A327877 -id:A327877 - cp's OEIS frontend

A190641 Numbers having exactly one non-unitary prime factor.

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164

Offset: 1

Reinhard Zumkeller, Dec 29 2012

nonn

Numbers k such that the powerful part of k, A057521(k), is a composite prime power (A246547). - Amiram Eldar, Aug 01 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio.
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, Iss. 1 (2011), pp. 52-66. See p. 61.

Subsequence of A013929 and of A327877.

Cf. A056170, A057521, A154945, A246547, A359466 (characteristic function).

a190641 n = a190641_list !! (n-1)
a190641_list = map (+ 1) $ elemIndices 1 a056170_list

Select[Range[164],Count[FactorInteger[#][[All, 2]], 1] == Length[FactorInteger[#]] - 1 &] (* Geoffrey Critzer, Feb 05 2015 *)

list(lim)=my(s=lim\4, v=List(), u=vectorsmall(s, i, 1), t, x); forprime(k=2, sqrtint(s), t=k^2; forstep(i=t, s, t, u[i]=0)); forprime(k=2, sqrtint(lim\1), for(e=2,logint(lim\1,k), t=k^e; for(i=1, #u, if(u[i] && gcd(k, i)==1, x=t*i; if(x>lim, break); listput(v, x))))); Set(v) \\ Charles R Greathouse IV, Aug 02 2016

isok(n) = my(f=factor(n)); #select(x->(x>1), f[,2]) == 1; \\ Michel Marcus, Jul 30 2017

A056170(a(n)) = 1.

a(n) ~ k*n, where k = Pi^2/(6*A154945) = 2.9816096.... - Charles R Greathouse IV, Aug 02 2016

A333634 Numbers with an even number of non-unitary prime divisors.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100, 101, 102

Offset: 1

Amiram Eldar, May 23 2020

nonn

Numbers that have an even number of distinct prime factors raised to a power larger than 1.

The asymptotic density of this sequence is 0.661317... (A065493, Feller and Tornier, 1933).

			1 is a term since it has 0 prime divisors, and 0 is even.
180 is a term since 180 = 2^2 * 3^2 * 5 has 2 prime divisors, 2 and 3, with exponents larger than 1 in its prime factorization, and 2 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Willy Feller and Erhard Tornier, Mengentheoretische Untersuchung von Eigenschaften der Zahlenreihe, Mathematische Annalen, Vol. 107 (1933), pp. 188-232.
I. J. Schoenberg, On asymptotic distributions of arithmetical functions, Transactions of the American Mathematical Society, Vol. 39, No. 2 (1936), pp. 315-330. See p. 326.
Eric Weisstein's World of Mathematics, Feller-Tornier Constant.
Wikipedia, Feller-Tornier constant.

Cf. A056170, A065493, A190641, A327877 (complement).

Select[Range[100], EvenQ @ Count[FactorInteger[#][[;;,2]], u_ /; u > 1]  &]

Numbers k with A056170(k) == 0 (mod 2).

A338540 Numbers having exactly three non-unitary prime factors.

Original entry on oeis.org

900, 1764, 1800, 2700, 3528, 3600, 4356, 4500, 4900, 5292, 5400, 6084, 6300, 7056, 7200, 8100, 8712, 8820, 9000, 9800, 9900, 10404, 10584, 10800, 11025, 11700, 12100, 12168, 12348, 12600, 12996, 13068, 13500, 14112, 14400, 14700, 15300, 15876, 16200, 16900, 17100

Offset: 1

Amiram Eldar, Nov 01 2020

nonn

Numbers k such that A056170(k) = A001221(A057521(k)) = 3.
Numbers divisible by the squares of exactly three distinct primes.
Subsequence of A318720 and first differs from it at n = 123.
The asymptotic density of this sequence is (eta_1^3 - 3*eta_1*eta_2 + 2*eta_3)/Pi^2 = 0.0032920755..., where eta_j = Sum_{p prime} 1/(p^2-1)^j (Pomerance and Schinzel, 2011).

