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.

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A362200 Semiprimes k such that k+1, k+2, 2*k+1 and 2*k+3 are also semiprimes.

Original entry on oeis.org

11733, 15117, 17245, 28113, 32365, 34413, 48745, 78481, 93453, 101665, 102957, 105333, 108753, 134097, 143101, 157713, 163801, 170853, 190621, 208293, 212545, 233097, 273417, 274893, 294301, 300385, 323281, 346497, 354565, 363777, 390205, 405357, 470341, 500217, 501477, 542193, 555153, 561205
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Apr 10 2023

Keywords

Comments

Numbers k such that 2*k+1 and 2*k+3 are both in A092192.
All terms == 1 or 33 (mod 36).

Examples

			a(3) = 17245 is a term because 17245 = 5 * 3449, 17246 = 2 * 8623, 17247 = 3 * 5749, 2 * 17245 + 1 = 34491 = 3 * 11497 and 2 * 17245 + 3 = 34493 = 17 * 2029 are all semiprimes.

Links

Crossrefs

Cf. A001358, A092192.

Programs

  • Maple
    SP:= select(t -> numtheory:-bigomega(t)=2, {$1..2*10^6}):
A:= SP intersect map(`-`,SP,1) intersect map(`-`,SP,2):
SPO:= select(type,SP,odd):
A:= A intersect map(t -> (t-1)/2, SPO) intersect map(t -> (t-3)/2, SPO):
sort(convert(A,list));

A365448 Array read by antidiagonals: row 1 is the semiprimes A001358; for m > 1, row m is the semiprimes that are the sum of two successive terms of row m-1.

Original entry on oeis.org

4, 6, 10, 9, 15, 25, 10, 51, 146, 422, 14, 69, 201, 551, 973, 15, 77, 221, 667, 1858, 2831, 21, 85, 249, 1191, 89855, 312493, 127418369, 22, 95, 302, 1343, 110099, 2676567, 154171217
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Oct 03 2023

Keywords

Examples

			The first 7 rows are
4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ...
10, 15, 51, 69, 77, 85, 95, 106, 115, 134, ...
25, 146, 201, 221, 249, 302, 365, 529, 662, 681, ...
422, 551, 667, 1191, 1343, 2661, 6621, 11207, 13637, 14183, ...
973, 1858, 89855, 110099, 202394, 332377, 352147, 383507, 469231, 528923, ...
2831, 312493, 2676567, 3754285, 4027807, 9438362, 10568289, 20372991, 20590454, 21591014, ...
127418369, 154171217, 213938227, 242408953, 296917233, 325907227, 345235903, 367725381, ...
T(4,3) = 667 is a term because 667 = 23 * 29 is a semiprime and 667 = 392 + 365 where 302 = T(3,6) and 365 = T(3,7).

Crossrefs

Cf. A001358 (first row), A092192 (second row), A366167 (third row).

Programs

  • Maple
    R[1]:= select(t -> numtheory:-bigomega(t) = 2, [$1..5*10^6]): M[1]:= nops(R[1]):
for i from 2 do
  R[i]:= select(t -> numtheory:-bigomega(t) = 2, R[i-1][1..M[i-1]-1] + R[i-1][2..M[i-1]]);
  M[i]:= nops(R[i]);
  if M[i] = 0 then break fi
od:
L:= NULL:
for k from 2 to 8 do
  L:= L, seq(R[i][k-i],i=1..k-1)
od:
L;

A370687 a(n) is the first number that is the sum of k successive semiprimes for 1 <= k <= n.

Original entry on oeis.org

4, 10, 134, 2045, 2705, 16626281
Offset: 1

Views

Author

Robert Israel, Feb 27 2024

Keywords

Comments

a(7) > 5 * 10^8 if it exists.

Examples

			a(3) = 134 because 134 = 2 * 67 is a semiprime, the sum of two successive semiprimes 65 = 5 * 13 and 69 = 3 * 23, and the sum of three successive semiprimes 39 = 3 * 13, 46 = 2 * 23, 49 = 7 * 7, and is the least such number.

Crossrefs

Cf. A001358, A092192, A370162, A370685.

Programs

  • Maple
    N:= 2*10^7: # for terms <= N
P:= select(isprime, [2, seq(i, i=3..N/2, 2)]):
nP:= nops(P):
SP:= 0:
for i from 1 while P[i]^2 <= N do
  m:= ListTools:-BinaryPlace(P, N/P[i]);
  SP:= SP, op(P[i]*P[i..m]);
od:
SP:= sort([SP]):
SS:= ListTools:-PartialSums(SP):
V:= Vector(6):
SI:= Vector(6):
II:= Vector(6,1):
for i from 1 to 6 do SI[i]:= SS[i+1]-SS[1] od:
count:= 1: V[1]:= 4: m:= 4: im:= {1}:
while count < 6 do
  for j in im do
    II[j]:= II[j]+1;
    SI[j]:= SS[II[j]+j] - SS[II[j]];
  od;
  m:= min(SI);
  im:= select(j -> SI[j] = m, {$1..20});
  for k from 1 to 20 while {$1..k} subset im do
    if V[k] = 0 then V[k]:= m; count:= count+1 fi
  od;
od:
convert(V,list);

A380348 Tetraprimes (or products of exactly four distinct prime numbers) that are the sum of two successive tetraprimes.

Original entry on oeis.org

4785, 11739, 13035, 14685, 17535, 17690, 24115, 24871, 26061, 28203, 33605, 34419, 35061, 37515, 37765, 37851, 38335, 40803, 41205, 48202, 48685, 48895, 49215, 52535, 52955, 55605, 58179, 58245, 59015, 59345, 59595, 62643, 62895, 64785, 66815, 70091, 71205, 71355, 72215
Offset: 1

Views

Author

Massimo Kofler, Jan 22 2025

Keywords

Examples

			4785 = 3*5*11*29 is a member because 4785 = 2370+2415, sum of two successive tetraprime numbers.
11739 = 3*7*13*43 is a member because 11739 = 5865+5874, sum of two successive tetraprime numbers.

Crossrefs

Cf. A001043, A046386, A092192, A380316.

Programs

  • Mathematica
    tetQ[n_] := FactorInteger[n][[;; , 2]] == {1, 1, 1, 1}; Select[MovingMap[Total, Select[Range[40000], tetQ], 1], tetQ] (* Amiram Eldar, Jan 22 2025 *)
  • PARI
    ist4(n) = omega(n)==4 && bigomega(n)==4; \\ A046386
lista(nn) = my(v=select(ist4, [1..nn])); select(ist4, vector(#v-1, k, v[k]+v[k+1])); \\ Michel Marcus, Jan 22 2025
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