A060687 -id:A060687 - cp's OEIS frontend

A271971 Decimal expansion of (6/Pi^2) Sum_{p prime} 1/(p(p+1)), a Meissel-Mertens constant related to the asymptotic density of certain sequences of integers.

2, 0, 0, 7, 5, 5, 7, 2, 2, 0, 1, 9, 2, 6, 5, 9, 8, 6, 9, 9, 6, 2, 5, 0, 7, 2, 3, 1, 1, 4, 4, 0, 4, 7, 6, 5, 8, 5, 3, 5, 3, 5, 5, 5, 5, 3, 5, 2, 5, 6, 1, 9, 1, 6, 1, 5, 9, 7, 6, 3, 2, 9, 8, 3, 6, 5, 2, 5, 4, 0, 7, 4, 7, 4, 7, 9, 6, 4, 9, 7, 9, 1, 2, 1, 1, 9, 0, 9, 4, 2, 6, 8, 4, 5, 0, 3, 5, 9, 4, 6

Offset: 0

Jean-François Alcover, Apr 17 2016

This is the density of A060687, the numbers with one 2 and the rest 1s in the exponents of its prime factorization. - Charles R Greathouse IV, Aug 03 2016

			0.200755722019265986996250723114404765853535555352561916...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens Constants, p. 95.

Cf. A059956, A060687, A085548, A136141, A179119, A222056.

digits = 100; S = (6/Pi^2)*NSum[(-1)^n PrimeZetaP[n], {n, 2, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+5]; RealDigits[ S, 10, digits] // First

eps()=2.>>bitprecision(1.)
primezeta(s)=my(t=s*log(2)); sum(k=1, lambertw(t/eps())\t, moebius(k)/k*log(abs(zeta(k*s))))
sumalt(k=2, (-1)^k*primezeta(k))*6/Pi^2 \\ Charles R Greathouse IV, Aug 03 2016

sumeulerrat(1/(p*(p+1)))/zeta(2) \\ Amiram Eldar, Mar 18 2021

Equals (6/Pi^2)*A179119.

A320632 Numbers k such that there exists a pair of factorizations of k into factors > 1 where no factor of one divides any factor of the other.

Original entry on oeis.org

36, 60, 72, 84, 90, 100, 108, 120, 126, 132, 140, 144, 150, 156, 168, 180, 196, 198, 200, 204, 210, 216, 220, 225, 228, 234, 240, 252, 260, 264, 270, 276, 280, 288, 294, 300, 306, 308, 312, 315, 324, 330, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 390

Offset: 1

Gus Wiseman, Dec 09 2018

nonn

Positions of nonzero terms in A322437 or A322438.
Mats Granvik has conjectured that these are all the positive integers k such that sigma_0(k) - 2 > (bigomega(k) - 1) * omega(k), where sigma_0 = A000005, omega = A001221, and bigomega = A001222. - Gus Wiseman, Nov 12 2019
Numbers with more semiprime divisors than prime divisors. - Wesley Ivan Hurt, Jun 10 2021

			An example of such a pair for 36 is (4*9)|(6*6).

Antti Karttunen, Table of n, a(n) for n = 1..23437
Christophe Cordero, Factorizations of Cyclic Groups and Bayonet Codes, arXiv:2301.13566 [math.CO], 2023, p. 20.

Cf. A001055, A050336, A285572, A303362, A305149, A305193, A317144, A322435, A322437, A322439, A322440, A322441, A322442.
The following are additional cross-references relating to Granvik's conjecture.
bigomega(n) * omega(n) is A113901(n).
(bigomega(n) - 1) * omega(n) is A307409(n).
sigma_0(n) - bigomega(n) * omega(n) is A328958(n).
sigma_0(n) - 2 - (omega(n) - 1) * nu(n) is A328959(n).
Cf. A060687, A070175, A124010, A323023, A328956, A328957, A328960, A328961, A328963.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Select[Subsets[facs[#],{2}],And[!Or@@Divisible@@@Tuples[#],!Or@@Divisible@@@Reverse/@Tuples[#]]&]!={}&]

factorizations(n, m=n, f=List([]), z=List([])) = if(1==n, listput(z,Vec(f)); z, my(newf); fordiv(n, d, if((d>1)&&(d<=m), newf = List(f); listput(newf,d); z = factorizations(n/d, d, newf, z))); (z));
is_ndf_pair(fac1,fac2) = { for(i=1,#fac1,for(j=1,#fac2,if(!(fac1[i]%fac2[j])||!(fac2[j]%fac1[i]),return(0)))); (1); };
has_at_least_one_ndfpair(z) = { for(i=1,#z,for(j=i+1,#z,if(is_ndf_pair(z[i],z[j]),return(1)))); (0); };
isA320632(n) = has_at_least_one_ndfpair(Vec(factorizations(n))); \\ Antti Karttunen, Dec 10 2020

A195087 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 3.

