A062729 -id:A062729 - cp's OEIS frontend

A224380 Table read by antidiagonals of numbers of form (2^n -1)*2^(m+2) + 3 where n>=1, m>=1.

11, 19, 27, 35, 51, 59, 67, 99, 115, 123, 131, 195, 227, 243, 251, 259, 387, 451, 483, 499, 507, 515, 771, 899, 963, 995, 1011, 1019, 1027, 1539, 1795, 1923, 1987, 2019, 2035, 2043, 2051, 3075, 3587, 3843, 3971, 4035, 4067, 4083, 4091, 4099, 6147, 7171, 7683, 7939, 8067, 8131, 8163, 8179

Offset: 1

Brad Clardy, Apr 05 2013

The table has row labels 2^n - 1 and column labels 2^(m+2). The table entry is row*col + 3. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the antidiagonals starting from the top left corner. Using the lexicographic ordering of A057555  the sequence is:
A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
+3  |    8    16    32    64   128    256    512 ...
----|-------------------------------------------
1   |   11    19    35    67   131    259    515
3   |   27    51    99   195   387    771   1539
7   |   59   115   227   451   899   1795   3587
15  |  123   243   483   963  1923   3843   7683
31  |  251   499   995  1987  3971   7939  15875
63  |  507  1011  2019  4035  8067  16131  32259
127 | 1019  2035  4067  8131 16259  32515  65027
...
All of these numbers have the following property: let m be a member of A(n); if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then the differences between successive members of B(n) is an alternating series of 1's and 3's with the last difference in the pattern m. The number of alternating 1's and 3's in the pattern is 2^(j+1) - 1, where j is the column index.
As an example consider A(1) which is 11, the sequence B(n) where i XOR 10 = i - 10 starts as 10, 11, 14, 15, 26, 27, 30, 31, 42, ... (A214864) with successive differences of 1, 3, 1, 11.
Main diagonal is A191341, the largest k such that k-1 and k+1 in binary representation have the same number of 1's and 0's

Brad Clardy, Table of n, a(n) for n = 1..1000

Cf. A057555(lexicographic ordering), A214864(example), A224195.
Rows: A062729(i=1), A147595(2 n>=5), A164285(3 n>=3).
Cols: A168616(j=1 n>=4).
Diagonal: A191341.

//program generates values in a table form,row labels of 2^i -1
for i:=1 to 10 do
    m:=2^i - 1;
    m, [ m*2^n +1 : n in [1..10]];
end for;
//program generates sequence in lexicographic ordering of A057555, read
//along antidiagonals from top. Primes in the sequence are marked with *.
for i:=2 to 18 do
    for j:=1 to i-1 do
       m:=2^j -1;
       k:=m*2^(2+i-j) + 3;
       if IsPrime(k) then k, "*";
          else k;
       end if;;
    end for;
end for;

a(n) = 2^(A057555(2*n - 1))*2^(A057555(2*n) + 2) + 3 for n>=1.

A284203 Number of twin prime (A001097) divisors of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 2, 1, 0, 1, 2, 1, 1, 2, 1, 1, 2, 0, 0, 1, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 0, 2, 2, 0, 1, 1, 2, 1, 1, 1, 0, 2, 1, 2, 2, 0, 1, 1, 1, 0, 2, 2, 1, 2, 1, 0, 2, 2, 0, 2, 0, 2, 1, 0, 1, 2, 1, 1, 2, 1, 1, 3, 0, 1, 1, 1, 2

Offset: 1

Ilya Gutkovskiy, Mar 22 2017

nonn

			--------------------------------------------
| n | divisors of n | twin prime    | a(n) |
|   |               | divisors of n |      |
|------------------------------------------
| 1 | {1}           |      {-}      |  0   |
| 2 | {1, 2}        |      {-}      |  0   |
| 3 | {1, 3}        |      {3}      |  1   |
| 4 | {1, 2, 4}     |      {-}      |  0   |
| 5 | {1, 5}        |      {5}      |  1   |
| 6 | {1, 2, 3, 6}  |      {3}      |  1   |
| 7 | {1, 7}        |      {7}      |  1   |
| 8 | {1, 2, 4, 8}  |      {-}      |  0   |
| 9 | {1, 3, 9}     |      {3}      |  1   |
--------------------------------------------

Eric Weisstein's World of Mathematics, Twin Primes.

Cf. A001097, A001221, A005087, A037074, A062729, A065421, A284599.

