A179702 -id:A179702 - cp's OEIS frontend

A137493 Numbers with 30 divisors.

720, 1008, 1200, 1584, 1620, 1872, 2268, 2352, 2448, 2592, 2736, 2800, 3312, 3564, 3888, 3920, 4050, 4176, 4212, 4400, 4464, 4608, 5200, 5328, 5508, 5808, 5904, 6156, 6192, 6768, 6800, 7452, 7500, 7600, 7632, 7938, 8112, 8496, 8624, 8784, 9200, 9396

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^29 (subset of A122970), p*q^2*r^4 (A179669), p^4*q^5 (A179702), p^2*q^9 (like 4608) or p*q^14, where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A137492 (29 divs), A139571 (31 divs).

Select[Range[10000],DivisorSigma[0,#]==30&]  (* Harvey P. Dale, Feb 18 2011 *)

is(n)=numdiv(n)==30 \\ Charles R Greathouse IV, Jun 19 2016

list(lim)=
{
  my(f=(v,s)->concat(v,listsig(lim,s,1)));
  Set(fold(f, [[], [29], [5, 4], [9, 2], [14, 1], [4, 2, 1]]));
}
listsig(lim, sig, coprime)=
{
  my(e=sig[1]);
  if(#sig<2,
    if(#sig==0 || sig[1]==0, return(if(lim<1,[],[1])));
    my(P=primes([2,sqrtnint(lim\1,e)]));
    if(coprime==1, return(if(e>1,apply(p->p^e,P),P)));
    P=select(p->gcd(p,coprime)==1, P);
    if(e>1, P=apply(p->p^e, P));
    return(P);
  );
  my(v=List(),ss=sig[2..#sig],t=leastOfSig(ss));
  forprime(p=2,sqrtnint(lim\t,e),
    if(coprime%p,
        my(u=listsig(lim\p^e,ss,coprime*p));
        for(i=1,#u, listput(v,p^e*u[i]));
    )
  );
  Vec(v);
} \\ Charles R Greathouse IV, Nov 18 2021

A000005(a(n))=30.

A382292 Numbers k such that A382290(k) = 1.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 64, 72, 88, 96, 104, 108, 120, 125, 135, 136, 152, 160, 168, 184, 189, 192, 200, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 320, 328, 343, 344, 351, 352, 360, 375, 376, 378, 392, 408, 416, 424, 432, 440, 448, 456, 459, 472, 480, 486, 488, 500

Offset: 1

Amiram Eldar, Mar 21 2025

First differs from A374590 and A375432 at n = 25: A374590(25) = A375432(25) = 216 is not a term of this sequence.
Numbers k such that A382291(k) = 2, i.e., numbers whose number of infinitary divisors is twice the number of their unitary divisors.
Numbers whose prime factorization has a single exponent that is a sum of two distinct powers of 2 (A018900) and all the other exponents, if they exist, are powers of 2. Equivalently, numbers of the form p^e * m, where p is a prime, e is a term in A018900, and m is a term in A138302 that is coprime to p.
If k is a term then k^2 is also a term. If m is a term in A138302 that is coprime to k then k * m is also a term. The primitive terms, i.e., the terms that cannot be generated from smaller terms using these rules, are the numbers of the form p^(2^i+1), where p is prime and i >= 1.
Analogous to A060687, which is the sequence of numbers k with prime excess A046660(k) = 2.
The asymptotic density of this sequence is A271727 * Sum_{p prime} (((1 - 1/p)/f(1/p)) * Sum_{k>=1} 1/p^A018900(k)) = 0.11919967112489084407..., where f(x) = 1 - x^3 + Sum_{k>=2} (x^(2^k)-x^(2^k+1)).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A018900, A046660, A060687, A271727, A374590, A375432, A382290, A382291.

Subsequences (numbers of the form): A030078 (p^3), A050997 (p^5), A030516 (p^6), A179665 (p^9), A030629 (p^10), A030631 (p^12), A065036 (p^3*q), A178740 (p^5*q), A189987 (p^6*q), A179692 (p^9*q), A143610 (p^2*q^3), A179646 (p^5*q^2), A189990 (p^2*q^6), A179702 (p^4*q^5), A179666 (p^4*q^3), A190464 (p^4*q^6), A163569 (p^3*q^2*r), A189975 (p*q*r^3), A190115 (p^2*q^3*r^4), A381315, A048109.

f[p_, e_] := DigitCount[e, 2, 1] - 1; q[1] = False; q[n_] := Plus @@ f @@@ FactorInteger[n] == 1; Select[Range[500], q]

isok(k) = vecsum(apply(x -> hammingweight(x) - 1, factor(k)[, 2])) == 1;

A376249 Numbers that are not prime powers and have a unique largest prime exponent that is larger than the second-largest prime exponent by 1.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 260, 261, 268, 275, 276, 279, 284, 292, 294

