A213930 -id:A213930 - cp's OEIS frontend

A277730 Irregular triangle read by rows: T(n,k) = number of times a gap of 2k occurs between the first n successive odd primes.

1, 2, 2, 1, 3, 1, 3, 2, 4, 2, 4, 3, 4, 3, 1, 5, 3, 1, 5, 3, 2, 5, 4, 2, 6, 4, 2, 6, 5, 2, 6, 5, 3, 6, 5, 4, 7, 5, 4, 7, 5, 5, 7, 6, 5, 8, 6, 5, 8, 6, 6, 8, 7, 6, 8, 7, 7, 8, 7, 7, 1, 8, 8, 7, 1, 9, 8, 7, 1, 9, 9, 7, 1, 10, 9, 7, 1, 10, 10, 7, 1, 10, 10, 7, 1, 0, 0, 1, 10, 11, 7, 1, 0, 0, 1

Offset: 2

N. J. A. Sloane, Nov 06 2016

T(m, 2) = max{T(m, k): k >= 1} for m >= 100. - Ya-Ping Lu, Dec 25 2024

			Triangle begins:
  1,
  2,
  2, 1,
  3, 1, <- gaps in 3,5,7,11,13 are 2 (3 times), 4 (once)
  3, 2,
  4, 2,
  4, 3,
  4, 3, 1,
  5, 3, 1,
  5, 3, 2,
  5, 4, 2,
  6, 4, 2,
  6, 5, 2,
  6, 5, 3,
  6, 5, 4,
  7, 5, 4,
  7, 5, 5,
  7, 6, 5,
  8, 6, 5,
  ...

Lars Blomberg, Table of n, a(n) for n = 2..9997 (the first 703 rows)

Cf. A277729, A213930.

from sympy import nextprime; p = 3; L = []
for n in range(2, 32):
    np = nextprime(p); k = (np - p)//2
    if len(L) < k: {L.append(0) for i in range(len(L), k-1)}; L.append(1)
    else: L[k-1] += 1
print(*L, sep =", ", end = ", "); p = np  # Ya-Ping Lu, Dec 25 2024

A277729 Irregular triangle read by rows: T(n,k) = number of times a gap of k occurs between the first n successive primes.

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane, Nov 06 2016

			Triangle begins:
1,
1, 1,
1, 2,
1, 2, 0, 1,
1, 3, 0, 1, <- gaps in 2,3,5,7,11,13 are 1, 2 (3 times), 4 (once)
1, 3, 0, 2,
1, 4, 0, 2,
1, 4, 0, 3,
1, 4, 0, 3, 0, 1,
1, 5, 0, 3, 0, 1,
1, 5, 0, 3, 0, 2,
1, 5, 0, 4, 0, 2,
1, 6, 0, 4, 0, 2,
1, 6, 0, 5, 0, 2,
1, 6, 0, 5, 0, 3,
1, 6, 0, 5, 0, 4,
1, 7, 0, 5, 0, 4,
1, 7, 0, 5, 0, 5,
1, 7, 0, 6, 0, 5,
1, 8, 0, 6, 0, 5,
...

Robert Israel, Table of n, a(n) for n = 2..10032 (rows 2 to 410, flattened)

Cf. A277730, A213930.

N:= 30: # for rows 2 to N
res:= 1; p:= 3; R:= <1>;
for n from 2 to N do
  pp:= nextprime(p);
  d:= pp - p;
  p:= pp;
  if d <= LinearAlgebra:-Dimension(R) then
    R[d]:= R[d]+1
  else
    R(d):= 1
  fi;
  res:= res, op(convert(R,list));
od:
res;  # Robert Israel, Nov 16 2016

A213949 One half of largest prime gap up to 10^n.

Original entry on oeis.org

1, 4, 10, 18, 36, 57, 77, 110, 141, 177, 232, 270, 337, 402, 453, 566, 610, 721

Offset: 1

Washington Bomfim, Jun 26 2012

Row lengths of A213930.

Index entries for primes, gaps between

Cf. A213930, A038460.

a(n) = A038460(n)/2.

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A277730 Irregular triangle read by rows: T(n,k) = number of times a gap of 2k occurs between the first n successive odd primes.

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A277729 Irregular triangle read by rows: T(n,k) = number of times a gap of k occurs between the first n successive primes.

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A213949 One half of largest prime gap up to 10^n.

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