A062301 -id:A062301 - cp's OEIS frontend

A107770 Index of greater of twin primes in the primes.

3, 4, 6, 8, 11, 14, 18, 21, 27, 29, 34, 36, 42, 44, 46, 50, 53, 58, 61, 65, 70, 82, 84, 90, 99, 105, 110, 114, 117, 121, 141, 143, 145, 149, 153, 172, 174, 177, 179, 183, 191, 202, 207, 210, 213, 216, 226, 231, 235, 237, 254, 257, 263, 266, 269, 278, 287

Offset: 1

Roger L. Bagula, Jun 11 2005

Numbers k such that prime(k) - prime(k-1) = 2.

Numbers k such that A062301(k) is 1. - Vincenzo Librandi, Apr 04 2018

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

Cf. A062301.

[n: n in [1..450]| IsPrime(NthPrime(n)-2)]; // Vincenzo Librandi, Apr 04 2018

select(n->ithprime(n)-ithprime(n-1)=2, [$2..300]); # Muniru A Asiru, Apr 05 2018

a = Flatten[Table[If[Prime[j] - Prime[j - 1] == 2, j, {}], {j, 2, 200}]]
Flatten[Position[Partition[Prime[Range[400]],2,1],?(#[[2]]-#[[1]] == 2&),{1},Heads->False]]+1 (* _Harvey P. Dale, Jun 10 2014 *)

for(n=3, 1e3, if(isprime(prime(n)-2), print1(n, ", "))); \\ Altug Alkan, Apr 05 2018

a(n) = A029707(n) + 1. - Juri-Stepan Gerasimov, Dec 16 2009

a(n) = A000720(A006512(n)).

Incorrect comment removed by Charles R Greathouse IV, Mar 19 2010

More terms from Harvey P. Dale, Jun 10 2014

A100810 a(n) = 0 if prime(n) + 2 = prime(n+1), otherwise 1.

Original entry on oeis.org

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

Offset: 1

Giovanni Teofilatto, Jan 05 2005

			a(2) = 0 because prime(2) + 2 = 5 = prime(3).
a(3) = 0 because prime(3) + 2 = 7 = prime(4).

a:= n-> `if`(isprime(ithprime(n)+2), 0, 1):
seq(a(n), n=1..105);  # Alois P. Heinz, Oct 02 2020

Table[If[Prime[n] + 2 == Prime[n + 1], 0, 1], {n, 120}] (* Ray Chandler, Jan 09 2005 *)
If[#[[2]]-#[[1]]==2,0,1]&/@Partition[Prime[Range[110]],2,1] (* Harvey P. Dale, Mar 05 2016 *)

a(n) = 1 - A100821(n) = 1 - A062301(n+1).

Corrected and extended by Ray Chandler, Jan 09 2005

A100821 a(n) = 1 if prime(n) + 2 = prime(n+1), otherwise 0.

Original entry on oeis.org

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

Offset: 1

Giovanni Teofilatto, Jan 06 2005

Same as A062301 except for starting point.

a(n)=1 iff prime(n) is the smaller of a pair of twin primes, else a(n)=0. This sequence can be derived from the sequence b(n)=1 iff n and n+2 are both prime, else b(n)=0. This latter sequence has as its inverse Moebius transform the sequence c(n) = the number of distinct factors of n which are the smaller of a pair of twin primes. For example, c(15)=2 because 15 is divisible by 3 and 5, each of which is the smaller of a pair of twin primes. - Jonathan Vos Post, Jan 07 2005

Table[If[Prime[n] + 2 == Prime[n + 1], 1, 0], {n, 120}] (* Ray Chandler, Jan 09 2005 *)

a(n) = A062301(n+1) = 1 - A100810(n).

