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 1-7 of 7 results.

A139571 Numbers with 31 divisors.

Original entry on oeis.org

1073741824, 205891132094649, 931322574615478515625, 22539340290692258087863249, 17449402268886407318558803753801, 2619995643649944960380551432833049
Offset: 1

Views

Author

Omar E. Pol, May 07 2008

Keywords

Comments

30th powers of primes. The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime.
Start of 31st row of A073915. - R. J. Mathar, Jun 27 2009, Jun 28 2009

Links

Crossrefs

Cf. A073915, A122971, A137493 (30 divs), A175742 (32 divs).

Programs

Formula

a(n) = A000040(n)^(31-1) = A000040(n)^30.
a(n) = A122971(A000040(n)). - R. J. Mathar, Jun 27 2009

A179669 Products of form p^4*q^2*r where p, q and r are three distinct primes.

Original entry on oeis.org

720, 1008, 1200, 1584, 1620, 1872, 2268, 2352, 2448, 2736, 2800, 3312, 3564, 3920, 4050, 4176, 4212, 4400, 4464, 5200, 5328, 5508, 5808, 5904, 6156, 6192, 6768, 6800, 7452, 7500, 7600, 7632, 7938, 8112, 8496, 8624, 8784, 9200, 9396, 9648, 9680, 10044
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 23 2010

Keywords

Links

Crossrefs

Cf. A137493.

Programs

  • Maple
    N:= 20000: # for terms < N
P:= select(isprime,[2,seq(i,i=1..N/(3^2*2^4),2)]):
R:= NULL:
for i from 1 while P[i]^4 * 2^2*3 < N do
  for j from 1 while P[i]^4 * P[j]^2 *2 < N do
    if j = i then next fi;
    m:= ListTools:-BinaryPlace(P,N/P[i]^4/P[j]^2);
    R:= R, seq(P[i]^4*P[j]^2*P[k],k={$1..m} minus {i,j});
od od:
sort([R]); # Robert Israel, Mar 28 2025
  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,4}; Select[Range[10000], f]
  • PARI
    list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\12)^(1/4), t1=p^4;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=2, lim\t2, if(p==r||q==r, next);listput(v,t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179669(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(x//(p**4*q**2)) for p in primerange(integer_nthroot(x,4)[0]+1) for q in primerange(isqrt(x//p**4)+1))+sum(primepi(integer_nthroot(x//p**4,3)[0]) for p in primerange(integer_nthroot(x,4)[0]+1))+sum(primepi(isqrt(x//p**5)) for p in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(x//p**6) for p in primerange(integer_nthroot(x,6)[0]+1))-(primepi(integer_nthroot(x,7)[0])<<1)
    return bisection(f,n,n) # Chai Wah Wu, Mar 28 2025

A166546 Natural numbers n such that d(n) + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85, 86, 87, 89, 90
Offset: 1

Views

Author

Giovanni Teofilatto, Oct 16 2009

Keywords

Comments

Natural numbers n such that d(d(n)+1)= 2. - Giovanni Teofilatto, Oct 26 2009
The complement is the union of A001248, A030514, A030516, A030626, A030627, A030629, A030631, A030632, A030633 etc. - R. J. Mathar, Oct 26 2009

Links

Crossrefs

Cf. A000005.
Cf. A073915. - R. J. Mathar, Oct 26 2009

Programs

  • Magma
    [n: n in [1..100] | IsPrime(NumberOfDivisors(n)+1)]; // Vincenzo Librandi, Jan 20 2019
  • Mathematica
    Select[Range@90, PrimeQ[DivisorSigma[0, #] + 1] &] (* Vincenzo Librandi, Jan 20 2019 *)
  • PARI
    isok(n) = isprime(numdiv(n)+1); \\ Michel Marcus, Jan 20 2019

Formula

{1} U A000040 U A030513 U A030515 U A030628 U A030630 U A030634 U A030636 U A137485 U A137491 U A137493 U ... . - R. J. Mathar, Oct 26 2009

A179702 Numbers of the form p^4*q^5 where p and q are two distinct primes.

Original entry on oeis.org

2592, 3888, 20000, 50000, 76832, 151875, 253125, 268912, 468512, 583443, 913952, 1361367, 2576816, 2672672, 3557763, 4170272, 5940688, 6940323, 7503125, 8954912, 10504375, 13045131, 20295603, 22632992, 22717712, 29552672, 30074733
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

Keywords

Comments

Subsequence of A046312 and of A137493. - R. J. Mathar, Jul 27 2010

Links

Crossrefs

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693, A179694, A179695, A179696, A179698, A179699, A179700.
Cf. A085964, A085965, A085969.

Programs

  • Mathematica
    fQ[n_] := Sort[Last /@ FactorInteger @n] == {4, 5}; Select[ Range@ 31668000, fQ] (* fixed by Robert G. Wilson v, Aug 26 2010 *)
lst = {}; Do[ If[p != q, AppendTo[lst, Prime@p^4*Prime@q^5]], {p, 12}, {q, 10}]; Take[ Sort@ Flatten@ lst, 27] (* Robert G. Wilson v, Aug 26 2010 *)
Take[Union[First[#]^4 Last[#]^5&/@Flatten[Permutations/@Subsets[ Prime[ Range[30]],{2}],1]],30] (* Harvey P. Dale, Jan 01 2012 *)
  • PARI
    list(lim)=my(v=List(),t);forprime(p=2, (lim\16)^(1/5), t=p^5;forprime(q=2, (lim\t)^(1/4), if(p==q, next);listput(v,t*q^4))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011
  • Python
    from sympy import primepi, integer_nthroot, primerange
def A179702(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(integer_nthroot(x//p**5,4)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Formula

Sum_{n>=1} 1/a(n) = P(4)*P(5) - P(9) = A085964 * A085965 - A085969 = 0.000748..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Extensions

Edited and extended by Ray Chandler and R. J. Mathar, Jul 26 2010

A175742 Numbers with 32 divisors.

