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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A307342 Products of four primes, except fourth powers of primes.

Original entry on oeis.org

24, 36, 40, 54, 56, 60, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 184, 189, 196, 198, 204, 210, 220, 225, 228, 232, 234, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 328, 330, 340, 342, 344, 348, 350, 351, 364, 372, 375, 376, 380, 390
Offset: 1

Views

Author

Kalle Siukola, Apr 02 2019

Keywords

Comments

Numbers with exactly four prime factors (counted with multiplicity) and more than one distinct prime factor.
Numbers n such that bigomega(n) = 4 and omega(n) > 1.

Crossrefs

Setwise difference of A014613 and A030514.
Union of A046386, A065036, A085986 and A085987.
Cf. A307682.

Programs

  • Mathematica
    Select[Range@ 400, And[! PrimePowerQ@ #, PrimeOmega@ # == 4] &] (* Michael De Vlieger, Apr 21 2019 *)
Select[Range[400],PrimeOmega[#]==4&&PrimeNu[#]>1&] (* Harvey P. Dale, Aug 27 2021 *)
  • PARI
    isok(n) = (bigomega(n)==4) && (omega(n) > 1); \\ Michel Marcus, Apr 03 2019
  • Python
    import sympy
def bigomega(n): return sympy.primeomega(n)
def omega(n): return len(sympy.primefactors(n))
print([n for n in range(1, 1000) if bigomega(n) == 4 and omega(n) > 1])

A327829 Squarefree numbers with a prime number of prime factors.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161
Offset: 1

Views

Author

Sebastian F. Orellana, Sep 26 2019

Keywords

Comments

210 is the first integer in A120944 but not here: it has 4 prime factors.

Links

Crossrefs

Subsequence of A120944.
A006881, A007304, A046387 are subsequences.
A046386, A067885 are not subsequences.

Programs

  • Mathematica
    Select[Range@ 161, And[SquareFreeQ@ #, PrimeQ@ PrimeNu@ #] &] (* Michael De Vlieger, Sep 29 2019 *)
  • PARI
    isok(n) = issquarefree(n) && isprime(omega(n)); \\ Michel Marcus, Sep 27 2019

Formula

A006881 UNION A007304 UNION A046387 UNION A123321 UNION .... - R. J. Mathar, Oct 13 2019

Extensions

Corrected and extended by Michel Marcus, Sep 27 2019

A356683 a(n) is the smallest positive k such that the count of squarefree numbers <= k that have n prime factors is equal to the count of squarefree numbers <= k that have n-1 prime factors (and the count is positive).

Original entry on oeis.org

2, 39, 1279786, 8377774397163159586
Offset: 1

Views

Author

Jon E. Schoenfield, Nov 22 2022

Keywords

Examples

			The first two squarefree numbers are 1 and 2; 1 has 0 prime factors and 2 has 1 prime factor, so a(1)=2.
At k=39, in the interval [1..k], there are 12 squarefree numbers with 1 prime factor (i.e., 12 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37), and 12 squarefree numbers with 2 prime factors (i.e., 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39). k=39 is the smallest such positive number for which these two counts are the same (and are positive), so a(2)=39.
At k=1279786, the interval [1..k] includes 265549 squarefree numbers with 2 prime factors and the same number of squarefree numbers with 3 prime factors, and there is no smaller positive number k that has this property (where the counts are positive), so a(3)=1279786. There are 75 numbers with this property, the last one being 1281378.
At k=8377774397163159586, the interval [1..k] includes 1356557942402075858 squarefree numbers with 3 prime factors and the same number of squarefree numbers with 4 prime factors, and there is no smaller positive number k that has this property (where the counts are positive), so a(4)=8377774397163159586. There are 14 numbers with this property, the last one being 8377774397163162544. - _Henri Lifchitz_, Jan 31 2025

Crossrefs

Cf. A005117, A072047.
Cf. 1 to 5 distinct primes: A000040, A006881, A007304, A046386, A046387.
Cf. 6 to 10 distinct primes: A067885, A123321, A123322, A115343, A281222.
Cf. A340316.

Programs

  • PARI
    a(n) = my(nbm = 0, nbn = 0); for (k=1, oo, if (issquarefree(k), my(o=omega(k)); if (o==n, nbn++); if (o==n-1, nbm++); if (nbm && (nbn==nbm), return(k)))); \\ Michel Marcus, Nov 25 2022
  • Python
    from itertools import count
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A356683(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(k,n): return sum(primepi(k//prod(c[1] for c in a))-a[-1][0] for a in g(k,0,1,1,n)) if n>1 else primepi(k)
    return 2 if n==1 else next(k for k in count(1) if (x:=f(k,n-1))>0 and x==f(k,n)) # Chai Wah Wu, Aug 31 2024

Extensions

a(4) from Henri Lifchitz, Jan 31 2025

A358988 Oblong numbers which are products of four distinct primes.