Corrected and extended by Ray Chandler, Jan 09 2005

A309332 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the sum of two positive triangular numbers.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 2, 0, 3, 0, 0, 1, 1, 3, 0, 0, 1, 0, 1, 0, 0, 3, 1, 1, 0, 1, 3, 0, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 2, 1, 1, 1, 1, 2, 1, 0, 3, 1, 1, 1, 0, 3, 1, 1, 0, 0, 2, 0, 1, 1, 1, 1, 1, 5, 0, 1, 1, 0, 1, 0, 0, 3, 0, 3, 1, 0, 3, 1, 3, 1, 3, 3, 0, 1, 0, 0, 3, 0, 2, 0, 1

Offset: 1

Alois P. Heinz, Aug 01 2019

nonn

The order doesn't matter. 21 = 6+15 = 15+6 are not counted as distinct solutions. - N. J. A. Sloane, Feb 22 2020

			a(3) = 1: 2*3/2 + 2*3/2 = 3*4/2.
a(21) = 2: 6*7/2 + 20*21/2 = 12*13/2 + 17*18/2 = 21*22/2.
a(23) = 3: 9*10/2 + 21*22/2 = 11*12/2 + 20*21/2 = 14*15/2 + 18*19/2 = 23*24/2.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000217, A001652, A012132, A027861, A046080 (the same for squares), A053141, A062301 (the same for primes), A108769, A309507.

a:= proc(n) local h, j, r, w; h, r:= n*(n+1), 0;
      for j from n-1 by -1 do w:= j*(j+1);
        if 2*w

a[n_] := Module[{h = n(n+1), j, r = 0, w}, For[j = n-1, True, j--, w = j(j+1); If[2w < h, Break[]]; If[ IntegerQ[Sqrt[4(h-w)+1]], r++]]; r];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Nov 16 2022, after Alois P. Heinz *)

a(n) > 0 <=> n in { A012132 }.
a(n) = 0 <=> n in { A027861 }.
a(n) = 1 <=> n in { A108769 }.

A322975 Number of divisors d of n such that d-2 is prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 2, 2, 0, 0, 1, 2, 1, 1, 2, 0, 2, 1, 1, 1, 0, 2, 2, 0, 1, 2, 2, 0, 2, 1, 1, 4, 0, 0, 1, 2, 2, 0, 2, 0, 1, 2, 2, 1, 0, 0, 3, 1, 1, 4, 1, 2, 1, 0, 1, 1, 2, 0, 2, 1, 0, 4, 2, 1, 2, 0, 2, 2, 0, 0, 3, 2, 1, 0, 1, 0, 4, 3, 1, 1, 0, 2, 1, 0, 2, 3, 3, 0, 0, 1, 2, 5

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

			10395 has 32 divisors: [1, 3, 5, 7, 9, 11, 15, 21, 27, 33, 35, 45, 55, 63, 77, 99, 105, 135, 165, 189, 231, 297, 315, 385, 495, 693, 945, 1155, 1485, 2079, 3465, 10395]. When 2 is subtracted from each, as 1-2 = -1, 3-2 = 1, 5-2 = 3, etc, the only differences that are primes are: [3, 5, 7, 13, 19, 31, 43, 53, 61, 97, 103, 163, 229, 313, 383, 691, 1153, 1483, 3463], thus (a10395) = 19.

Antti Karttunen, Table of n, a(n) for n = 1..10395
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A010051, A062301, A067513, A072627, A322358, A322976, A322977, A322978.

Array[DivisorSum[#, 1 &, PrimeQ[# - 2] &] &, 105] (* Michael De Vlieger, Jan 04 2019 *)

A322975(n) = sumdiv(n, d, isprime(d-2));

a(n) = Sum_{d|n, d>2} A010051(d-2).

a(A000040(n)) = A062301(n).

A336409 Distance from prime(n) to the nearest odd composite that is < prime(n).

Original entry on oeis.org

2, 4, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 2, 2, 4, 2, 4, 2, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 2, 2, 2, 4

Offset: 5

Clark Kimberling, Sep 06 2020

nonn

			Beginning with prime(5) = 11:  11-9 = 2, 13-9 = 4, 17-15 = 2, 19-15 = 4.

Cf. A175191, A014076, A336410, A336411, A062301.

A336409 := proc(n)
    local p;
    p := ithprime(n) ;
    for a from p-2 by -2 do
        if not isprime(a) then
            return p-a ;
        end if;
    end do:
end proc:
seq(A336409(n),n=5..100) ; # R. J. Mathar, Oct 02 2020
# second Maple program:
a:= n-> `if`(isprime(ithprime(n)-2), 4, 2):
seq(a(n), n=5..100);  # Alois P. Heinz, Oct 02 2020

z = 5000; d = Select[Range[2, z], ! PrimeQ@# && OddQ@# &];  (* A014076 *)
f[n_] := Select[d, # < Prime[n] &];
t = Table[Prime[n] - Max[f[n]], {n, 5, 300}]  (* A336409 *)
Flatten[Position[t, 2]]  (* A336410 *)
Flatten[Position[t, 4]]  (* A336411 *)

a(n) = 2 * A175191(n-1). - Alois P. Heinz, Oct 02 2020

a(n) = 2 * (A062301(n) + 1). - Hugo Pfoertner, Oct 02 2020

A340001 Number of ways prime(n) is a sum of five distinct primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 5, 6, 11, 14, 16, 25, 29, 39, 57, 68, 75, 88, 92, 109, 169, 198, 235, 240, 322, 331, 379, 437, 497, 565, 635, 634, 803, 798, 896, 888, 1091, 1328, 1477, 1444, 1616, 1753, 1730, 2080, 2262, 2452, 2627, 2588, 2790, 3043, 3004, 3535

Offset: 1

Michel Lagneau, Dec 26 2020

nonn

Conjecture: all primes >= 43 are the sum of five distinct primes.
The sequence of the prime numbers that are the sum of five distinct prime numbers begins with 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, ...
The primes in the sequence are 2, 5, 11, 29, 109, 331, 379, 1091, 1753, ...
The squares in the sequence are 0, 1, 16, 25, 169, 1444, ...

			a(14) = 1 because prime(14) = 43 = 3 + 5 + 7 + 11 + 17.
a(17) = 5 because prime(17) = 59 = 3 + 5 + 7 + 13 + 31 = 3 + 5 + 11 + 17 + 23 = 3 + 7 + 13 + 17 + 19 = 5 + 7 + 11 + 13 + 23 = 5 + 7 + 11 + 17 + 19.

Alois P. Heinz, Table of n, a(n) for n = 1..3000
Wikipedia, Goldbach's weak conjecture

Cf. A000040, A002372, A002375, A024684, A062301, A178041, A219199, A224534.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+(p-> `if`(p>n, 0,
       x*b(n-p, i-1)))(ithprime(i)))), x, 6)
    end:
a:= n-> coeff(b(ithprime(n), n), x, 5):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 30 2020

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 1, 0, b[n, i - 1] + Function[p, If[p > n, 0,
     x*b[n - p, i - 1]]][Prime[i]]]], {x, 0, 6}];
a[n_] := SeriesCoefficient[b[Prime[n], n], {x, 0, 5}];
Array[a, 100] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[p,{5}],AllTrue[#,PrimeQ]&&Length[Union[#]]==5&]],{p,Prime[Range[70]]}] (* Harvey P. Dale, Jul 07 2024 *)

a(n) = A219199(A000040(n)).

a(n) = [x^prime(n)*y^5] Product_{i>=1} (1+x^prime(i)*y). - Alois P. Heinz, Dec 30 2020

cp's OEIS Frontend

A107770 Index of greater of twin primes in the primes.

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A100810 a(n) = 0 if prime(n) + 2 = prime(n+1), otherwise 1.

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A100821 a(n) = 1 if prime(n) + 2 = prime(n+1), otherwise 0.

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A309332 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the sum of two positive triangular numbers.

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A322975 Number of divisors d of n such that d-2 is prime.

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PARI

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A336409 Distance from prime(n) to the nearest odd composite that is < prime(n).

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A340001 Number of ways prime(n) is a sum of five distinct primes.

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Maple

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