Original entry on oeis.org

16, 48, 72, 80, 81, 108, 112, 162, 176, 200, 208, 240, 272, 304, 336, 360, 368, 392, 405, 464, 496, 500, 504, 528, 540, 560, 567, 592, 600, 624, 625, 656, 675, 688, 752, 756, 792, 810, 816, 848, 880, 891, 900, 912, 936, 944, 968, 976

Offset: 1

Harvey P. Dale, Sep 08 2011

nonn

The asymptotic density of this sequence is (Sum_{p prime} 1/(p^3*(p+1)) + Sum_{p != q primes} 1/(p^2*(p+1)*q*(q+1)) + Sum_{p < q < r primes} 1/(p*(p+1)*q*(q+1)*r*(r+1)))/zeta(2) = 0.04761... . - Amiram Eldar, Sep 03 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A001222, A060687, A195069, A195086, A195088, A195089, A195090, A195091, A195092, A195093.

Cf. A025487, A046660, A257851, A261256, A264959.

a195087 n = a195087_list !! (n-1)
a195087_list = filter ((== 3) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[1000],PrimeOmega[#]-PrimeNu[#]==3&]

is(n)=bigomega(n)-omega(n)==3 \\ Charles R Greathouse IV, Sep 14 2015

A001222(a(n)) - A001221(a(n)) = 3.

A046660(a(n)) = 3. - Reinhard Zumkeller, Nov 29 2015

A195089 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 5.

Original entry on oeis.org

64, 192, 288, 320, 432, 448, 648, 704, 729, 800, 832, 960, 972, 1088, 1216, 1344, 1440, 1458, 1472, 1568, 1856, 1984, 2000, 2016, 2112, 2160, 2240, 2368, 2400, 2496, 2624, 2752, 3008, 3024, 3168, 3240, 3264, 3392, 3520, 3600, 3645, 3648, 3744, 3776, 3872, 3904

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0118439..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195088, A195090, A195091, A195092, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195089 n = a195089_list !! (n-1)
a195089_list = filter ((== 5) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[4000],PrimeOmega[#]-PrimeNu[#]==5&]

is(n)=bigomega(n)-omega(n)==5 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 5. - Reinhard Zumkeller, Nov 29 2015

A195091 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 7.

Original entry on oeis.org

256, 768, 1152, 1280, 1728, 1792, 2592, 2816, 3200, 3328, 3840, 3888, 4352, 4864, 5376, 5760, 5832, 5888, 6272, 6561, 7424, 7936, 8000, 8064, 8448, 8640, 8748, 8960, 9472, 9600, 9984, 10496, 11008, 12032, 12096, 12672, 12960, 13056, 13122, 13568

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0029589..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 24 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A060687, A195069, A195086, A195087, A195088, A195089, A195090, A195092, A195093.

Cf. A025487, A046660, A059956, A112526, A257851, A261256, A264959.

a195091 n = a195091_list !! (n-1)
a195091_list = filter ((== 7) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[14000],PrimeOmega[#]-PrimeNu[#]==7&]

is(n)=bigomega(n)-omega(n)==7 \\ Charles R Greathouse IV, Sep 14 2015

A046660(a(n)) = 7. - Reinhard Zumkeller, Nov 29 2015

A257851 Triangle read by rows: row n contains the first n+1 numbers m such that A046660(m) = n.

Original entry on oeis.org

1, 4, 9, 8, 24, 27, 16, 48, 72, 80, 32, 96, 144, 160, 216, 64, 192, 288, 320, 432, 448, 128, 384, 576, 640, 864, 896, 1296, 256, 768, 1152, 1280, 1728, 1792, 2592, 2816, 512, 1536, 2304, 2560, 3456, 3584, 5184, 5632, 6400, 1024, 3072, 4608, 5120, 6912, 7168, 10368, 11264, 12800, 13312

Offset: 0

Reinhard Zumkeller, Nov 29 2015

At the suggestion of Michel Marcus's remark in Carlos Eduardo Olivieri's A261256.

			0:    1
1:    4     9
2:    8    24      27
3:   16    48      72    80
4:   32    96     144   160     216
5:   64   192     288   320     432   448
6:  128   384     576   640     864   896    1296
7:  256   768    1152  1280    1728  1792    2592   2816
8:  512  1536    2304  2560    3456  3584    5184   5632    6400
--  ------------------------------------------------------------
0:  1
1:  2^2   3^2
2:  2^3 2^3*3     3^3
3:  2^4 2^4*3 2^3*3^2 2^4*5
4:  2^5 2^5*3 2^4*3^2 2^5*5 2^3*3^3
5:  2^6 2^6*3 2^5*3^2 2^6*5 2^4*3^3 2^6*7
6:  2^7 2^7*3 2^6*3^2 2^7*5 2^5*3^3 2^7*7 2^4*3^4
7:  2^8 2^8*3 2^7*3^2 2^8*5 2^6*3^3 2^8*7 2^5*3^4 2^8*11
8:  2^9 2^9*3 2^8*3^2 2^9*5 2^7*3^3 2^9*7 2^6*3^4 2^9*11 2^8*5^2

Reinhard Zumkeller, Rows n = 0..20 of triangle, flattened

Cf. A046660, A151821, A261256.

