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.

A295102 Squarefree numbers n that are sqrt(n)-smooth: if prime p|n then p^2 <= n.

Original entry on oeis.org

1, 30, 70, 105, 154, 165, 182, 195, 210, 231, 273, 286, 330, 357, 374, 385, 390, 399, 418, 429, 442, 455, 462, 494, 510, 546, 561, 570, 595, 598, 627, 646, 663, 665, 690, 714, 715, 741, 759, 770, 782, 798, 805, 858, 870, 874, 897, 910, 935, 957, 966, 969
Offset: 1

Views

Author

Max Alekseyev, Nov 14 2017

Keywords

Comments

If n is in the sequence and m < n is squarefree and coprime to n, then m*n is in the sequence. - Robert Israel, Aug 12 2019

Links

Crossrefs

Squarefree terms of A048098, and also of A063539.
Besides initial 1, subsequence of A164596.
Together with A001248, forms indices of records in A295101.
Cf. A005117.

Programs

  • Maple
    filter:= proc(n) local t;
  andmap(t -> t[2]=1 and t[1]^2<=n, ifactors(n)[2])
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 12 2019
  • Mathematica
    selQ[n_] := SquareFreeQ[n] && AllTrue[FactorInteger[n][[All, 1]], #^2<=n&];
Select[Range[1000], selQ] (* Jean-François Alcover, Jul 30 2020 *)
  • PARI
    isok(n) = (n==1) || (issquarefree(n) && (vecmax(factor(n)[,1])^2 < n)); \\ Michel Marcus, Aug 12 2019

A381736 Integers k = p*q*r, where p < q < r are distinct primes and p*q > r.

Original entry on oeis.org

30, 70, 105, 154, 165, 182, 195, 231, 273, 286, 357, 374, 385, 399, 418, 429, 442, 455, 494, 561, 595, 598, 627, 646, 663, 665, 715, 741, 759, 782, 805, 874, 897, 935, 957, 969, 986, 1001, 1015, 1023, 1045, 1054, 1085, 1102, 1105, 1131, 1173, 1178, 1209
Offset: 1

Views

Author

Matthew Goers, Mar 05 2025

Keywords

Comments

These are squarefree 3-almost-primes, called sphenic numbers, that are greater than the square of the largest of its prime factors. As all sphenic numbers are, by definition, less than the cube of their largest prime factor, numbers in this sequence satisfy r^2 < k < r^3, where k = p*q*r, p < q < r.

Examples

			30 = 2*3*5 and 2*3 > 5, so 30 is in the sequence.
70 = 2*5*7 and 2*5 > 7, so 70 is in the sequence.
110 = 2*5*11 but 2*5 < 11, so 110 is not in the sequence.

Links

Crossrefs

Intersection of A007304 (sphenic numbers) and A164596.
Cf. A382022.

Programs

  • Maple
    N:= 2000: # for terms < N
P:= select(isprime, [2,seq(i,i=3..isqrt(N),2)]):
R:= NULL:
for k from 1 to nops(P) do
  for i from 1 to k-2 while P[i]*P[i+1]*P[k] < N do
     jmin:= max(i+1,ListTools:-BinaryPlace(P,P[k]/P[i])+1);
     jmax:= min(k-1,ListTools:-BinaryPlace(P,N/(P[i]*P[k])));
     R:= R, seq(P[i]*P[j]*P[k],j=jmin .. jmax);
od od:
sort([R]); # Robert Israel, Mar 28 2025
  • Mathematica
    q[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1, 1} && f[[1, 1]]*f[[2, 1]] > f[[3, 1]]]; Select[Range[1500], q] (* Amiram Eldar, Mar 20 2025 *)
  • PARI
    is_a381736(n) = my(F=factor(n)); omega(F)==3 && bigomega(F)==3 && F[1,1]*F[2,1]>F[3,1] \\ Hugo Pfoertner, Mar 08 2025
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A381736(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(min(x//(p*q),p*q-1))-b for a,p in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,q in enumerate(primerange(p+1,isqrt(x//p)+1),a+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 28 2025

A291046 Minimal multiplicative semigroup of numbers n > 1 such that in the prime factorization of n an initial product of primes is greater than a later prime in the factorization.

Original entry on oeis.org

30, 60, 70, 90, 105, 120, 140, 150, 154, 165, 180, 182, 195, 210, 231, 240, 270, 273, 280, 286, 300, 308, 315, 330, 350, 357, 360, 364, 374, 385, 390, 399, 418, 420, 429, 442, 450, 455, 462, 480, 490, 494, 495, 510, 525, 540, 546, 560, 561, 570, 572, 585, 595, 598, 600, 616, 627
Offset: 1

Views

Author

Richard Locke Peterson, Aug 16 2017

Keywords

Comments

Definition: Let a number n>1 have prime factorization n=p1^e1*...*pi^ei*..*pm^em, with the primes written in ascending order and the ei>0. If an initial product p1*..*pi is greater than some later prime p(i+1), then n is in the sequence. The definition contains a more restrictive requirement than A289484 does, so it is a proper subsemigroup of A289484. It can be seen that if s and t are in the sequence, the so is s*t. More strongly, if n is in the sequence, so is every multiple of n. Any number in it is divisible by at least 3 primes, although that is not a sufficient condition.
Differs from A212666 first at a(93), because 930=2*3*5*31 is in this sequence but not in A212666. - R. J. Mathar, Sep 02 2018
Numbers whose squarefree kernel (A007947) is in A164596. - Peter Munn, Feb 05 2024

Links

Crossrefs

Cf. A007947, A164596, A212666, A289484.

Programs

  • Maple
    filter:= proc(n) local S,p,i;
  S:= sort(convert(numtheory:-factorset(n),list));
  p:= 1;
  for i from 1 to nops(S)-1 do
    p:= p*S[i];
    if p > S[i+1] then return true fi;
  od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 26 2018
Showing 1-3 of 3 results.

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