60, 84, 360, 384, 480, 564, 840, 864, 1284, 1320, 1620, 2040, 2064, 2100, 2460, 2580, 2904, 2964, 3864, 4260, 4284, 4680, 5100, 5940, 6600, 6660, 6720, 6780, 7080, 7644, 7704, 8040, 8544, 8964, 10464, 10560, 10884, 11004, 11280, 11484, 11700, 12264, 12540

Offset: 1

Wesley Ivan Hurt, Jan 01 2022

nonn

			8 is in the sequence since 8 = 3+5 = 7+1.
12 is in the sequence since 12 = 5+7 = 11+1.
24 is in the sequence since 24 = 11+13 = 23+1.

Cf. A008864, A054735 (sums of twin prime pairs).

A350500 Even numbers that are both the sum of a twin prime pair and the sum of 1 and a semiprime.

Original entry on oeis.org

36, 120, 144, 204, 216, 300, 396, 624, 696, 924, 1044, 1140, 1200, 1644, 1656, 1764, 2124, 2184, 2604, 2856, 3216, 3240, 3444, 3744, 3756, 3900, 4056, 4164, 4224, 4536, 4620, 4764, 5184, 5316, 5460, 5580, 5604, 6000, 6240, 6504, 6516, 6744, 7116, 7344, 7836, 7860, 8004

Offset: 1

Wesley Ivan Hurt, Jan 01 2022

nonn

			36 is in the sequence since 36 = 17+19 = 1+35.
120 is in the sequence since 120 = 59+61 = 1+119.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Intersection of A054735 and A088707.

Cf. A349757.

Select[12 * Range[700], And @@ PrimeQ[#/2 + {-1, 1}] && PrimeOmega[# - 1] == 2 &] (* Amiram Eldar, Jan 02 2022 *)
Select[Total/@Select[Partition[Prime[Range[600]],2,1],#[[2]]-#[[1]]==2&],PrimeOmega[#-1]==2&] (* Harvey P. Dale, Feb 02 2025 *)

from sympy import isprime, factorint
def ok(n): return n%2 == 0 and isprime(n//2-1) and isprime(n//2+1) and sum(factorint(n-1).values()) == 2
print([k for k in range(8005) if ok(k)]) # Michael S. Branicky, Jan 02 2022

A087187 Number of ways to write n = ip + jq, where (p,q) is a twin prime pair and i,j>0.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Aug 23 2003

nonn

a(A054735(n)) > 0.

			n=50 = 5+5+5+5+5+5+5+3+3+3+3+3 = 5+5+5+5+3+3+3+3+3+3+3+3+3+3 =
5+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3 = 7+7+7+7+7+5+5+5 = 13+13+13+11, therefore
a(50)=5

Eric Weisstein's World of Mathematics, Prime Partition

Cf. A000607, A001359, A006512.

A135284 Sum of staircase twin primes according to the rule: top + bottom - next top.

Original entry on oeis.org

3, 1, 7, 7, 19, 25, 49, 43, 97, 79, 127, 121, 169, 187, 169, 217, 211, 259, 253, 277, 277, 409, 403, 403, 475, 541, 583, 595, 625, 511, 799, 817, 799, 835, 745, 1009, 1015, 1039, 1033, 1033, 1075, 1183, 1267, 1279, 1285, 1213, 1405, 1423, 1477, 1369, 1597, 1573

Offset: 1

Cino Hilliard, Dec 03 2007

nonn

The case for bottom - top + next top produces A006512(n+1), the upper twin primes > 5.

g(n) = for(x=1,n,y=twinu(x) + twinl(x) - twinl(x+1);print1(y",")) twinl(n) = / *The n-th lower twin prime. */ { local(c,x); c=0; x=1; while(c

We list the twin primes in staircase fashion as in A135283. Then a(n) = tl(n) + tu(n) + (-tl(n+1)).

a(n) = A054735(n)-A001359(n+1). - R. J. Mathar, Sep 10 2016

A270245 Lesser of twin primes such that sum of twin prime pair is the sum of 2 nonzero squares.

Original entry on oeis.org

3, 179, 521, 809, 1619, 1871, 2087, 2339, 3257, 3329, 4049, 4337, 4931, 5651, 5849, 6569, 6659, 6947, 7487, 8009, 8387, 8819, 8999, 10529, 10889, 11699, 12239, 14561, 15137, 16361, 16451, 16649, 17657, 17747, 19079, 19889, 19961, 20231, 20771, 20807, 21059, 22481, 22697, 23039, 23201

Offset: 1

Altug Alkan, Mar 13 2016

nonn

			3 is a term because 3 + 5 = 2^2 + 2^2.
179 is a term because 179 + 181 = 6^2 + 18^2.
521 is a term because 521 + 523 = 12^2 + 30^2.
809 is a term because 809 + 811 = 18^2 + 36^2.

Cf. A000404, A001359, A054735.

Select[Select[Prime@ Range@ 2700, PrimeQ[# + 2] &], Length[PowersRepresentations[2 # + 2, 2, 2] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Mar 15 2016 *)

isA000404(n)={ for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
t(n,p=3) = {while( p+2 < (p=nextprime( p+1 )) || n-->0, ); p-2}
for(n=1, 1e3, if(isA000404(2*t(n)+2), print1(t(n), ", ")));

A305558 If (p1,p2) is the n-th twin prime pair and p the prime before p1 and q the prime after p2 then a(n) = p + q - (p1 + p2).

