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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A353027 Tetrahedral (or triangular pyramidal) numbers which are products of four distinct primes.

Original entry on oeis.org

1330, 6545, 16215, 23426, 35990, 39711, 47905, 52394, 57155, 79079, 105995, 138415, 198485, 221815, 246905, 366145, 477191, 762355, 1004731, 1216865, 1293699, 1373701, 1587986, 1633355, 1726669, 1823471, 1975354, 2246839, 2862209, 2997411, 3208094, 3580779, 4149466, 4590551
Offset: 1

Views

Author

Massimo Kofler, Apr 18 2022

Keywords

Comments

A squarefree subsequence of tetrahedral numbers.

Examples

			   1330 = 19*20*21/6 = 2 *  5 *  7 * 19;
   6545 = 33*34*35/6 = 5 *  7 * 11 * 17;
  16215 = 45*46*47/6 = 3 *  5 * 23 * 47;
  23426 = 51*52*53/6 = 2 * 13 * 17 * 53.

Links

Crossrefs

Intersection of A000292 and A046386.
Subsequence of A070755.

Programs

  • Maple
    filter:= proc(n) local F;
  F:= ifactors(n,easy)[2];
  F[..,2] = [1,1,1,1]
end proc:
select(filter, [seq(n*(n+1)*(n+2)/6,n=1..1000)]); # Robert Israel, Apr 18 2023
  • Mathematica
    Select[Table[n*(n + 1)*(n + 2)/6, {n, 1, 300}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Apr 18 2022 *)
  • Python
    from sympy import factorint
from itertools import count, islice
def agen():
    for t in (n*(n+1)*(n+2)//6 for n in count(1)):
        f = factorint(t, multiple=True)
        if len(f) == len(set(f)) == 4: yield t
print(list(islice(agen(), 34))) # Michael S. Branicky, May 28 2022

A359642 Number of numbers <= 10^n that are products of 4 distinct primes.

Original entry on oeis.org

0, 0, 16, 429, 7039, 92966, 1103888, 12364826, 133702610, 1413227318, 14709861824, 151469044739, 1547593008310, 15721130285808, 159006397271949, 1602820838558101, 16114386617828822, 161673560523193369, 1619352576852638084, 16197963371445222701
Offset: 1

Views

Author

Peter Dolland, Jan 09 2023

Keywords

Examples

			a(3) = 16 = #{210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 930, 966}

Crossrefs

Cf. A006880, A046386, A036351, A215218.

Programs

  • PARI
    a(n) = my(N=10^n); (f(m, p, k, j=1)=my(s=sqrtnint(N\m, k), count=0); if(k==2, forprime(q=p, s, count += primepi(N\(m*q)) - j; j+=1); return(count)); forprime(q=p, s, count += f(m*q, q+1, k-1, j+1); j+=1); count); f(1, 2, 4); \\ Daniel Suteu, Jan 11 2023

Extensions

a(14)-a(15) from Daniel Suteu, Jan 11 2023
a(16)-a(20) from Henri Lifchitz, Jan 31 2025

A361796 Prime numbers preceded by two consecutive numbers which are products of four distinct primes (or tetraprimes).

Original entry on oeis.org

8647, 15107, 20407, 20771, 21491, 23003, 23531, 24767, 24971, 27967, 29147, 33287, 34847, 36779, 42187, 42407, 42667, 43331, 43991, 46807, 46867, 51431, 52691, 52747, 53891, 54167, 58567, 63247, 63367, 69379, 71711, 73607, 73867, 74167, 76507, 76631, 76847, 80447, 83591, 84247, 86243
Offset: 1

Views

Author

Massimo Kofler, Apr 26 2023

Keywords

Examples

			8647 (prime), 8646 = 2*3*11*131 and 8645 = 5*7*13*19.
15107 (prime), 15106 = 2*7*13*83 and 15105 = 3*5*19*53.
20407 (prime), 20406 = 2*3*19*179 and 20405 = 5*7*11*53.

Links

Crossrefs

Cf. A000040, A046386, A140078 and A362578.

Programs

  • Maple
    N:= 10^5: # for terms <= N
TP:= NULL:
P:= select(isprime, [2,seq(i,i=3..N/30,2)]):
for i from 1 to nops(P) do
  for j from 1 to i-1 while P[i]*P[j] <= N/6 do
    for k from 1 to j-1 while P[i]*P[j]*P[k] <= N/2 do
      TP:= TP, op(select(`<=`,map(`*`,P[1..k-1],P[i]*P[j]*P[k]),N));
od od od:
TP:= {TP}:
TTP:= TP intersect map(`-`,TP,1):
sort(convert(select(isprime, map(`+`,TTP,2)),list)); # Robert Israel, Apr 28 2023
  • Mathematica
    q[n_] := FactorInteger[n][[;; , 2]] == {1, 1, 1, 1}; Select[Prime[Range[10^4]], AllTrue[# - {1, 2}, q] &] (* Amiram Eldar, Apr 26 2023 *)

A363167 Products of four distinct strong primes.

Original entry on oeis.org

200651, 222343, 283679, 319957, 363341, 385033, 408221, 428417, 452353, 463573, 483923, 491249, 513689, 526031, 544357, 546601, 547723, 580261, 605693, 671143, 688721, 696377, 698819, 739211, 740333, 742951, 743699, 747881, 771661, 774367, 783343, 790801, 808027, 820369
Offset: 1

Views

Author

Massimo Kofler, Sep 07 2023

Keywords

Comments

Strong primes: prime(n) > (prime(n-1) + prime(n+1))/2.

Examples

			200651 = 11*17*29*37 and 11 > (7+13)/2, 17 > (13+19)/2, 29 > (23+31)/2, 37 > (31+41)/2.
222343 = 11*17*29*41 and 11 > (7+13)/2, 17 > (13+19)/2, 29 > (23+31)/2, 41 > (37+43)/2.
283679 = 11*17*37*41 and 11 > (7+13)/2, 17 > (13+19)/2, 37 > (31+41)/2, 41 > (37+43)/2.

Crossrefs

Cf. A046386, A051634, A364778, A363782.

Programs

  • Mathematica
    strongQ[p_] := p > 2 && 2*p > Total[NextPrime[p, {-1, 1}]]; Select[Range[1, 10^6, 2], (f = FactorInteger[#])[[;; , 2]] == {1, 1, 1, 1} && AllTrue[f[[;; , 1]], strongQ] &] (* Amiram Eldar, Sep 08 2023 *)

A378717 Products of 4 distinct prime numbers (or tetraprimes) that are deficient.

Original entry on oeis.org

1155, 1365, 1785, 1995, 2145, 2415, 2618, 2805, 2926, 3003, 3045, 3094, 3135, 3255, 3315, 3458, 3542, 3705, 3795, 3885, 3910, 3927, 4186, 4305, 4370, 4389, 4466, 4485, 4515, 4522, 4641, 4774, 4785, 4810, 4845, 4862, 4930, 4935, 5005, 5115, 5187, 5270, 5278, 5313, 5330, 5434, 5474, 5510, 5565, 5590
Offset: 1

Views

Author

Massimo Kofler, Dec 05 2024

Keywords

Examples

			1155 is a term because 1155=3*5*7*11 is the product of four distinct primes and it is larger than the sum of its proper divisors (1+3+5+7+11+15+21+33+35+55+77+105+165+231+385=1149).
1365 is a term because 1365=3*5*7*13 is the product of four distinct primes and it is larger than the sum of its proper divisors (1+3+5+7+13+15+21+35+39+65+91+105+195+273+455=1323).

Crossrefs

Intersection of A005100 and A046386. A046390 is a subsequence.
Cf. A378480.

Programs

  • Mathematica
    q[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1, 1, 1} && Times @@ (1 + 1/f[[;; , 1]]) < 2]; Select[Range[6000], q] (* Amiram Eldar, Dec 05 2024 *)
  • PARI
    catpr(~v, lim, mult, startAt)=forprime(p=startAt,lim\mult, listput(v,mult*p))
list(lim)=my(v=List()); forprime(p=3,sqrtnint(lim\=1,4), forprime(q=p+2,sqrtnint(lim\p,3), forprime(r=q+2,sqrtint(lim\p\q), catpr(~v,lim,p*q*r, r+2)))); forprime(p=11,sqrtnint(lim\2,3), forprime(q=13,sqrtint(lim\2\p), catpr(~v, lim, 2*p*q, q+2))); forprime(p=13,sqrtint(lim\14), catpr(~v,lim,14*p,p+2)); forprime(p=19,sqrtint(lim\10), catpr(~v,lim, 10*p, p+2)); catpr(~v, lim, 154, 17); catpr(~v, lim, 110, 59); catpr(~v, lim, 130, 37); catpr(~v, lim, 170, 23); Set(v) \\ Charles R Greathouse IV, Dec 06 2024