Cf. A005117, A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A195069.

a257851 n k = a257851_tabl !! n !! k
a257851_row n = a257851_tabl !! n
a257851_tabl = map
   (\x -> take (x + 1) $ filter ((== x) . a046660) [1..]) [0..]

T[n_] := Reap[For[m = 1; k = 1, k <= n+1, If[PrimeOmega[m] - PrimeNu[m] == n, Sow[m]; k++]; m++]][[2, 1]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Sep 17 2021 *)

T(n,0) = A151821(n+1);
T(n,n-1) = A261256(n) for n > 0;
T(n,n) = A264959(n).
T(0,0) = A005117(1);
T(1,k) = A060687(k+1), k = 0..1;
T(2,k) = A195086(k+1), k = 0..2;
T(3,k) = A195087(k+1), k = 0..3;
T(4,k) = A195088(k+1), k = 0..4;
T(5,k) = A195089(k+1), k = 0..5;
T(6,k) = A195090(k+1), k = 0..6;
T(7,k) = A195091(k+1), k = 0..7;
T(8,k) = A195092(k+1), k = 0..8;
T(9,k) = A195093(k+1), k = 0..9;
T(10,k) = A195069(k+1), k = 0..10.

A261256 Let S_k denote the sequence of numbers j such that A001222(j) - A001221(j) = k. Then a(n) is the n-th term of S_n.

Original entry on oeis.org

4, 24, 72, 160, 432, 896, 2592, 5632, 12800, 26624, 61440, 124416, 278528, 622592, 1376256, 2949120, 5971968, 12058624, 25690112, 60817408, 130023424, 262144000, 528482304, 1107296256, 2264924160, 4586471424, 9395240960, 19864223744, 40265318400, 83751862272

Offset: 1

Carlos Eduardo Olivieri, Aug 12 2015

nonn

S_0 would correspond to the squarefree numbers (A005117), that is, numbers j such that A001222(j) = A001221(j). Note that S_0 is excluded from the scheme. - Michel Marcus, Sep 21 2015

			For n = 1, S_1 = {4, 9, 12, 18, 20, 25, ...}, so a(1) = S_1(1) = 4.
For n = 2, S_2 = {8, 24, 27, 36, 40, 54, ...}, so a(2) = S_2(2) = 24.
For n = 3, S_3 = {16, 48, 72, 80, 81, 108, ...}, so a(3) = S_3(3) = 72.
For n = 4, S_4 = {32, 96, 144, 160, 216, 224, ...}, so a(4) = S_4(4) = 160.
For n = 5, S_5 = {64, 192, 288, 320, 432, 448, ...}, so a(5) = S_5(5) = 432.

Charlie Neder, Table of n, a(n) for n = 1..500

Cf. A001221, A001222.
Cf. A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093.
Cf. A046660, A257851, A264959.

a261256 n = a257851 n (n - 1)  -- Reinhard Zumkeller, Nov 29 2015

OutSeq = {}; For[i = 1, i <= 16, i++, l = Select[Range[10^2*2^i], PrimeOmega[#] - PrimeNu[#] == i &]; AppendTo[OutSeq, l[[i]]]]; OutSeq

a(n) = {ik = 1; nbk = 0; while (nbk != n, ik++; if (bigomega(ik) == omega(ik) + n, nbk++);); ik;} \\ Michel Marcus, Oct 06 2015

a(n+1) > 2*a(n).
a(n) >= 2^prime(n) for n < 5.
a(n) = A257851(n,n-1). - Reinhard Zumkeller, Nov 29 2015
a(n) = b(n)*2^(n+1), where b(n) consists of the values of k/2^excess(k) over odd k, sorted in ascending order. In particular, a(n) <= prime(n)*2^(n+1), with equality only when n = 2. - Charlie Neder, Jan 31 2019

a(17)-a(21) from Jon E. Schoenfield, Sep 12 2015

More terms from Charlie Neder, Jan 31 2019

A328956 Numbers k such that sigma_0(k) = omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 104, 106, 111, 112, 115, 116, 117

Offset: 1

Gus Wiseman, Nov 01 2019

nonn

First differs from A084227 in having 60.

			The sequence of terms together with their prime indices begins:
   6: {1,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  33: {2,5}
  34: {1,7}
  35: {3,4}
  38: {1,8}
  39: {2,6}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}

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

Zeros of A328958.
The complement is A328957.
Prime signature is A124010.
Omega-sequence is A323023.
omega(n) * Omega(n) is A113901(n).
(Omega(n) - 1) * omega(n) is A307409(n).
sigma_0(n) - omega(n) * Omega(n) is A328958(n).
sigma_0(n) - 2 - (Omega(n) - 1) * omega(n) is A328959(n).
Cf. A000040, A005117, A060687, A070175, A090858, A112798, A303555, A320632, A328960, A328961, A328962, A328963, A328964, A328965.

Select[Range[100],DivisorSigma[0,#]==PrimeOmega[#]*PrimeNu[#]&]

is(k) = {my(f = factor(k)); numdiv(f) == omega(f) * bigomega(f);} \\ Amiram Eldar, Jul 28 2024

A000005(a(n)) = A001222(a(n)) * A001221(a(n)).

A048107 Numbers k such that the number of unitary divisors of k (A034444) is larger than the number of non-unitary divisors of k (A048105).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86

Offset: 1

Labos Elemer

nonn

Numbers with at most one 2 and no 3s or higher in their prime exponents. - Charles R Greathouse IV, Aug 25 2016

A disjoint union of A005117 and A060687. The asymptotic density of this sequence is (6/Pi^2) * (1 + Sum_{p prime} 1/(p*(p+1))) = A059956 * (1 + A179119) = A059956 + A271971 = 0.8086828238... - Amiram Eldar, Nov 07 2020

			n = 420 = 2*2*3*5*7, 4 distinct prime factors, 24 divisors of which 16 are unitary and 8 are not; ud(n) > nud(n) and 2^(4+1) = 32 is larger than d, the number of divisors.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Complement of A048108.
A072357 is a subsequence.
Cf. A000005, A001221, A005117, A034444, A048105, A059956, A060687, A179119, A271971, A325973.

Select[Range[500], 2^(1 + PrimeNu[#]) > DivisorSigma[0, #] &] (* G. C. Greubel, May 05 2017 *)

is(n)=my(f=factor(n)[, 2], t); for(i=1, #f, if(f[i]>1, if(t||f[i]>2, return(0), t=1))); 1 \\ Charles R Greathouse IV, Sep 17 2015

is(n)=n==1 || factorback(factor(n)[,2])<3 \\ Charles R Greathouse IV, Aug 25 2016

Numbers for which 2^(r(n)+1) > d(n), where r = A001221, d = A000005.

A195069 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 10.

Original entry on oeis.org

2048, 6144, 9216, 10240, 13824, 14336, 20736, 22528, 25600, 26624, 30720, 31104, 34816, 38912, 43008, 46080, 46656, 47104, 50176, 59392, 63488, 64000, 64512, 67584, 69120, 69984, 71680, 75776, 76800, 79872, 83968, 88064, 96256, 96768, 101376, 103680, 104448

Offset: 1

Harvey P. Dale, Sep 08 2011

The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} f(a(k)) = 0.0003698..., where f(k) = A112526(k) * Product_{p|k} p/(p+1). - Amiram Eldar, Sep 25 2024

			14336 = 2^11 * 7^1, so it has 12 prime factors (counted with multiplicity) and 2 distinct prime factors, and 12-2 = 10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A025487, A060687, A195087, A195088, A195089, A195090, A195091, A195092, A195093.

Cf. A046660, A059956, A112526, A257851, A261256, A264959.

a195069 n = a195069_list !! (n-1)
a195069_list = filter ((== 10) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

op(select(n->bigomega(n)-nops(factorset(n))=10, [$1..104448])); # Paolo P. Lava, Jul 03 2018

Select[Range[200000], PrimeOmega[#] - PrimeNu[#] == 10&]

isok(n) = bigomega(n) - omega(n) == 10; \\ Michel Marcus, Jul 03 2018

A046660(a(n)) = 10. - Reinhard Zumkeller, Nov 29 2015

cp's OEIS Frontend

A271971 Decimal expansion of (6/Pi^2) Sum_{p prime} 1/(p(p+1)), a Meissel-Mertens constant related to the asymptotic density of certain sequences of integers.

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A320632 Numbers k such that there exists a pair of factorizations of k into factors > 1 where no factor of one divides any factor of the other.

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A195087 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 3.

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A195089 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 5.

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A195091 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 7.

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A257851 Triangle read by rows: row n contains the first n+1 numbers m such that A046660(m) = n.

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A261256 Let S_k denote the sequence of numbers j such that A001222(j) - A001221(j) = k. Then a(n) is the n-th term of S_n.

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