			900 = 2^2 * 3^2 * 5^2 is a term since it has exactly 3 prime factors, 2, 3 and 5, that are non-unitary.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, No. 1 (2011), pp. 52-66. See pp. 61-62.

Subsequence of A013929, A318720 and A327877.
Cf. A001221, A056170, A057521, A190641, A338539, A338541, A338542.
Cf. A154945 (eta_1), A324833 (eta_2), A324834 (eta_3).

Select[Range[17000], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] == 3 &]

A338542 Numbers having exactly five non-unitary prime factors.

Original entry on oeis.org

5336100, 7452900, 10672200, 12744900, 14905800, 15920100, 16008300, 18404100, 21344400, 22358700, 23328900, 25489800, 26680500, 29811600, 31472100, 31840200, 32016600, 36072036, 36808200, 37088100, 37264500, 37352700, 38234700, 39312900, 42380100, 42688800, 43956900

Offset: 1

Amiram Eldar, Nov 01 2020

nonn

Numbers k such that A056170(k) = A001221(A057521(k)) = 5.
Numbers divisible by the squares of exactly five distinct primes.
The asymptotic density of this sequence is (eta_1^5 - 10*eta_1^3*eta_2 + 15*eta_1*eta_2^2 + 20*eta_1^2*eta_3 - 20*eta_2*eta_3 - 30*eta_1*eta_4 + 24*eta_5)/(20*Pi^2) = 0.0000015673..., where eta_j = Sum_{p prime} 1/(p^2-1)^j (Pomerance and Schinzel, 2011).

			5336100 = 2^2 * 3^2 * 5^2 * 7^2 * 11^2 is a term since it has exactly 5 prime factors, 2, 3, 5, 7 and 11, that are non-unitary.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, No. 1 (2011), pp. 52-66. See pp. 61-62.

Subsequence of A013929, A318720 and A327877.
Cf. A001221, A056170, A057521, A190641, A338539, A338540, A338541.
Cf. A154945 (eta_1), A324833 (eta_2), A324834 (eta_3), A324835 (eta_4), A324836 (eta_5).

Select[Range[2*10^7], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] == 5 &]

A359468 Numbers that are either multiples of 4 with their odd part squarefree, or that are not multiples of 4 and not squarefree.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 169, 171, 172, 175, 176, 184, 188, 189, 192, 198, 204, 207, 208, 212, 220, 224, 225, 228

Offset: 1

Antti Karttunen, Jan 02 2023

nonn

Numbers k for which the sum A166486(k)+A353627(k) [equally: A166486(k)+A355689(k)] is odd.

The asymptotic density of this sequence is 3/4 - 4/Pi^2 = 0.344715... . - Amiram Eldar, Jan 24 2023

			8 is included because it is a multiple of 4, and A000265(8) = 1 is squarefree.
12 is included because it is a multiple of 4, and A000265(12) = 3 is squarefree.
225 = 3^2 * 5^2 is included because it is not a multiple of 4, and it is not squarefree.

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

Cf. A000265, A166486, A355689, A359467 (characteristic function).
Positions of odd terms in A342419.
Differs from A190641 and A327877 for the first time at n=77, as a(77) = 225 is not included in them.

q[n_] := Module[{e = IntegerExponent[n, 2], o}, o = n/2^e; sqf = SquareFreeQ[o]; (e > 1 && sqf) || (e < 2 && ! sqf)]; Select[Range[250], q] (* Amiram Eldar, Jan 24 2023 *)

PARI
```
isA359468(n) = A359467(n);
```

cp's OEIS Frontend

A190641 Numbers having exactly one non-unitary prime factor.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

A333634 Numbers with an even number of non-unitary prime divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A338540 Numbers having exactly three non-unitary prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A338542 Numbers having exactly five non-unitary prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A359468 Numbers that are either multiples of 4 with their odd part squarefree, or that are not multiples of 4 and not squarefree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI