A105093 -id:A105093 - cp's OEIS frontend

A263203 Consider the numbers m such that m = prime(k) + prime(k+2i+1) = prime(k+i) + prime(k+i+1) for some i and k. The sequence lists the number of pairs (i,k) giving the same value m = A105093(n).

1, 2, 2, 1, 4, 3, 4, 2, 2, 2, 1, 1, 2, 6, 5, 4, 1, 2, 1, 4, 4, 5, 7, 3, 6, 7, 1, 2, 1, 7, 10, 7, 7, 2, 6, 1, 5, 10, 12, 5, 10, 3, 5, 11, 9, 9, 8, 2, 6, 2, 2, 3, 10, 1, 5, 11, 10, 7, 7, 5, 3, 5, 5, 1, 4, 2, 4, 2, 5, 7, 4, 5, 8, 7, 6, 5, 3, 7, 13, 1, 1, 9, 5, 1

Offset: 1

Michel Lagneau, Oct 12 2015

nonn

The form m = prime(k) + prime(k+2i+1) = prime(k+i) + prime(k+i+1) is a generalization of A105093 (form prime(k) + prime(k+3) = prime(k+1) + prime(k+2)), and the set of the numbers m is exactly A105093(n).

			a(6) = 3 because A105093(6)= 84 and:
for (i,k)=(1,12), prime(12)+ prime(15)= prime(13)+ prime(14)=37+47=41+43=84;
for (i,k)=(2,11), prime(11)+ prime(16)= prime(12)+ prime(15)=31+53=37+47=84;
for (i,k)=(4,9), prime(9)+ prime(18)= prime(13)+ prime(14)=23+61=41+43=84.
So, we find 3 pairs (i,k) giving m = 84.

Cf. A105093.

with(numtheory):nn:=5000:
A105093:={18,24,30,36,60,84,120,162,204,210,216,240,288,330,372,390,456,520,540,624,726,762,798,840,852,882,924,978,990,1104,1140,1164,1200,1392,1410,1428,1530,1632,1650,1716,1740,1764,1794,1848,1974,2052,2100,2112,2184,2226,2334,2460,2580,2604,2688,2856,2970,2976,3054,3102,3138,3150,3180,3240,3500,3536,3612,3744,3750,3882,3966,3996,4056,4092,4170,4242,4680,4698,4728,4782,4810,5100,5376,5460}:n0:=nops(A105093):
  for n from 1 to n0 do:
   ii:=0:it:=0:q:=A105093[n]:
   for i from 1 to 100 do:
      for k from 1 to nn do:
        s1:=ithprime(k)+ithprime(k+2*i+1):
        s2:= ithprime(k+i)+ithprime(k+i+1):
        if s1=s2 and s1=q
        then
        it:=it+1:
        else
        fi:
       od:
      od:
       printf(`%d, `,it):
     od:

A333122 Numbers m such that m = prime(k) + prime(k+5) = prime(k+1) + prime(k+4) for some k.

Original entry on oeis.org

24, 30, 60, 84, 102, 210, 234, 288, 330, 378, 420, 426, 496, 528, 588, 594, 624, 690, 1050, 1156, 1200, 1218, 1302, 1336, 1410, 1470, 1484, 1638, 1650, 1680, 1686, 1716, 1734, 1740, 1746, 1788, 1848, 1908, 1918, 1930, 2052, 2154, 2226, 2364, 2410, 2580, 2892, 2934, 3168, 3524, 4080

Offset: 1

Marcin Barylski, Mar 08 2020

nonn

Terms are always even because all primes used in this sequence are odd.

Conjecture: this sequence is infinite.

			a(1)=24 because prime(3)+prime(8)=prime(4)+prime(7)=5+19=7+17.

Cf. A022889 (the prime(k) primes), A105093 (similar sequence).

(#[[1]] + #[[6]]) & /@ Select[ Partition[ Prime@ Range@ 320, 6, 1], #[[1]] + #[[6]] == #[[2]] + #[[5]] &] (* Giovanni Resta, Mar 08 2020 *)

from sympy import nextprime
A333122_list, plist = [], [2,3,5,7,11,13]
while len(A333122_list) < 10000:
    m = plist[0]+plist[5]
    if m == plist[1]+plist[4]:
        A333122_list.append(m)
    plist = plist[1:] + [nextprime(plist[-1])] # Chai Wah Wu, Mar 30 2020

A358028 Primes p = prime(9t+1) such that the 9 consecutive primes prime(9t+1) .. prime(9*t+9) arranged in a 3 X 3 array have at least 2 equal sums along the rows, columns or main diagonals.

Original entry on oeis.org

2, 29, 67, 107, 157, 257, 311, 367, 541, 599, 709, 769, 829, 967, 1021, 1549, 1741, 1811, 1879, 1973, 2609, 2677, 3019, 3541, 3677, 4051, 4217, 4271, 4517, 4597, 4663, 4931, 5227, 5303, 5399, 5449, 5623, 5683, 5839, 6079, 6229, 6301, 6361, 6451, 6949, 7253, 7351, 7477, 7537, 7589, 7673

Offset: 1

Saish S. Kambali, Nov 12 2022

nonn

Primes are taken in successive blocks of 9 and arranged, for t>=0,
   | prime(9*t+1) | prime(9*t+2) | prime(9*t+3)  |
   | prime(9*t+4) | prime(9*t+5) | prime(9*t+6)  |
   | prime(9*t+7) | prime(9*t+8) | prime(9*t+9)  |
There are 8 lines altogether: 3 rows, 3 columns, and 2 main diagonals.
The sum of the first row is never duplicated since any other line has a greater sum.
The sum of the last row is never duplicated since any other line has a smaller sum.

			2 is a term since its block of 9 primes is
  | 2  | 3  | 5  |
  | 7  | 11 | 13 |
  | 17 | 19 | 23 |
which has among its lines (3 + 11 + 19) = (17 + 11 + 5).
67 is a term since its block of 9 primes (the 3rd block) is 67..103,
  | 67 | 71 | 73 |
  | 79 | 83 | 89 |
  | 97 | 101| 103|
which has 67+83+103 = 97+83+73.

Cf. A105093.

Subsequence of A031918 (by definition).

a = {}
row = {{1, 4, 7}, {2, 5, 8}, {3, 6, 9}};
col = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};
dia = {{1, 3}, {5, 5}, {9, 7}};
Duplicates[l_] :=
Block[{i}, i[n_] := (i[n] = n; Unevaluated@Sequence[]); i /@ l]
Do[If[Duplicates[
    Flatten[{Total[Prime[row + 9 n]], Total[Prime[col + 9 n]],
      Total[Prime[dia + 9 n]]}]] != {},
  AppendTo[a, Prime[9 n + 1]]], {n, 0, 110}]
a (* Gerry Martens, Nov 12 2022 *)

More terms from Gerry Martens, Nov 12 2022

cp's OEIS Frontend

A263203 Consider the numbers m such that m = prime(k) + prime(k+2i+1) = prime(k+i) + prime(k+i+1) for some i and k. The sequence lists the number of pairs (i,k) giving the same value m = A105093(n).

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A333122 Numbers m such that m = prime(k) + prime(k+5) = prime(k+1) + prime(k+4) for some k.

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A358028 Primes p = prime(9t+1) such that the 9 consecutive primes prime(9t+1) .. prime(9*t+9) arranged in a 3 X 3 array have at least 2 equal sums along the rows, columns or main diagonals.

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cp's OEIS Frontend

A263203 Consider the numbers m such that m = prime(k) + prime(k+2i+1) = prime(k+i) + prime(k+i+1) for some i and k. The sequence lists the number of pairs (i,k) giving the same value m = A105093(n).

Views

Author

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Comments

Examples

Crossrefs

Programs

Maple

A333122 Numbers m such that m = prime(k) + prime(k+5) = prime(k+1) + prime(k+4) for some k.

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Author

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Examples

Crossrefs

Programs

Mathematica

Python

A358028 Primes p = prime(9*t+1) such that the 9 consecutive primes prime(9*t+1) .. prime(9*t+9) arranged in a 3 X 3 array have at least 2 equal sums along the rows, columns or main diagonals.

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Author

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Comments

Examples

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Programs

Mathematica

Extensions

A358028 Primes p = prime(9t+1) such that the 9 consecutive primes prime(9t+1) .. prime(9*t+9) arranged in a 3 X 3 array have at least 2 equal sums along the rows, columns or main diagonals.