A143121 -id:A143121 - cp's OEIS frontend

A194939 Table T read by rows, where T(n, k) is the sum of the largest k primes up to and including prime(n), for 1 <= k <= n.

2, 3, 5, 5, 8, 10, 7, 12, 15, 17, 11, 18, 23, 26, 28, 13, 24, 31, 36, 39, 41, 17, 30, 41, 48, 53, 56, 58, 19, 36, 49, 60, 67, 72, 75, 77, 23, 42, 59, 72, 83, 90, 95, 98, 100, 29, 52, 71, 88, 101, 112, 119, 124, 127, 129, 31, 60, 83, 102, 119, 132, 143, 150, 155, 158, 160

Offset: 1

Alonso del Arte, Sep 07 2011

From the left, the second column gives the sums of two consecutive primes, the third column gives the sums of three consecutive primes, etc. Thus, from the right, the rightmost column gives the running sum of all prime numbers up to that row.
This triangle is the mirror image of A143121: left border are the primes (right border in the other one) while the right border is the sum of the first n primes (A007504, left border in the other one). Row sums are given by A014285, just like the other triangle.
On odd numbered rows, the central entry is exactly the same as the corresponding position in A143121: T(n, (n + 1)/2) = A143121(n, (n + 1)/2). The rest of the row is of course the reverse.

			First few rows of triangle are:
2
3,   5
5,   8, 10
7,  12, 15, 17
11, 18, 23, 26, 28
...
T(5, 2) = 18 because the sum of the fourth and fifth primes (two consecutive primes) is 7 + 11 = 18.
T(5, 3) = 23 because the sum of the third, fourth and fifth primes (three consecutive primes) is 5 + 7 + 11 = 23.

Michel Marcus, Table of n, a(n) for n = 1..5050 (first 100 rows)

Cf. A143121 (rows reversed), A014285 (row sums).
Cf. A000040 (column k=1), A007504 (main diagonal).
Cf. A067377.

a[n_, k_] := a[n, k] = Plus@@Prime[Range[n - k + 1, n]]; Column[Table[a[n, k], {n, 15}, {k, n}], Center]

T(n, k) = Sum_{i = n-k+1..n} prime(i), where prime(i) is the i-th prime number.

More terms from Michel Marcus, Aug 31 2020

New name from David A. Corneth, Aug 31 2020

A339269 a(n) is the least number that is the product of n primes (not necessarily distinct) and is the sum of n consecutive primes, or 0 if there are none.

Original entry on oeis.org

2, 0, 425, 36, 243, 756, 29889, 4704, 207765, 8448, 4465125, 108864, 13640319, 1022976, 146146275, 3010560, 6500054871, 4259840, 60767621145, 36864000, 454444233597, 167215104, 8664659236485, 796262400, 49362406764957, 7935623168, 2310430513712625, 4160749568, 4216955409197811, 28538044416

Offset: 1

J. M. Bergot and Robert Israel, Nov 29 2020

nonn

Conjecture: a(n) > 0 for every n > 2.

If n > 2 is odd, the sum of n consecutive odd primes is odd, so (if nonzero) a(n) >= 3^n.

			a(3)=425 because 425 = 5^2*7 is the product of three primes and 425 = 137+139+149 is the sum of three consecutive primes, and no smaller number has this property.

Robert Israel, Table of n, a(n) for n = 1..50

Cf. A001222, A143121, A339185.

sumofconsecprimes:= proc(x, n)
  local P, k, p, q, t;
  P:= nextprime(floor(x/n));
  p:= P; q:= P;
  for k from 1 to n-1 do
    if k::even or q = 2 then p:= nextprime(p); P:= P, p;
    else q:= prevprime(q); P:= q, P;
    fi
  od;
  P:= [P];
  t:= convert(P, `+`);
  if t = x then return P fi;
  if t > x then
    while t > x do
      if q = 2 then return false fi;
      q:= prevprime(q);
      t:= t + q - p;
      P:= [q, op(P[1..-2])];
      p:= P[-1];
      if t = x then return P fi;
    od
  else
    while t < x do
      p:= nextprime(p);
      t:= t + p - q;
      P:= [op(P[2..-1]), p];
      q:= P[1];
      if t = x then return P fi;
    od
  fi;
  false
end proc:
children:= proc(r) local L, x, p, q, t, R;
  x:= r[1];
  L:= r[2];
  t:= L[-1];
  p:= t[1]; q:= nextprime(p);
  if t[2]=1 then t:= [q, 1];
  else t:= [p, t[2]-1], [q, 1]
  fi;
  R:= [x*q/p, [op(L[1..-2]), t]];
  if nops(L) >= 2 then
    p:= L[-2][1];
    q:= L[-1][1];
    if L[-2][2]=1 then t:= [q, L[-1][2]+1]
    else t:= [p, L[-2][2]-1], [q, L[-1][2]+1]
    fi;
    R:= R, [x*q/p, [op(L[1..-3]), t]]
  fi;
  [R]
end proc:
f:= proc(n) local Q, t, x, v;
      uses priqueue;
      initialize(Q);
      if n::even then insert([-2^n, [[2, n]]], Q)
      else insert([-3^n, [[3, n]]], Q)
      fi;
      do
        t:= extract(Q);
        x:= -t[1];
        v:= sumofconsecprimes(x, n);
        if v <> false then return x fi;
        for t in children(t) do insert(t, Q) od;
      od
   end proc:
f(1):= 2:
f(2):= 0:
map(f, [$1..34]);

a(n) = A143121(A339185(n)+n, A339185(n)).

A339185 a(n) is the least prime p such that the sum of n consecutive primes starting with p has exactly n prime factors, counted with multiplicity, or 0 if no such p exists.

Original entry on oeis.org

2, 0, 137, 5, 41, 109, 4253, 569, 23057, 821, 405863, 9013, 1049173, 73009, 9742969, 188017, 382355863, 236527, 3198295691, 1843111, 21640201361, 7600499, 376724314301, 33177461, 1974496270177, 305216017, 85571500507397, 148597987, 145412255489161, 951267841, 2609815945304401, 1140850357, 24575914221842531

Offset: 1

J. M. Bergot and Robert Israel, Nov 26 2020

nonn

Conjecture: Such p exists for every n > 2.

			a(3)=137 because the sum of 3 consecutive primes starting with 137 is 137+139+149=425=5^2*7 is the product of 3 primes counting multiplicity, and 137 is the least prime with this property.

Cf. A001222, A143121, A339269.

sumofconsecprimes:= proc(x,n)
  local P,k,p,q,t;
  P:= nextprime(floor(x/n));
  p:= P; q:= P;
  for k from 1 to n-1 do
    if k::even or q = 2 then p:= nextprime(p); P:= P,p;
    else q:= prevprime(q); P:= q,P;
    fi
  od;
  P:= [P];
  t:= convert(P,`+`);
  if t = x then return P fi;
  if t > x then
    while t > x do
      if q = 2 then return false fi;
      q:= prevprime(q);
      t:= t + q - p;
      P:= [q, op(P[1..-2])];
      p:= P[-1];
      if t = x then return P fi;
    od
  else
    while t < x do
      p:= nextprime(p);
      t:= t + p - q;
      P:= [op(P[2..-1]),p];
      q:= P[1];
      if t = x then return P fi;
    od
  fi;
  false
end proc:
children:= proc(r) local L,x,p,q,t,R;
  x:= r[1];
  L:= r[2];
  t:= L[-1];
  p:= t[1]; q:= nextprime(p);
  if t[2]=1 then t:= [q,1];
  else t:= [p,t[2]-1],[q,1]
  fi;
  R:= [x*q/p,[op(L[1..-2]),t]];
  if nops(L) >= 2 then
    p:= L[-2][1];
    q:= L[-1][1];
    if L[-2][2]=1 then t:= [q,L[-1][2]+1]
    else t:= [p,L[-2][2]-1],[q,L[-1][2]+1]
    fi;
    R:= R, [x*q/p, [op(L[1..-3]),t]]
  fi;
  [R]
end proc:
f:= proc(n) local Q,t,x,v;
      uses priqueue;
      initialize(Q);
      if n::even then insert([-2^n,[[2,n]]],Q)
      else insert([-3^n,[[3,n]]],Q)
      fi;
      do
        t:= extract(Q);
        x:= -t[1];
        v:= sumofconsecprimes(x,n);
        if v <> false then return v[1] fi;
        for t in children(t) do insert(t,Q) od;
      od
   end proc:
f(1):= 2:
f(2):= 0:
map(f, [$1..34]);

A339269(n) = A143121(a(n)+n, a(n)).

cp's OEIS Frontend

A194939 Table T read by rows, where T(n, k) is the sum of the largest k primes up to and including prime(n), for 1 <= k <= n.

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A339269 a(n) is the least number that is the product of n primes (not necessarily distinct) and is the sum of n consecutive primes, or 0 if there are none.

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A339185 a(n) is the least prime p such that the sum of n consecutive primes starting with p has exactly n prime factors, counted with multiplicity, or 0 if no such p exists.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula