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 1-4 of 4 results.

A361215 Intersection of A361073 and 2 * A361611.

Original entry on oeis.org

8, 20, 50, 1406, 1516, 1558, 1868, 1898, 1948, 1978, 1986, 5862, 5972, 6014, 7122, 7966, 7996, 8270, 8348, 8366, 8548, 8618, 21092, 31804, 31822, 32158, 33092, 33162, 33316, 33414, 37124, 37190, 37292, 37394, 39164, 39214, 39316, 39346, 39484, 39562, 39604, 39622, 39692, 39794, 45044, 45244
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Apr 09 2023

Keywords

Comments

If A361073(j) = 2*A361611(k) then x = 2*A361611(k+1) has the property that x, x - A361073(j) and x + A361073(j) are triprimes, so x >= A361073(j+1), with equality if and only if A361073(j+1) is even.

Examples

			a(4) = 1406 is a term because 1406 = A361073(20) = 2*A361611(17).

Links

Crossrefs

Cf. A361073, A361611.

Programs

  • Maple
    A:= {8}: lasta:= 8:
for i from 2 to 1000 do
  for x from lasta+8 do
    if numtheory:-bigomega(x) = 3 and numtheory:-bigomega(x-lasta) = 3 and numtheory:-bigomega(x+lasta) = 3 then
       A:= A union {x}; lasta:= x; break
    fi
od od:
R:= {8}: lastb:= 4:
while 2*lastb < lasta do
for x from lastb+4 do
  if numtheory:-bigomega(x) = 2 and numtheory:-bigomega(x-lastb) = 2 and numtheory:-bigomega(x+lastb) = 2 then
     if member(2*x,A) then R:= R union {2*x} fi;
     lastb:= x; break
  fi
od od:
sort(convert(R,list));

A361073 Lexicographically least increasing sequence of triprimes (A014612) a(n) such that a(n) - a(n-1) and a(n) + a(n-1) are also triprimes.

Original entry on oeis.org

8, 20, 50, 125, 279, 426, 531, 539, 814, 822, 897, 1002, 1010, 1076, 1146, 1209, 1325, 1353, 1398, 1406, 1516, 1558, 1868, 1898, 1948, 1978, 1986, 2013, 2225, 2233, 2397, 2527, 2547, 2575, 2763, 2783, 2810, 2908, 2938, 2946, 3009, 3054, 3081, 3414, 3422, 3452, 3522, 3567, 3714, 3759, 3786, 3813
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Apr 09 2023

Keywords

Examples

			a(3) = 50 because 50 = 2^2*5, 50 - a(2) = 30 = 2*3*5 and 50 + a(2) = 70 = 2*5*7 are all products of 3 (not necessarily distinct) primes, and 50 is the least number that works.

Links

Crossrefs

Cf. A014612, A361611, A361215.

Programs

  • Maple
    A[1]:= 8:
for i from 2 to 100 do
  for x from A[i-1]+8 do
    if numtheory:-bigomega(x) = 3 and numtheory:-bigomega(x-A[i-1]) = 3 and numtheory:-bigomega(x+A[i-1]) = 3 then
       A[i]:= x; break
    fi
od od:
seq(A[i],i=1..100);
  • Mathematica
    s = {m = 8}; Do[p = m + 8; While[{3, 3, 3} != PrimeOmega[{p, m + p,
p - m}], p++];  AppendTo[s, m = p], {50}]; s

A363513 a(1) = 2, then a(n) is the least prime p > a(n - 1) such that p + a(n-1) and p - a(n-1) have the same number of prime factors counted with multiplicity.

Original entry on oeis.org

2, 5, 13, 31, 61, 103, 157, 173, 181, 193, 211, 223, 239, 269, 313, 337, 353, 419, 487, 499, 577, 613, 631, 647, 677, 709, 727, 827, 857, 947, 1039, 1093, 1117, 1231, 1283, 1303, 1319, 1483, 1499, 1553, 1609, 1627, 1657, 1693, 1721, 1733, 1823, 1913, 1933, 1951, 2003, 2027, 2039, 2129, 2161, 2203
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Jun 07 2023

Keywords

Examples

			a(2) = 5 because A001222(5-2) = A001222(5+2) = 1.
a(3) = 13 because A001222(13-5) = A001222(13+5) = 3.

Links

Crossrefs

Cf. A001222, A361611.

Programs

  • Maple
    R:= 2: r:= 2:
for i from 1 to 100 do
    p:= nextprime(r);
    while numtheory:-bigomega(r+p) <> numtheory:-bigomega(p-r) do
      p:= nextprime(p)
    od;
    R:= R,p; r:= p;
od:
R;
  • Mathematica
    s = {p = 2}; Do[q = NextPrime[p]; While[PrimeOmega[p + q]
!= PrimeOmega[q - p], q = NextPrime[q]]; AppendTo[s, p = q], {200}]; s

Formula

A001222(a(n) - a(n-1)) = A001222(a(n) + a(n-1)).

A368648 Lexicographically earliest increasing sequence of semiprimes such that a(n) + a(n+1) is a semiprime, with a(0) = 4.

Original entry on oeis.org

4, 6, 9, 25, 26, 39, 46, 49, 57, 58, 65, 69, 74, 85, 93, 94, 111, 115, 122, 143, 146, 155, 159, 187, 194, 201, 202, 205, 206, 209, 213, 214, 237, 265, 278, 287, 299, 323, 326, 329, 358, 365, 381, 382, 403, 415, 427, 451, 454, 469, 482, 497, 501, 502, 505, 537, 542, 559, 562, 573, 581, 586, 591, 611
Offset: 0

Views

Author

Zak Seidov and Robert Israel, Jan 02 2024

Keywords

Comments

The only case where two successive terms are even is 4, 6 at the start, since if 2*p and 2*q are semiprimes where p and q are odd primes, 2*p + 2*q is divisible by 4.

Examples

			a(2) = 6 because 6 = 2 * 3 and 4 + 6 = 10 = 2 * 5 are semiprimes.
a(3) = 9 because 9 = 3 * 3 and 6 + 9 = 15 = 3 * 5 are semiprimes.
a(4) = 25 because 25 = 5 * 5 and 9 + 25 = 34 = 2 * 17 are semiprimes.

Links

Crossrefs

Cf. A001358, A361611.

Programs

  • Maple
    R:= 4: x:= 4: count:= 1:
for y from 5 do
    if numtheory:-bigomega(y) = 2 and numtheory:-bigomega(x+y) = 2 then
      R:= R,y; x:= y; count:= count+1;
      if count = 100 then break fi
    fi
od:
R;
Showing 1-4 of 4 results.

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