Cf. A048599 (positions of records).

nmax = 110; Rest[CoefficientList[Series[Sum[Boole[PrimeQ[k] && (PrimeQ[k - 2] || PrimeQ[k + 2])] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Length[Select[Divisors[n], PrimeQ[#] && (PrimeQ[# - 2] || PrimeQ[# + 2]) &]], {n, 110}]

concat([0, 0],Vec(sum(k=1, 110, (isprime(k) && (isprime(k - 2) || isprime(k + 2)))* x^k/(1 - x^k)) + O(x^111))) \\ Indranil Ghosh, Mar 22 2017

a(n) = sumdiv(n, d, isprime(d) && (isprime(d-2) || isprime(d+2))); \\ Amiram Eldar, Jun 03 2024

from sympy import isprime, divisors
print([len([i for i in divisors(n) if isprime(i) and (isprime(i - 2) or isprime(i + 2))]) for n in range(1, 111)]) # Indranil Ghosh, Mar 22 2017

G.f.: Sum_{k>=1} x^A001097(k)/(1 - x^A001097(k)).
a(A062729(n)) = 0. - Ilya Gutkovskiy, Apr 02 2017
From Amiram Eldar, Jun 03 2024: (Start)
a(A048599(n)) = n.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A065421 - 1/5 = 1.7021605... . (End)
Additive with a(p^e) = 1 if p is in A001097, and 0 otherwise. - Amiram Eldar, May 15 2025
a(A037074(n)) = 2. - Michel Marcus, May 15 2025

A284599 Sum of twin prime (A001097) divisors of n.

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 3, 5, 11, 3, 13, 7, 8, 0, 17, 3, 19, 5, 10, 11, 0, 3, 5, 13, 3, 7, 29, 8, 31, 0, 14, 17, 12, 3, 0, 19, 16, 5, 41, 10, 43, 11, 8, 0, 0, 3, 7, 5, 20, 13, 0, 3, 16, 7, 22, 29, 59, 8, 61, 31, 10, 0, 18, 14, 0, 17, 3, 12, 71, 3, 73, 0, 8, 19, 18, 16, 0, 5, 3, 41, 0, 10, 22, 43, 32, 11, 0, 8

Offset: 1

Ilya Gutkovskiy, Mar 30 2017

nonn

			a(15) = 8 because 15 has 4 divisors {1, 3, 5, 15} among which 2 are twin primes {3, 5} therefore 3 + 5 = 8.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes.
Index entries for sequences related to sums of divisors.

Cf. A001097, A008472, A062729, A284203.

N:= 200: # to get a(1)..a(N)
P:= select(isprime, {seq(i,i=3..N+2)}):
TP:= P intersect map(`-`,P,2):
TP:= TP union map(`+`,TP,2):
V:= Vector(N):
for p in TP do
  pm:= [seq(i,i=p..N,p)];
  V[pm]:= map(`+`,V[pm],p);
od:
convert(V,list); # Robert Israel, Mar 30 2017

Table[Total[Select[Divisors[n], PrimeQ[#1] && (PrimeQ[#1 - 2] || PrimeQ[#1 + 2]) &]], {n, 80}]

a(n) = sumdiv(n, d, d*(isprime(d) && (isprime(d-2) || isprime(d+2)))); \\ Michel Marcus, Apr 02 2017

from sympy import divisors, isprime
def a(n): return sum([i for i in divisors(n) if isprime(i) and (isprime(i - 2) or isprime(i + 2))])
print([a(n) for n in range(1, 91)]) # Indranil Ghosh, Mar 30 2017

G.f.: Sum_{k>=1} A001097(k)*x^A001097(k)/(1 - x^A001097(k)).
a(n) = Sum_{d|n, d twin prime} d.
a(A062729(n)) = 0.
a(A001097(n)) = A001097(n).
Additive with a(p^e) = p if p is in A001097, and 0 otherwise. - Amiram Eldar, May 15 2025

A062506 n divisible by at least one twin prime (A001097).

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87

Offset: 1

Leroy Quet, Jul 09 2001

nonn

Complement of A062729.

			36 is included because 36 is divisible by 3 and 3+2 is a prime.

isok(n) = {my(f = factor(n)); for (i=1, #f~, p = f[i, 1]; if (isprime(p-2) || isprime(p+2), return (1));); return (0);} \\ Michel Marcus, May 20 2014

Offset changed to 1 by Michel Marcus, May 20 2014

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A224380 Table read by antidiagonals of numbers of form (2^n -1)*2^(m+2) + 3 where n>=1, m>=1.

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A284203 Number of twin prime (A001097) divisors of n.

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A284599 Sum of twin prime (A001097) divisors of n.

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Maple

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A062506 n divisible by at least one twin prime (A001097).

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