A046386 -id:A046386 - cp's OEIS frontend

A346014 Numbers whose average number of distinct prime factors of their divisors is an integer.

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202

Offset: 1

Amiram Eldar, Jul 01 2021

nonn

First differs from A030229 at n = 275. a(275) = 900 is the least term that is not squarefree and therefore not in A030229.
The least term whose exponents in its prime factorization are not all the same is 1080 = 2^3 * 3^3 * 5.
The least term whose exponents in its prime factorization are distinct is 1440 = 2^5 * 3^2 * 5.
Numbers k such that A000005(k) | A062799(k).
Numbers k such that A346010(k) = 1.
Numbers k such that if the prime factorization of k is Product_{i} p_i^e_i, then Sum_{i} e_i/(e_i + 1) is an integer.
Includes all the squarefree numbers with an even number of prime divisors (A030229), i.e., the union of A006881, A046386, A067885, A123322, ...
If k is squarefree with m prime divisors then k^(m-1) is a term. E.g., the squares of the sphenic numbers (A162143) are terms.

			6 is a term since it has 4 divisors, 1, 2, 3 and 6 and (omega(1) + omega(2) + omega(3) + omega(6))/4 = (0 + 1 + 1 + 2)/4 = 1 is an integer.

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

Cf. A000005, A001221, A062799, A346009, A346010.

Subsequences: A006881, A030229, A046386, A067885, A123322, A162143.

f[p_, e_] := e/(e + 1); d[1] = 1; d[n_] := Denominator[Plus @@ f @@@ FactorInteger[n]]; Select[Range[200], d[#] == 1 &]

A380476 Numbers k with at least 4 prime factors such that A380459(k) is in A048103, i.e., has no divisors of the form p^p.

Original entry on oeis.org

4686, 32406, 184866, 209166, 388086, 1099626, 1714866, 2111406, 2166846, 2356206, 3081606, 3303366, 6445806, 11366106, 21621606, 23022366, 39824466, 39826986, 42882846, 43197846, 46043826, 58216686, 61265886, 63603546, 66496506, 66611166, 87941706, 88968246, 92086746, 97117026, 101108706, 103367886, 118743306, 119658066

Offset: 1

Antti Karttunen, Feb 04 2025

nonn

Numbers m with four or more distinct prime factors such that their arithmetic derivative (A003415) can be formed as a carryless (or "carry-free") sum (in the primorial base, A049345) of the respective summands. See the example.
The terms are all squarefree and even (see A380468 and A380478 to find out why). Moreover, they are all multiples of six, because A380459(n) = Product_{d|n} A276086(n/d)^A349394(d) applied to a product of 2*p*q*r, with p, q, r three odd primes > 3 would yield three subproducts which would be multiples of 3 (consider A047247), so the 3-adic valuation of the whole product would be >= 3; hence the second smallest prime factor must be 3. For a similar reason, with terms that are product of four primes, the two remaining prime factors are either both of the form 6m+1 (A002476), or they are both of the form 6m-1 (A007528).
It is conjectured that there are no terms with more than four prime factors. See A380475 and A380528, A380530, also A380526.

			4686 = 2*3*11*71 and taking subproducts of three primes at time, we obtain 2*3*11 = 66, 2*3*71 = 426, 2*11*71 = 1562, 3*11*71 = 2343. Then A380459(4686) = A276086(66) * A276086(426) * A276086(1562) * A276086(2343) = 1622849599205985150 = 2^1 * 3^2 * 5^2 * 7^6 * 11^9 * 13^1, and because all the exponents are less than the corresponding primes, the product is in A048103.
Considering the primorial base expansions of the same summands (subproducts), we obtain
    2100  = A049345(66)
   20100  = A049345(426)
   73010  = A049345(1562)
  101011  = A049345(2343)
  ------
  196221  = A049345(A003415(4686)), with the summands adding together cleanly without any carries.
Note how the primorial base digits at the bottom are the exponents in the product A380459(4686) given above, read from the largest to the smallest prime factor

Antti Karttunen, Table of n, a(n) for n = 1..433; all terms <= 2^35
Index entries for sequences related to primorial base.

Intersection of A033987 and A380468.
Subsequence of A005117, A358673, A380478.
Conjectured to be a subsequence of A046386.
Cf. A003415, A001222, A048103, A276086, A349394, A380459, A380467.
Cf. A002476, A007528, A047247, A047257.
Cf. also A317836, A358235, A380475, A380526, A380528, A380530.

is_A380476(n) = (issquarefree(n) && (omega(n)>=4) && A380467(n)); \\ Note that issquarefree here is just an optimization as A380467(n) = 1 implies squarefreeness of n.

A097978 a(n) = least m such that m and m+n are both products of exactly n distinct primes.

Original entry on oeis.org

1, 2, 33, 102, 1326, 115005, 31295895, 159282123, 9617162170, 1535531452026, 1960347077019695, 16513791577659519, 271518698440871310

Offset: 0

Lekraj Beedassy, Sep 07 2004

Note that a(n) and a(n)+n are required to be squarefree (compare A135058). - David Wasserman, Feb 19 2008
If we change "exactly n" to "at least n", the sequence is still the same at least through a(12). - David Wasserman, Feb 19 2008
a(13) <= 592357638037885411965. - David Wasserman, Feb 19 2008

			a(2) = 33  because 33 and 35 are both in A006881.
a(3) = 102 because 102 and 105 are both in A007304.
a(4) = 1326 because 1326 and 1330 are both in A046386.

Cf. A098515. A135058 (without regard to multiplicity).

f[n_] := Block[{lst = FactorInteger[n], a, b}, a = Plus @@ Last /@ lst; b = Length[lst]; If[a == b, b, 0]]; g[n_] := Block[{k = Product[ Prime[i], {i, n}]}, While[ f[k] != n || f[k] != f[k + n], k++ ]; k]; Do[ Print[ g[n]], {n, 1, 6}] (* Robert G. Wilson v, Sep 11 2004 *)

a(n) = min{m: A001221(m) = A001222(m) = A001221(m+n) = A001222(m+n)= n}. - R. J. Mathar, Mar 01 2017

Edited and extended by Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 08 2004

More terms from David Wasserman, Feb 19 2008

A110475 Number of symbols '*' and '^' to write the canonical prime factorization of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 0, 2, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 3, 0, 1, 2, 1, 1, 2, 0, 2, 1, 2, 0, 3, 0, 1, 2, 2, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 2, 0, 3, 1, 2, 1, 1, 1, 2, 0, 2, 2, 3, 0, 2, 0, 2, 2

Offset: 1

Reinhard Zumkeller, Sep 08 2005

nonn

It is conjectured that 1,2,3,4,5,6,7,9,11 are the only positive integers which cannot be represented as the sum of two elements of indices n such that a(n) = 1. - Jonathan Vos Post, Sep 11 2005

a(n) = 2 iff n is a sphenic number (A007304) or n is a prime p times a prime power q^e with e > 1 and q not equal to p. a(n) = 3 iff n has exactly four distinct prime factors (A046386); or n is the product of two prime powers (p^e)*(q^f) with e > 1, f > 1 and p not equal to q; or n is a semiprime s times a prime power r^g with g > 1 and r relatively prime to s. For a(n) > 3, Reinhard Zumkeller's description is a simpler description than the above compound descriptions. - Jonathan Vos Post, Sep 11 2005

			a(208029250) = a(2*5^3*11^2*13*23^2) = 4 '*' + 3 '^' = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.

Cf. A050252, A001358, A025475, A000040, A238949.
Cf. A007304, A046386, A008578.
Cf. A001221, A056170, A118914, A077761, A085548.

a110475 1 = 0
a110475 n = length us - 1 + 2 * length vs where
            (us, vs) = span (== 1) $ a118914_row n
-- Reinhard Zumkeller, Mar 23 2014

A110475[n_] := 2*Length[#] - 1 - Count[#, 1] & [FactorInteger[n][[All, 2]]];
Array[A110475, 100] (* Paolo Xausa, Mar 10 2025 *)

a(n) = A001221(n) - 1 + A056170(n) for n > 1.
a(n) = 0 iff n=1 or n is prime: a(A008578(n)) = 0.
a(n) = 1 iff n is a semiprime or a prime power p^e with e > 1.
From Amiram Eldar, Sep 27 2024: (Start)
a(n) = A238949(n) - 1 for n >= 2.
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C - 1), where B is Mertens's constant (A077761) and C = Sum_{p prime} 1/p^2 (A085548). (End)

A176687 Numbers k such that k^2-1 is the product of 4 distinct primes.

