cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A373888 a(n) is the length of the longest arithmetic progression of primes ending with prime(n).

Original entry on oeis.org

1, 2, 2, 3, 3, 2, 3, 3, 4, 5, 3, 2, 4, 4, 3, 5, 4, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 4, 3, 4, 5, 5, 3, 4, 4, 4, 6, 4, 4, 5, 3, 4, 4, 4, 5, 4, 3, 4, 5, 4, 4, 4, 4, 5, 6, 4, 4, 5, 3, 4, 5, 5, 4, 6, 4, 4, 4, 3, 4, 4, 6, 4, 4, 5, 3, 4, 5, 5, 4, 4, 4, 5, 4, 4, 4, 5, 5, 4, 4, 6, 4, 5, 4, 4, 3, 4, 6, 5, 4
Offset: 1

Views

Author

Robert Israel, Aug 11 2024

Keywords

Comments

a(n) is the greatest k such that there exists d > 0 such that A000040(n) - j*d is prime for j = 0 .. k-1.
The first appearance of m in this sequence is at A000720(A005115(m)).
Conjectures: a(n) >= 3 for n >= 13.
Limit_{n -> oo} a(n) = oo.

Examples

			a(4) = 3 because the 4th prime is 7 and there is an arithmetic progression of 3 primes ending in 7, namely 3, 5, 7, and no such arithmetic progression of 4 primes.

Links

Crossrefs

Cf. A000040, A000720, A005115, A373887.

Programs

  • Maple
    f:= proc(n) local s,i,m,d,j;
  m:= 1;
  s:= ithprime(n);
  for i from n-1 to 1 by -1 do
    d:= s - ithprime(i);
    if s - m*d < 2 then return m fi;
    for j from 2 while isprime(s-j*d) do od;
    m:= max(m, j);
  od;
  m
end proc:
map(f, [$1..100]);

A375511 a(n) is the common difference in the longest arithmetic progression of semiprimes ending in the n-th semiprime. If there is more than one such arithmetic progression, the smallest difference is chosen.

Original entry on oeis.org

2, 3, 1, 4, 1, 6, 8, 3, 11, 12, 12, 1, 12, 1, 12, 14, 13, 17, 11, 12, 24, 7, 18, 35, 19, 24, 8, 24, 18, 29, 8, 36, 1, 24, 17, 30, 12, 4, 3, 48, 4, 36, 11, 48, 23, 24, 1, 30, 12, 13, 12, 36, 42, 24, 14, 16, 36, 14, 8, 32, 36, 7, 60, 42, 60, 60, 3, 4, 36, 46, 4, 12, 32, 4, 60, 16, 18, 44, 36, 16
Offset: 2

Views

Author

Robert Israel, Aug 18 2024

Keywords

Comments

a(n) is the smallest common difference in an arithmetic progression of A373887(n) semiprimes ending in A001358(n).

Examples

			The 5th semiprime is 14, A373887(5) = 3, and there are two arithmetic progressions of semiprimes of length 3 ending in 14, namely 6, 10, 14 with common difference 4 and 4, 9, 14 with common difference 5.  Therefore a(5) = min(4, 5) = 4.

Links

Crossrefs

Cf. A001358, A373887, A375386.

Programs

  • Maple
    S:= select(t -> numtheory:-bigomega(t)=2, [$1..10^5]):
f:= proc(n) local s, i, m, d, j, dm;
  m:= 1;
  s:= S[n];
  for i from n-1 to 1 by -1 do
    d:= s - S[i];
    if s - m*d < 4 then return dm fi;
    for j from 2 while ListTools:-BinarySearch(S, s-j*d) <> 0 do od;
    if j > m then m:= j; dm:= d fi;
  od;
dm;
end proc:
map(f, [$2..200]);

A376109 a(n) is the length of the longest arithmetic progression ending at n consisting of numbers with the same number of prime factors as n, counted with multiplicity.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 2, 3, 2, 1, 3, 2, 3, 2, 3, 3, 4, 2, 2, 3, 2, 3, 5, 2, 3, 1, 3, 3, 3, 2, 2, 3, 2, 2, 4, 3, 4, 3, 2, 4, 3, 2, 3, 2, 3, 3, 5, 2, 4, 3, 3, 5, 4, 2, 3, 3, 3, 1, 3, 3, 3, 3, 4, 3, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 2, 4, 4, 3, 3, 4, 5, 3, 3, 2, 4, 4, 4, 3, 3, 2, 4, 3, 3
Offset: 1

Views

Author

Robert Israel, Sep 10 2024

Keywords

Comments

a(n) is the greatest k such that there exists d >= 1 with A001222(n-i*d) = A001222(n) for 0 <= i < k.
If m divides n, then a(n) >= a(m).
a(n) = 1 if and only if n is a power of 2.

Examples

			a(7) = 3 because 7 is prime and there is an arithmetic progression of 3 primes, namely 3, 5, 7, ending with 7 but no such arithmetic progression of 4 primes.

Links

Crossrefs

Cf. A001222, A005115, A096003, A373887, A373888.

Programs

  • Maple
    M:= Array(1..10):
for n from 2 to 100 do
  v:= numtheory:-bigomega(n);
  if M[v] = 0 then M[v]:= n else M[v]:= M[v],n fi;
od:
for i from 1 to 10 do M[i]:= [M[i]] od:
f:= proc(s) local n,i,m,d,v,j;
   m:= 1;
   v:= numtheory:-bigomega(s);
   member(s,M[v],n);
   for i from n-1 to 1 by -1 do
     d:= s - M[v][i];
     if s - m*d < M[v][1] then return m fi;
     for j from 2 while ListTools:-BinarySearch(M[v],s-j*d) <> 0 do od:
     m:= max(m,j);
   od;
  m;
end proc:
f(1):= 1:
map(f, [$1..100]);
Showing 1-3 of 3 results.

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