Original entry on oeis.org

210, 462, 870, 930, 1122, 1190, 1482, 1722, 1806, 3306, 4422, 4970, 6162, 7310, 7482, 8742, 8930, 10302, 10506, 11990, 12882, 14042, 15006, 17030, 17822, 18906, 19182, 20022, 20306, 21170, 25122, 30102, 31506, 32942, 36290, 40602, 41006, 42230, 45582, 46010, 47306
Offset: 1

Views

Author

Massimo Kofler, Dec 10 2022

Keywords

Examples

			210 = 14*15 = 2*3*5*7.
1122 = 33*34 = 2*3*11*17.
10302 = 101*102 = 2*3*17*101.
20306 = 142*143 = 2*11*13*71.

Links

Crossrefs

Intersection of A002378 and A046386.

Programs

  • Mathematica
    Select[(#*(# + 1)) & /@ Range[220], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Dec 10 2022 *)
Select[Times@@@Partition[Range[400],2,1],PrimeNu[#]==PrimeOmega[#]==4&] (* Harvey P. Dale, Jul 08 2025 *)

A359845 Centered triangular numbers which are products of four distinct primes.

Original entry on oeis.org

9010, 20710, 39610, 67735, 91885, 108946, 163846, 191710, 197110, 237010, 243010, 293710, 307135, 342010, 349210, 409510, 461206, 466210, 474610, 488206, 526585, 544510, 619210, 625006, 640594, 656374, 678385, 688510, 698710, 708985, 789526, 890506, 934966, 985366, 992674, 1039585
Offset: 1

Views

Author

Massimo Kofler, Jan 15 2023

Keywords

Comments

A squarefree subsequence of centered triangular numbers.

Examples

			9010 = 2 * 5 * 17 * 53 = (3 * 78 * (78 - 1) / 2) + 1.
20710 = 2 * 5 * 19 * 109 = (3 * 118 * (118 - 1) / 2) + 1.
39610 = 2 * 5 * 17 * 233 = (3 * 163 * (163 - 1) / 2) + 1.

Crossrefs

Intersection of A005448 and A046386.

Programs

  • Mathematica
    Select[Table[3*n*(n - 1)/2 + 1, {n, 1, 1000}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Jan 15 2023 *)

A362578 Prime numbers followed by two consecutive numbers which are products of four distinct primes (or tetraprimes).

Original entry on oeis.org

8293, 16553, 17389, 18289, 22153, 26893, 29209, 33409, 35509, 36293, 39233, 39829, 40493, 41809, 45589, 48109, 58393, 59629, 59753, 59981, 60493, 60913, 64013, 64921, 65713, 66169, 69221, 71329, 74093, 75577, 75853, 77689, 77933, 79393, 79609, 82913, 84533, 85853, 87589, 87701, 88681
Offset: 1

Views

Author

Massimo Kofler, Apr 25 2023

Keywords

Examples

			8293 (prime), 8294 = 2*11*13*29 and 8295 = 3*5*7*79.
16553 (prime), 16554 = 2*3*31*89 and 16555 = 5*7*11*43.
17389 (prime), 17390 = 2*5*37*47 and 17391 = 3*11*17*31.

Crossrefs

Cf. A000040, A046386, A140078 and A361796.

Programs

  • Mathematica
    q[n_] := FactorInteger[n][[;; , 2]] == {1, 1, 1, 1}; Select[Prime[Range[10^4]], AllTrue[# + {1, 2}, q] &] (* Amiram Eldar, Apr 25 2023 *)
  • PARI
    is(n) = (omega(n)==4) && (bigomega(n)==4); \\ A046386
isok(p) = isprime(p) && is(p+1) && is(p+2); \\ Michel Marcus, Apr 25 2023

A364141 Products k of 4 distinct primes (or tetraprimes) such that k has no squarefree neighbors.

Original entry on oeis.org

3774, 5565, 6726, 8151, 10659, 10934, 11726, 11935, 12426, 13035, 13195, 13674, 13755, 14763, 15042, 15249, 15351, 15785, 16215, 16226, 17630, 17765, 17974, 17985, 18249, 18278, 18915, 18998, 19565, 20085, 21385, 21574, 21855, 22015, 23023, 23345, 23374, 23426, 24038, 24605, 25185
Offset: 1

Views

Author

Massimo Kofler, Jul 10 2023

Keywords

Examples

			3773 = 7^3 * 11, 3774 = 2 * 3 * 17 * 37, 3775 = 5^2 * 151, so 3774 is a term.
5564 = 2^2 * 13 * 107, 5565 = 3 * 5 * 7 * 53, 5566 = 2 * 11^2 * 23, so 5565 is a term.

Crossrefs

Cf. A013929, A046386.

Programs

  • Mathematica
    Select[Range[26000], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} && ! SquareFreeQ[# - 1] && ! SquareFreeQ[# + 1] &] (* Amiram Eldar, Jul 10 2023 *)

A369238 Tetraprime numbers differing by more than 3 from any other squarefree number.

Original entry on oeis.org

72474, 106674, 193026, 237522, 261478, 308649, 342066, 370785, 391674, 491322, 604878, 865974, 885477, 931022, 938598, 1005630, 1070727, 1152822, 1186926, 1206822, 1289978, 1306878, 1363326, 1371774, 1392726, 1412918, 1455249, 1528111, 1634227, 1654678, 1688478
Offset: 1

Views

Author

Massimo Kofler, Jan 19 2024

Keywords

Comments

Tetraprimes are the product of four distinct prime numbers (cf. A046386).

Examples

			72474 = 2 * 3 * 47 * 257 is a tetraprime; 72471 = 3 * 7^2 * 17 * 29, 72472 = 2^3 * 9059, 72473 = 23^2 * 137, 72475 = 5^2 * 13 * 223, 72476 = 2^2 * 18119, 72477 = 3^2 * 8053 are all nonsquarefree numbers, so 72474 is a term.

Links

Crossrefs

Cf. A046386, A013929. Subsequence of A268332.

Programs

  • Maple
    N:= 3*10^6: # for terms <= N
P:= select(isprime,[2,seq(i,i=3 .. N/30,2)]): nP:= nops(P):
filter:= proc(x) not ormap(numtheory:-issqrfree, [x-3,x-2,x-1,x+1,x+2,x+3]) end proc:
R:= NULL:
for i1 from 1 to nP do
  r1:= P[i1];
  for i2 from 1 to i1-1 do
    r2:= r1 * P[i2]; if r2 > N/6 then break fi;
    for i3 from 1 to i2-1 do
      r3:= r2 * P[i3]; if r3 > N/2 then break fi;
      for i4 from 1 to i3-1 do
        r:= r3 * P[i4];
        if r > N then break fi;
        if filter(r) then R:= R,r; fi
od od od od:
sort([R]); # Robert Israel, Jan 19 2025
  • Mathematica
    f[n_] := Module[{e = FactorInteger[n][[;; , 2]], p}, p = Times @@ e; If[p > 1, 0, If[e == {1, 1, 1, 1}, 1, -1]]]; SequencePosition[Array[f, 2*10^6], {0, 0, 0, 1, 0, 0, 0}][[;; , 1]] + 3 (* Amiram Eldar, Jan 19 2024 *)

A370795 Centered square numbers which are products of four distinct primes (or tetraprimes).

Original entry on oeis.org

99905, 107185, 242905, 350285, 363805, 372385, 433381, 569245, 590785, 630565, 692665, 752765, 907205, 942565, 1026745, 1076045, 1090765, 1105585, 1227745, 1275205, 1333345, 1467185, 1526005, 1647113, 1661665, 1761565, 1810705, 1911013, 2026085, 2317705, 2395861, 2470865
Offset: 1

Views

Author

Massimo Kofler, Mar 02 2024

Keywords

Comments

All terms are odd.

Examples

			99905 = 5 * 13 * 29 * 53 = (2 * 223 * 224) + 1.
107185 = 5 * 13 * 17 * 97 = (2 * 231 * 232) + 1.
242905 = 5 * 13 * 37 * 101 = (2 * 348 * 349) + 1

Crossrefs

Intersection of A046386 and A001844.

Programs

  • Mathematica
    Select[Table[2*n*(n + 1) + 1, {n, 0, 1200}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Mar 02 2024 *)

Formula

a(n) == 1 (mod 4).

A376352 Squarefree semiprimes k such that k+1 is the product of three distinct primes and k+2 is the product of four distinct primes.

Original entry on oeis.org

2413, 6193, 6697, 9469, 11065, 11233, 11893, 12153, 13333, 13393, 14005, 14089, 14233, 15293, 17113, 17533, 17833, 17869, 18613, 18653, 19693, 20053, 20557, 20613, 20733, 20893, 20993, 21145, 22033, 22285, 22405, 22693, 22753, 22969, 23329, 23413, 24033, 24493, 26101, 26453, 27113, 27553, 28117, 28453, 28741, 29053, 29353, 29713
Offset: 1

Views

Author

Massimo Kofler, Sep 21 2024

Keywords

Examples

			2413 is a term because 2413 = 19*127 is the product of two distinct primes, 2414 = 2*17*71 is the product of three distinct primes and 2415 = 3*5*7*23 is the product of four distinct primes.
6193 is a term because 6193 = 11*563 is the product of two distinct primes, 6194 = 2*19*163 is the product of three distinct primes and 6195 = 3*5*7*59 is the product of four distinct primes.

Links

Crossrefs

Cf. A006881, A007304, A046386, A328137.

Programs

  • Maple
    q:= n-> andmap(j-> map(i-> i[2], ifactors(n+j-2)[2])=[1$j], [$2..4]):
select(q, [$1..30000])[];  # Alois P. Heinz, Sep 21 2024
  • Mathematica
    Position[Partition[FactorInteger[#][[;; , 2]] & /@ Range[30000], 3, 1], {{1, 1}, {1, 1, 1}, {1, 1, 1, 1}}] // Flatten (* Amiram Eldar, Sep 21 2024 *)

Formula

a(n) == 1 (mod 4).
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