Original entry on oeis.org

34, 56, 86, 92, 94, 104, 106, 142, 144, 160, 164, 166, 184, 186, 194, 196, 202, 204, 214, 216, 218, 220, 230, 232, 236, 248, 256, 266, 272, 284, 300, 302, 304, 320, 322, 328, 340, 346, 358, 384, 392, 394, 398, 400, 412, 414, 416, 430, 434, 446, 452, 456, 464

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 23 2010

nonn

			34 is in the sequence, because 34^2 - 1 = 1155 = 3 * 5 * 7 * 11, so it's a product of 4 distinct primes.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A006881, A007304, A014574, A046386, A176686

Select[Range[7! ],Last/@FactorInteger[ #^2-1]=={1,1,1,1}&]
dp4Q[n_]:=Module[{c=n^2-1},PrimeNu[c]==PrimeOmega[c]==4]; Select[Range[ 500], dp4Q] (* Harvey P. Dale, Dec 31 2013 *)

A264887 Numbers in A007504 such that omega(a(n)) = Omega(a(n)) = 4.

Original entry on oeis.org

5830, 6870, 13490, 16401, 58406, 60146, 61910, 65534, 75130, 136114, 148827, 153178, 213538, 257358, 269074, 273054, 327198, 354102, 377310, 382038, 403611, 443685, 475323, 488774, 496905, 665130, 684510, 691026, 799846, 817563

Offset: 1

Debapriyay Mukhopadhyay, Nov 27 2015

nonn

Omega and omega are given in A001221 and A001222, respectively.
The corresponding numbers of prime summands, k(n), are 53, 57, 77, 84, 149, 151, 153, 157, 167, 219, 228, 231, 269, 293, 299, 301, 327, 339, 349, 351, 360, 376, 388, 393, 396, 453, 459, 461, 493, 498, ...
Intersection of A007504 and A046386 (products of four distinct primes). - Michel Marcus, Dec 15 2015

			For n = 1, k(n) = 53 and a(n) = A007504(53) = 5830 = 2*5*11*53.
For n = 2, k(n) = 57 and a(n) = A007504(57) = 6870 = 2*3*5*229.
For n = 3, k(n) = 77 and a(n) = A007504(77) = 13490 = 2*5*19*71.
For n = 4, k(n) = 84 and a(n) = A007504(84) = 16401 = 3*7*11*71.
For n = 5, k(n) = 149 and a(n) = A007504(149) = 58406 = 2*19*29*53.
For n = 6, k(n) = 151 and a(n) = A007504(151) = 60146 = 2*17*29*61.
Note that for each of the elements of the sequence, omega(a(n)) = Omega(a(n)) = 4, i.e., the number of prime factors of a(n) = the number of distinct prime factors of a(n) = 4.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A007504, A013918, A046386, A189072, A264885.

t = Accumulate@ Prime@ Range@ 600; Select[t, PrimeNu@ # == PrimeOmega@ # == 4 &] (* Michael De Vlieger, Nov 27 2015, after Zak Seidov at A007504 *)

lista(nn) = {my(s = 0); for (n=1, nn, s += prime(n); if ((omega(s) == 4) && (bigomega(s)==4), print1(s, ", ")););} \\ Michel Marcus, Nov 28 2015

A272078 Numbers k such that k^2 + 1 is product of 3 distinct primes.

Original entry on oeis.org

13, 17, 21, 23, 27, 31, 33, 37, 53, 55, 63, 67, 72, 75, 77, 81, 87, 89, 91, 97, 98, 103, 105, 109, 111, 112, 113, 115, 119, 122, 125, 127, 128, 129, 135, 137, 138, 142, 147, 148, 149, 151, 153, 155, 161, 162, 163, 167, 172, 174, 179, 185, 189, 192, 197, 200, 208

Offset: 1

K. D. Bajpai, Apr 19 2016

nonn

			13 appears in the list because 13^2 + 1 = 170 = 2 * 5 * 17.
21 appears in the list because 21^2 + 1 = 442 = 2 * 13 * 17.

Daniel Starodubtsev, Table of n, a(n) for n = 1..1000

Cf. A007304, A046386, A176686, A176687.

A272078 = {}; Do[ k = Last /@ FactorInteger[n^2 + 1]; If[k == {1, 1, 1}, AppendTo[A272078, n]], {n, 1000}]; A272078
Select[Range[1000], Last /@ FactorInteger[#^2 + 1] == {1, 1, 1} &]

isok(k) = my(x=k^2+1); (omega(x)==3) && (bigomega(x)==3); \\ Michel Marcus, Mar 11 2020

A275345 Characteristic polynomials of a square matrix based on A051731 where A051731(1,N)=1 and A051731(N,N)=0 and where N=size of matrix, analogous to the Redheffer matrix.

Original entry on oeis.org

1, 1, -1, -1, -1, 1, -1, 0, 2, -1, 0, 0, 2, -3, 1, -1, 2, 1, -5, 4, -1, 1, -3, 5, -8, 9, -5, 1, -1, 4, -4, -5, 15, -14, 6, -1, 0, -1, 6, -17, 29, -31, 20, -7, 1, 0, 0, 2, -13, 36, -55, 50, -27, 8, -1, 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1

Offset: 0

Mats Granvik, Jul 24 2016

From Mats Granvik, Sep 30 2017: (Start)
Conjecture: The largest absolute value of the eigenvalues of these characteristic polynomials appear to have the same prime signature in the factorization of the matrix sizes N.
In other words: Let b(N) equal the sequence of the largest absolute values of the eigenvalues of the characteristic polynomials of the matrices of size N. b(N) is then a sequence of truncated eigenvalues starting:
b(N=1..infinity)
= 1.00000, 1.61803, 1.61803, 2.00000, 1.61803, 2.20557, 1.61803, 2.32472, 2.00000, 2.20557, 1.61803, 2.67170, 1.61803, 2.20557, 2.20557, 2.61803, 1.61803, 2.67170, 1.61803, 2.67170, 2.20557, 2.20557, 1.61803, 3.08032, 2.00000, 2.20557, 2.32472, 2.67170, 1.61803, 2.93796, 1.61803, 2.89055, 2.20557, 2.20557, 2.20557, 3.21878, 1.61803, 2.20557, 2.20557, 3.08032, 1.61803, 2.93796, 1.61803, 2.67170, 2.67170, 2.20557, 1.61803, 3.45341, 2.00000, 2.67170, 2.20557, 2.67170, 1.61803, 3.08032, 2.20557, 3.08032, 2.20557, 2.20557, 1.61803, 3.53392, 1.61803, 2.20557, 2.67170, ...
It then appears that for n = 1,2,3,4,5,...,infinity we have the table:
Prime signature: b(Axxxxxx(n)) = Largest abs(eigenvalue):
p^0      : b(1)          = 1.0000000000000000000000000000...
p        : b(A000040(n)) = 1.6180339887498949025257388711...
p^2      : b(A001248(n)) = 2.0000000000000000000000000000...
p*q      : b(A006881(n)) = 2.2055694304005917238953315973...
p^3      : b(A030078(n)) = 2.3247179572447480566665944934...
p^2*q    : b(A054753(n)) = 2.6716998816571604358216518448...
p^4      : b(A030514(n)) = 2.6180339887498917939012699207...
p^3*q    : b(A065036(n)) = 3.0803227214906021558249449299...
p*q*r    : b(A007304(n)) = 2.9379558827528557962693867011...
p^5      : b(A050997(n)) = 2.8905508875432590620846440288...
p^2*q^2  : b(A085986(n)) = 3.2187765853016649941764626419...
p^4*q    : b(A178739(n)) = 3.4534111136673804054453285061...
p^2*q*r  : b(A085987(n)) = 3.5339198574905377192578725953...
p^6      : b(A030516(n)) = 3.1478990357047909043330946587...
p^3*q^2  : b(A143610(n)) = 3.7022736187975437971431347250...
p^5*q    : b(A178740(n)) = 3.8016448153137023524550386355...
p^3*q*r  : b(A189975(n)) = 4.0600260453688532535920785448...
p^7      : b(A092759(n)) = 3.3935083220984414431597997463...
p^4*q^2  : b(A189988(n)) = 4.1453038440113498808159420150...
p^2*q^2*r: b(A179643(n)) = 4.2413382309993874486053755390...
p^6*q    : b(A189987(n)) = 4.1311805192254587026923218218...
p*q*r*s  : b(A046386(n)) = 3.8825338629275134572083061357...
...
b(Axxxxxx(1)) in the sequences above, is given by A025487.
(End)
First column in the coefficients of the characteristic polynomials is the Möbius function A008683.
Row sums of coefficients start: 0, -1, 0, 0, 0, 0, 0, 0, 0, ...
Third diagonal is a signed version of A000096.
Most of the eigenvalues are equal to 1. The number of eigenvalues equal to 1 are given by A075795 for n>1.
The first three of the eigenvalues above can be calculated as nested radicals. The fourth eigenvalue 2.205569430400590... minus 1 = 1.205569430400590... is also a nested radical.

			{
{ 1},
{ 1, -1},
{-1, -1,  1},
{-1,  0,  2,  -1},
{ 0,  0,  2,  -3,  1},
{-1,  2,  1,  -5,  4,   -1},
{ 1, -3,  5,  -8,  9,   -5,   1},
{-1,  4, -4,  -5, 15,  -14,   6,  -1},
{ 0, -1,  6, -17, 29,  -31,  20,  -7,  1},
{ 0,  0,  2, -13, 36,  -55,  50, -27,  8, -1},
{ 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1}
}

Mats Granvik, Mathematica program to verify that eigenvalues determine prime signature
OEIS Wiki, Prime signatures
Eric Weisstein, Prime signature
Index to sequences related to prime signature

Cf. A051731, A008683, A000040, A001248, A006881, A030078, A030514, A054753, A000096, A001622, A272874, A075795.

Cf. A025487. - Mats Granvik, Sep 30 2017

Clear[x, AA, nn, s]; Monitor[AA = Flatten[Table[A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}]; MatrixForm[A]; a = A[[1, nn]]; A[[1, nn]] = A[[nn, nn]]; A[[nn, nn]] = a; CoefficientList[CharacteristicPolynomial[A, x], x], {nn, 1, 10}]], nn]

A307341 Products of four primes, not all distinct.

Original entry on oeis.org

16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 184, 189, 196, 198, 204, 220, 225, 228, 232, 234, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 328, 340, 342, 344, 348, 350, 351, 364, 372, 375, 376, 380, 414

Offset: 1

Kalle Siukola, Apr 02 2019

Numbers with exactly four prime factors (counted with multiplicity) but fewer than four distinct prime factors.

Numbers n such that bigomega(n) = 4 and omega(n) < 4.

Kalle Siukola, Table of n, a(n) for n = 1..10000

Setwise difference of A014613 and A046386.

Union of A030514, A065036, A085986 and A085987.

isok(n) = (bigomega(n) == 4) && (omega(n) < 4); \\ Michel Marcus, Apr 03 2019

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) < 4])

A353022 Products of four distinct primes between twin primes.

Original entry on oeis.org

462, 570, 858, 1230, 1290, 1302, 1482, 1722, 2130, 3390, 3930, 4002, 4218, 4242, 4422, 5010, 5478, 5502, 5658, 6690, 6870, 7458, 7878, 8430, 8862, 9042, 9462, 9858, 9930, 10038, 10302, 11058, 11118, 11490, 11778, 13002, 13710, 13830, 13902, 14010, 15270, 15738

Offset: 1

Massimo Kofler, Apr 17 2022

nonn

			462 = 2*3*7*11 between 461 and 463 (twin primes);
570 = 2*3*5*19 between 569 and 571;
858 = 2*3*11*13 between 857 and 859;
1230 = 2*3*5*41 between 1229 and 1231.

Intersection of A014574 and A046386.

Cf. A001097.

Select[Range[20000], And @@ PrimeQ[# + {-1, 1}] && FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Apr 17 2022 *)

cp's OEIS Frontend

A346014 Numbers whose average number of distinct prime factors of their divisors is an integer.

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A380476 Numbers k with at least 4 prime factors such that A380459(k) is in A048103, i.e., has no divisors of the form p^p.

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A097978 a(n) = least m such that m and m+n are both products of exactly n distinct primes.

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A110475 Number of symbols '*' and '^' to write the canonical prime factorization of n.

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A176687 Numbers k such that k^2-1 is the product of 4 distinct primes.

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A264887 Numbers in A007504 such that omega(a(n)) = Omega(a(n)) = 4.

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A272078 Numbers k such that k^2 + 1 is product of 3 distinct primes.

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A275345 Characteristic polynomials of a square matrix based on A051731 where A051731(1,N)=1 and A051731(N,N)=0 and where N=size of matrix, analogous to the Redheffer matrix.

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