Original entry on oeis.org

1, 2, 0, 0, 0, 0, 0, 2, 0, 0, 4, -4, 4, -6, 8, 0, 4, 0, 6, 0, -6, 0, -4, 0, 6, 0, 0, 8, -6, 6, -2, -6, 6, 0, 0, 4, -4, 0, -4, 0, -12, 0, -14, 0, 0, -6, 0, 2, -6, 0, -2, 0, 20, 6, -2, 8, 0, 6, -2, 6, 0, 0, -8, 6, 4, -10, 6, -12, -12, 10, 0, 2, 0, 4, -6, 0, 2, 0, -6, 12, 22, -18, 6, 8, -18, 8, -22, 6, -2, 6, 0, 0, 18, -6

Offset: 1

Dimitris Valianatos, Jun 21 2018

			For n = 8, the 8th prime pair is (71, 73), the prime before 71 is 67 and prime after 73 is 79. So a(8) = 67 + 79 - 71 - 73 = 2.

Cf. A001097, A054735, A036263.

Map[#1 + #4 - (#2 + #3) & @@ # &, Select[Partition[Prime@ Range[500], 4, 1], And[#3 - #2 == 2] & @@ # &]] (* Michael De Vlieger, Jun 30 2018 *)

{
print1(2+7-(5+3)", ");
forstep(n=6,100,6,
        if(isprime(n-1)&&isprime(n+1),
           a=precprime(n-2);b=nextprime(n+2);
           print1(a+b-2*n", ")
          )
       )
}

a(n) = A000040(A029707(n)-1) + A000040(A107770(n)+1) - (A001359(n) + A006512(n)). - Jianing Song, Jun 22 2018

Definition clarified by Jianing Song, Jun 22 2018

A329590 Odd numbers k that cannot be expressed as k = p+q+r, with p prime and (q, r) a pair of twin primes.

Original entry on oeis.org

1, 3, 5, 7, 9, 33, 57, 93, 99, 129, 141, 153, 177, 183, 195, 213, 225, 243, 255, 261, 267, 273, 297, 309, 327, 333, 351, 369, 393, 411, 423, 435, 453, 477, 489, 501, 513, 519, 525, 537, 561, 573, 591, 597, 603, 633, 645, 657, 663, 675, 687, 693, 705, 711, 723

Offset: 1

Antonio Roldán, Feb 13 2020

nonn

			33 can be expressed as the sum of three primes in 9 different ways:
33 = 11 + 11 + 11 = 13 + 13 + 7 = 17 + 11 + 5 = 17 + 13 + 3 = 19 + 7 + 7 = 19 + 11 + 3 = 23 + 5 + 5 = 23 + 7 + 3 = 29 + 2 + 2;
there is no pair of twin primes in the addends, so 33 is a term.

Cf. A001359, A034961, A054735, A068307, A124868.

for(n = 0, 500, m = 2*n+1; v = 0; forprime(i = 3, m-8, j = (m-i)/2; if(isprime(j-1) && isprime(j+1), v = 1)); if(v == 0, print1(m,", ")))

isok(k) = {if (! (k % 2), return (0)); forprime(p=3, k, if (isprime((k-p)\2-1) && isprime((k-p)\2+1), return(0));); return (1);} \\ Michel Marcus, Feb 16 2020

A334295 Integers k such that the sum of k twin primes pairs starting from (5,7) is a perfect power.

Original entry on oeis.org

2, 5, 9, 352, 165421, 356928514, 795471483

Offset: 1

Devansh Singh, Apr 21 2020

			a(1) = 2 as 5+7 + 11+13 = 36 = 6^2;
a(2) = 5 as 5+7 + 11+13 + 17+19 + 29+31 + 41+43 = 216 = 6^3.
From _Michel Marcus_, Apr 27 2020: (Start)
Table of results, with k, greatest prime and corresponding sum:
   2, 13, 36 = 6^2;
   5, 43, 216 = 6^3;
   9, 109, 900 = 30^2;
   352, 20749, 6290064 = 2508^2;
   165421, 32841799, 5048685437184 = 2246928^2. (End)
From _Giovanni Resta_, Apr 27 2020: (Start)
The next two entries of the table above are:
   356928514, 165800305423, 56622416174760209796 = 7524786786^2;
   795471483, 396030375733, 301922786495024336196 = 17375925486^2. (End)

Cf. A001097 (twin primes), A054735 (sum of twin prime pairs).

lista(nn) = {my(s = 0, nb = 0); forprime(p=5, nn, if (isprime(p+2), s += 2*p+2; nb++; if (ispower(s), print1(nb, ", "));););} \\ Michel Marcus, Apr 22 2020

a(4)-a(5) from Michel Marcus, Apr 22 2020
a(6) from Jinyuan Wang, Apr 24 2020
a(7) from Giovanni Resta, Apr 27 2020

cp's OEIS Frontend

A335303 Numbers k such that k divides sum of k-th twin prime pair.

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A335304 Primes k such that k divides sum of k-th twin prime pair.

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A349757 Even numbers that are both the sum of a twin prime pair and the sum of 1 and a prime.

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A350500 Even numbers that are both the sum of a twin prime pair and the sum of 1 and a semiprime.

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A087187 Number of ways to write n = i*p + j*q, where (p,q) is a twin prime pair and i,j>0.

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A135284 Sum of staircase twin primes according to the rule: top + bottom - next top.

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A270245 Lesser of twin primes such that sum of twin prime pair is the sum of 2 nonzero squares.

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A305558 If (p1,p2) is the n-th twin prime pair and p the prime before p1 and q the prime after p2 then a(n) = p + q - (p1 + p2).

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A329590 Odd numbers k that cannot be expressed as k = p+q+r, with p prime and (q, r) a pair of twin primes.

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A087187 Number of ways to write n = ip + jq, where (p,q) is a twin prime pair and i,j>0.