Formula

a(n) ~ A046390(n) ~ A046386(n) ~ A014613(n) ~ 6n log n / (log log n)^3. - Charles R Greathouse IV, Dec 06 2024

A379532 Ulam numbers that are products of exactly four distinct primes (or tetraprimes).

Original entry on oeis.org

390, 546, 690, 798, 1155, 1230, 1770, 2010, 2090, 2418, 2618, 2814, 3090, 3290, 3390, 3930, 4326, 4370, 4470, 4578, 4602, 4641, 6110, 6870, 7170, 7490, 7735, 7930, 8294, 9834, 10110, 10545, 10738, 11102, 11346, 11390, 11454, 11622, 11715, 11886, 12270, 12441, 12470, 12570
Offset: 1

Views

Author

Massimo Kofler, Dec 24 2024

Keywords

Comments

Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.

Examples

			390 is a term because 390=2*3*5*13 is the product of 4 distinct primes and 390 is an Ulam number.
546 is a term because 546=2*3*7*13 is the product of 4 distinct primes and 546 is an Ulam number.
1155 is a term because 1155=3*5*7*11 is the product of 4 distinct primes and 1155 is an Ulam number.

Links

Crossrefs

Intersection of A002858 and A046386.
Cf. A068820, A378795, A379162.

Programs

  • Maple
    N:= 20000: # for terms <= N
U:= [1,2]: V:= Vector(N): V[3]:= 1: R:= NULL: count:= 0:
for i from 3 do
   for k from U[-1]+1 to N do
     if V[k] = 1 then
       J:= select(`<=`,U +~ k, N);
       V[J]:= V[J] +~ 1;
       U:= [op(U),k];
       F:= ifactors(k)[2]:
       if F[..,2] = [1,1,1,1] then R:= R,k; count:= count+1;  fi;
       break
     fi
   od;
   if k > N then break fi;
od:
R; # Robert Israel, Dec 25 2024
  • Mathematica
    seq[numUlams_] := Module[{ulams = {1, 2}}, Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[DeleteCases[Intersection[ulams, n - ulams], n/2, 1, 1]] != 2]; n], {numUlams}]; Select[ulams, FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &]]; seq[1200] (* Amiram Eldar, Dec 24 2024, after Jean-François Alcover at A002858 *)

A046444 Composite numbers whose 4 prime factors are distinct in length.

Original entry on oeis.org

2241998, 2250886, 2264218, 2268662, 2286394, 2290882, 2295326, 2295458, 2308658, 2309054, 2313586, 2330878, 2335322, 2336246, 2340778, 2354374, 2357542, 2361986, 2375186, 2375318, 2377034, 2381566, 2384602, 2398726
Offset: 1

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Comments

Equivalently, the products of 4 prime numbers that have distinct lengths when written in decimal. - Peter Munn, May 30 2023

Examples

			2241998 because 2241998 = 2 * 11 * 101 * 1009, which have 1, 2, 3, and 4 digits, respectively.

Crossrefs

Subsequence of A046386.

Programs

  • Mathematica
    Select[Range[24*10^5],PrimeNu[#]==4&&Length[Union[IntegerLength/@ FactorInteger[ #][[All,1]]]]==4&] (* Harvey P. Dale, Jan 12 2020 *)

A190377 Numbers with prime factorization p^2*q^2*r^2*s^2 where p, q, r, and s are distinct primes.