Offset: 1

Amiram Eldar, Sep 16 2024

First differs from A325241 at n = 36: A325241(36) = 2^2 * 3^2 * 5 is not a term of this sequence. Also, a(71) = 360 = 2^3 * 3^2 * 5 is the least term that is not a term of A325241.
Numbers whose unordered prime signature (i.e., sorted, see A118914) ends with two consecutive integers: {..., k, k+1} for some k >= 1.
The asymptotic density of this sequence is Sum_{k >= 1, p prime} (d(k+1, p) - d(k, p))/p^(k+1) = 0.21831645263800520483..., where d(k, p) = 0 for k = 1, and (1-1/p)/((1-1/p^k)*zeta(k)) for k > 1, is the density of terms that have in their prime factorization a prime p with the largest exponent that is > k.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A356862.
Subsequences: A054753, A072357, A095990, A096156, A143610, A163569, A179666, A179702.
Cf. A118914, A246655, A325241.

q[k_] := Module[{e = Sort[FactorInteger[k][[;; , 2]]]}, Length[e] > 1 && e[[-1]] == e[[-2]] + 1]; Select[Range[300], q]

is(k) = {my(e = vecsort(factor(k)[, 2])); #e > 1 && e[#e] == e[#e-1] + 1;}

A355462 Powerful numbers divisible by exactly 2 distinct primes.

Original entry on oeis.org

36, 72, 100, 108, 144, 196, 200, 216, 225, 288, 324, 392, 400, 432, 441, 484, 500, 576, 648, 675, 676, 784, 800, 864, 968, 972, 1000, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1728, 1936, 1944, 2000, 2025, 2116, 2304, 2312, 2500

Offset: 1

Amiram Eldar, Jul 03 2022

nonn

First differs from A286708 at n = 25.
Number of the form p^i * q^j, where p != q are primes and i,j > 1.
Numbers k such that A001221(k) = 2 and A051904(k) >= 2.
The possible values of the number of the divisors (A000005) of terms in this sequence is any composite number that is not 8 or twice a prime (A264828 \ {1, 8}).
675 = 3^3*5^2 and 676 = 2^2*13^2 are 2 consecutive integers in this sequence. There are no other such pairs below 10^22 (the lesser members of such pairs are terms of A060355).

			36 is a term since 36 = 2^2 * 3^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index to sequences related to prime signature.

Intersection of A001694 and A007774.
Subsequence of A286708.
Subsequences: A085986, A143610, A162142, A179646, A179666, A179671, A179689, A179694, A179699, A179702, A179705, A189988, A189990, A189991, A190464, A190465, A303661.
Cf. A000005, A001221, A051904, A060355, A136141, A264828.

Select[Range[2500], Length[(e = FactorInteger[#][[;; , 2]])] == 2 && Min[e] > 1 &]

is(n) = {my(f=factor(n)); #f~ == 2 && vecmin(f[,2]) > 1};

Sum_{n>=1} 1/a(n) = ((Sum_{p prime} (1/(p*(p-1))))^2 - Sum_{p prime} (1/(p^2*(p-1)^2)))/2 = 0.1583860791... .

A375143 Numbers whose prime factorization has a minimum exponent that is larger than 1 and is 1 less than the maximum exponent.

Original entry on oeis.org

72, 108, 200, 392, 432, 500, 648, 675, 968, 1125, 1323, 1352, 1372, 1800, 2000, 2312, 2592, 2700, 2888, 3087, 3267, 3528, 3888, 4232, 4500, 4563, 5000, 5292, 5324, 5400, 5488, 6125, 6728, 7688, 7803, 8575, 8712, 8788, 9000, 9747, 9800, 10125, 10584, 10952, 11979

Offset: 1

Amiram Eldar, Aug 01 2024

Numbers k such that 2 <= A051904(k) = A051903(k) - 1.

Numbers that are product of two coprime nonsquarefree powers of squarefree numbers (A072777) with consecutive exponents.

			72 = 2^3 * 3^2 is a term since A051904(72) = 2 is larger than 1 and is 1 less than A051903(72) = 3.

Cf. A051903, A051904, A072777.
Subsequence of A001694.
Subsequences: A143610, A167747 \ {1, 2, 12}, A093136 \ {1, 2, 20}, A179666, A179702, A190472, A375073.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 2 <= Min[e] == Max[e] - 1]; Select[Range[12000], q]

is(k) = {my(e = factor(k)[,2]); k > 1 && 2 <= vecmin(e) && vecmin(e) + 1 == vecmax(e);}

Sum_{n>=1} 1/a(n) = Sum_{k>=2} f(k) = 0.053695635500385312854..., where f(k) = Product_{p prime} (1 + 1/p^k + 1/p^(k+1)) - zeta(k)/zeta(2*k) - zeta(k+1)/zeta(2*k+2) + 1 is the sum of reciprocals of the subset of numbers m with A051904(m) = k.

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A137493 Numbers with 30 divisors.

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A382292 Numbers k such that A382290(k) = 1.

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A376249 Numbers that are not prime powers and have a unique largest prime exponent that is larger than the second-largest prime exponent by 1.

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A355462 Powerful numbers divisible by exactly 2 distinct primes.

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A375143 Numbers whose prime factorization has a minimum exponent that is larger than 1 and is 1 less than the maximum exponent.

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