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 1-5 of 5 results.

A079890 Least number > n having one more prime factor than n, not necessarily distinct.

Original entry on oeis.org

2, 4, 4, 8, 6, 8, 9, 16, 12, 12, 14, 16, 14, 18, 18, 32, 21, 24, 21, 24, 27, 27, 25, 32, 27, 27, 36, 36, 33, 36, 33, 64, 42, 42, 42, 48, 38, 42, 42, 48, 46, 54, 46, 54, 54, 50, 49, 64, 50, 54, 52, 54, 55, 72, 63, 72, 63, 63, 62, 72, 62, 63, 81, 128, 66, 81, 69, 81, 70, 81, 74, 96
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 14 2003

Keywords

Comments

A001222(a(n)) = A001222(n) + 1;
a(2^k) = 2^(k+1).
a(A076156(n)) = A076156(n)+1. - Reinhard Zumkeller, Feb 01 2008

Links

Crossrefs

Cf. A079891, A079892.
Cf. A165712.

Programs

  • Haskell
    a079890 n = head [x | x <- [n + 1 ..], a001222 x == 1 + a001222 n]
-- Reinhard Zumkeller, Aug 29 2013
  • Mathematica
    lng[n_]:=Module[{x=n+1,pon=PrimeOmega[n]},While[PrimeOmega[x]-pon!=1, x++]; x]; Array[lng,80] (* Harvey P. Dale, Nov 09 2011 *)

A369139 Numbers k such that Omega(k) = 1 + Omega(k + 1).

Original entry on oeis.org

4, 6, 8, 10, 20, 22, 45, 46, 50, 58, 68, 76, 80, 82, 92, 104, 105, 106, 110, 114, 117, 152, 154, 165, 166, 178, 182, 186, 189, 212, 226, 236, 246, 258, 260, 261, 262, 266, 273, 286, 290, 315, 318, 322, 325, 333, 338, 342, 344, 345, 346, 354, 357, 358, 370, 382, 385, 402, 406, 410, 412, 424, 426
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Jan 14 2024

Keywords

Comments

Numbers k that have one more prime divisor (counted by multiplicity) than k + 1.

Examples

			a(3) = 8 is a term because 8 = 2^3 has 3 prime divisors (counted by multiplicity) and 8 + 1 = 9 = 3^2 has 2.

Links

Crossrefs

Cf. A001222, A045920, A076156. Contains A077065.

Programs

  • Maple
    N:= 1000: # for terms <= N
V:= map(numtheory:-bigomega, [$1..N+1]):
select(t -> V[t] = 1 + V[t+1], [$1..N]);
  • Mathematica
    s = {}; Do[If[PrimeOmega[k] == 1 + PrimeOmega[k + 1], AppendTo[s, k]], {k, 500}]; s

A322300 a(n) is the least k such that A001222(k)=n and A001222(k+1)=n+1.

Original entry on oeis.org

1, 3, 26, 99, 495, 728, 1215, 6560, 309824, 1896128, 1043199, 15752960, 178149375, 399112191, 4226550272, 7219625984, 45990608895, 558743781375, 1565795778560, 28996228218879, 63685431525375, 45922887663615, 1956754664980479, 30987856352641023
Offset: 0

Views

Author

Robert Israel and J. M. Bergot, Dec 02 2018

Keywords

Comments

Dickson's conjecture implies that a(n) exists for each n.

Examples

			a(3)= 99 because 99=3^2*11 has three prime factors (counted with multiplicity) and 99+1=2^2*5^2 has four, and 99 is the least number with those properties.

Crossrefs

Cf. A001222, A076156, A079482.

Programs

  • Maple
    b:= 0:
for n from 2 do
  a:= b;
  b:= numtheory:-bigomega(n);
  if b = a+1 and not assigned(A[a]) then
     A[a]:= n-1;
     if a = 9 then break fi
  fi
od:
seq(A[i],i=0..9);
  • Mathematica
    a[n_] := Module[{k = 1}, While[PrimeOmega[k] != n || PrimeOmega[k + 1] != n + 1, k++]; k]; Array[a, 10, 0] (* Amiram Eldar, Dec 03 2018 *)
  • PARI
    isok(n,k) = bigomega(k) == n && bigomega(k+1) == n+1;
a(n) = for(k=1, oo, if(isok(n,k), return(k))); \\ Daniel Suteu, May 05 2022
  • PARI
    generate(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, if(bigomega(m*q-1) == k, listput(list, m*q-1))), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 2, n)));
a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n+1, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Jul 09 2023

Extensions

a(9)-a(13) from Rémy Sigrist, Dec 03 2018
a(14)-a(18) from Giovanni Resta, Jun 11 2020
a(19)-a(21) from Daniel Suteu, May 05 2022
a(22)-a(23) from Daniel Suteu, Jul 09 2023

A335652 Numbers k such that Omega(k+1) = Omega(k) + 2, where Omega(k) = A001222(k) is the number of prime factors of k with multiplicity.

Original entry on oeis.org

7, 11, 15, 17, 19, 29, 35, 39, 41, 43, 55, 67, 87, 97, 101, 109, 113, 134, 137, 155, 163, 173, 175, 181, 183, 203, 207, 209, 211, 219, 229, 241, 242, 247, 249, 257, 259, 279, 281, 283, 295, 305, 314, 317, 327, 329, 331, 337, 339, 341, 351, 353, 371, 373, 401, 404, 409, 413, 433, 455
Offset: 1

Views

Author

Zak Seidov, Jun 16 2020

Keywords

Examples

			7 is prime, Omega(7) = 1, 7 + 1 = 8 = 2*2*2, Omega(8) = 3.

Links

Crossrefs

Omega(k+1) = Omega(k) + m: A045920 (m = 0), A076156 (m = 1).
Cf. A001222, A335655. Contains A063639.

Programs

A335655 Numbers k such that Omega(k+1) = Omega(k) + m, where Omega(k) = A001222(k) is the number of prime factors of k with multiplicity, case m = 3.

Original entry on oeis.org

23, 53, 59, 63, 83, 89, 103, 111, 119, 131, 139, 149, 151, 161, 197, 227, 233, 293, 299, 303, 307, 347, 349, 377, 379, 389, 391, 395, 399, 407, 443, 461, 487, 491, 509, 519, 521, 539, 551, 557, 563, 566, 569, 571, 591
Offset: 1

Views

Author

Zak Seidov, Jun 16 2020

Keywords

Examples

			23 is in the sequence since Omega(24) = 4 = 1 + 3 = Omega(23) + 3.

Crossrefs

Cf. A001222, A076156 (case m=1), A335652 (case m=2).

Programs

  • Mathematica
    m = 3; s = {}; Do[If[PrimeOmega[x + 1] == PrimeOmega[x] + m, AppendTo[s, x]], {x, 600}]; s
Showing 1-5 of 5 results.

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