Original entry on oeis.org

44100, 108900, 152100, 213444, 260100, 298116, 324900, 476100, 509796, 592900, 636804, 736164, 756900, 828100, 864900, 933156, 1232100, 1258884, 1334025, 1416100, 1483524, 1512900, 1572516, 1664100, 1695204, 1758276, 1768900, 1863225
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 09 2011

Keywords

Links

Crossrefs

Cf. A190108, A190109.
Cf. A046386, A085548, A085964, A085966, A085968.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,2,2};Select[Range[3000000],f]
  • PARI
    list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtint(lim\900), t1=p^2; forprime(q=2,sqrtint(lim\(36*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,sqrtint(lim\(4*t2)), if(r==p || r==q, next); t3=r^2*t2; forprime(s=2,sqrtint(lim\t3), if(s==p || s==q || s==r, next); listput(v, t3*s^2))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A190377(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 int(n+x-sum(primepi(x//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(x,4)[0]+1),1) for b,m in enumerate(primerange(k+1,integer_nthroot(x//k,3)[0]+1),a+1) for c,r in enumerate(primerange(m+1,isqrt(x//(k*m))+1),b+1)))
    return bisection(f,n,n)**2 # Chai Wah Wu, Mar 27 2025

Formula

Sum_{n>=1} 1/a(n) = (P(2)^4 - 6*P(2)^2*P(4) + 8*P(2)*P(6) + 3*P(4)^2 - 6*P(8))/24 = 0.00010511750849230980748..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024
a(n) = A046386(n)^2. - Chai Wah Wu, Mar 27 2025

A206096 Fibonacci numbers with 4 distinct prime divisors, each of multiplicity 1.

Original entry on oeis.org

6765, 196418, 317811, 2178309, 32951280099, 139583862445, 1304969544928657, 5527939700884757, 259695496911122585, 679891637638612258, 12200160415121876738, 83621143489848422977, 483162952612010163284885, 22698374052006863956975682
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 03 2012

Keywords

Comments

Intersection of A000045 and A046386. - Michel Marcus, Sep 11 2014

Links

Crossrefs

Cf. A000045, A053409, A137563

Programs

  • Maple
    filter:= proc(t)
local F;
F:= ifactors(t)[2];
map(f -> f[2],F) = [1,1,1,1];
end proc:
select(filter, [seq(combinat:-fibonacci(n),n=1..200)]); # Robert Israel, Sep 07 2014
  • Mathematica
    Select[Fibonacci[Range[200]],Last/@FactorInteger[#]=={1,1,1,1}&]
  • PARI
    Vec(select(k -> omega(k)==4 && bigomega(k)==4, vector(100, i, fibonacci(i)))) \\ Edward Jiang, Sep 11 2014

A282141 a(n)=least number strictly greater than n with an equivalent prime tower factorization.

Original entry on oeis.org

3, 5, 27, 7, 10, 11, 9, 25, 14, 13, 20, 17, 15, 21, 7625597484987, 19, 24, 23, 28, 22, 26, 29, 50, 32, 33, 3125, 44, 31, 42, 37, 49, 34, 35, 38, 100, 41, 39, 46, 45, 43, 66, 47, 52, 56, 51, 53, 80, 121, 98, 55, 54, 59, 68, 57, 63, 58, 62, 61, 84, 67, 65, 75
Offset: 2

Views

Author

Rémy Sigrist, Feb 07 2017

Keywords

Comments

The prime tower factorization of a number is defined in A182318.
The prime tower factorization equivalence classes are described in A279686.
For any n>1, a(n)=least k>n such that A279690(n)=A279690(k).
This sequence is a permutation of the complement of A279686.
This sequence is to prime tower factorization what A081761 is to prime signature.

Links

Crossrefs

Cf. A000040, A006881, A007304, A046386, A046387, A051674, A067885, A081761, A182318, A279686, A279690

Programs

  • PARI
    a(n) = my (c=a279690(n)); my (k=n+1); while (c!=a279690(k), k++); k

Formula

a(A000040(n)) = A000040(n+1) for any n>0.
a(A006881(n)) = A006881(n+1) for any n>0.
a(A051674(n)) = A051674(n+1) for any n>0.
a(A007304(n)) = A007304(n+1) for any n>0.
a(A046386(n)) = A046386(n+1) for any n>0.
a(A046387(n)) = A046387(n+1) for any n>0.
a(A067885(n)) = A067885(n+1) for any n>0.
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