Original entry on oeis.org

840, 1080, 1320, 1512, 1560, 1848, 1890, 1920, 2040, 2184, 2280, 2310, 2376, 2688, 2730, 2760, 2808, 2856, 2970, 3000, 3080, 3192, 3432, 3456, 3480, 3510, 3570, 3640, 3672, 3720, 3864, 3990, 4104, 4158, 4224, 4290, 4440, 4480, 4488, 4590, 4760, 4830, 4872
Offset: 1

Views

Author

Jaroslav Krizek, Aug 27 2010

Keywords

Comments

Numbers of the form p^31, p^15*q^1, p^7*q^3, p^7*q^1*r^1, p^3*q^3*r^1, p^3*q^1*r^1*s^1 and p^1*q^1*r^1*s^1*t^1, where p, q, r, s and t are distinct primes.

Links

Crossrefs

Cf. A137491, A137492, A137493, A139571.
Cf. A046303 (a subsequence). - Michel Marcus, Apr 06 2017

Programs

Formula

A000005(a(n))=32.

Extensions

Extended by T. D. Noe, May 09 2011

A336595 Numbers whose number of divisors is divisible by 5.

Original entry on oeis.org

16, 48, 80, 81, 112, 144, 162, 176, 208, 240, 272, 304, 324, 336, 368, 400, 405, 432, 464, 496, 512, 528, 560, 567, 592, 624, 625, 648, 656, 688, 720, 752, 784, 810, 816, 848, 880, 891, 912, 944, 976, 1008, 1040, 1053, 1072, 1104, 1134, 1136, 1168, 1200, 1232
Offset: 1

Views

Author

Amiram Eldar, Jul 26 2020

Keywords

Comments

The asymptotic density of this sequence is 1 - zeta(5)/zeta(4) = 0.0419426259... (Sathe, 1945).

Examples

			16 is a term since A000005(16) = 5 is divisible by 5.

References

  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 63.

Links

Crossrefs

Cf. A000005, A008587, A059269, A336596.
Cf. A030514, A030628, A113849, A178739, A179665, A030633, A030638, A137488, A137493, A175745.

Programs

  • Maple
    q:= n-> is(irem(numtheory[tau](n), 5)=0):
select(q, [$1..1300])[];  # Alois P. Heinz, Jul 26 2020
  • Mathematica
    Select[Range[1300], Divisible[DivisorSigma[0, #], 5] &]

Formula

A030514 UNION A030628 \ {1} UNION A030633 UNION A030638 UNION A137488 UNION A137493 UNION A175745 UNION A175749 UNION A175752 UNION A175756 UNION ... - R. J. Mathar, May 05 2023

A274365 Numbers n such that n and n+1 both have 30 divisors.

Original entry on oeis.org

180224, 257499, 579375, 1075599, 1990575, 2353616, 5598800, 10320624, 11560400, 13975983, 16951599, 17213552, 17651600, 17672499, 17784207, 20626991, 20660624, 21041775, 21912848, 22252400, 24533199, 24953103, 26161875, 26604207, 29232175, 29253392
Offset: 1

Views

Author

Charles R Greathouse IV, Jun 19 2016

Keywords

Links

Crossrefs

Intersection of A005237 and A137493.
Cf. A000005, A274357.

Programs

  • Mathematica
    SequencePosition[Table[If[DivisorSigma[0,n]==30,1,0],{n,3*10^7}],{1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 10 2018 *)
  • PARI
    is(n)=numdiv(n)==30 && numdiv(n+1)==30
  • PARI
    list(lim)=my(v=List(),t); forprime(p=2,sqrtnint(lim\2,14), my(p14=p^14); forprime(q=2,lim\p14, if(p==q, next); t=p14*q; if(numdiv(t+1)==30, listput(v,t)); if(numdiv(t-1)==30, listput(v,t-1)))); forprime(p=2,sqrtnint(lim\4,9), my(p9=p^9); forprime(q=2,sqrtint(lim\p9), if(p==q, next); t=p9*q^2; if(numdiv(t+1)==30, listput(v,t)); if(numdiv(t-1)==30, listput(v,t-1)))); forprime(p=2,sqrtnint(lim\16,5), my(p5=p^5); forprime(q=2,sqrtnint(lim\p5,4), if(p==q, next); t=p5*q^4; if(numdiv(t+1)==30, listput(v,t)); if(numdiv(t-1)==30, listput(v,t-1)))); forprime(p=2,sqrtnint(lim\12,4), my(p4=p^4); forprime(q=2,sqrtint(lim\p4\2), if(p==q, next); my(q2=q^2); forprime(r=2,lim\p4\q2, if(p==r || q==r, next); t=p4*q2*r; if(numdiv(t+1)==30, listput(v,t)); if(numdiv(t-1)==30, listput(v,t-1))))); Set(v)
Showing 1-7 of 7 results.

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