A119288 -id:A119288 - cp's OEIS frontend

A351566 Radix of the second least significant nonzero digit in the primorial base expansion of n, or 1 if there is no such digit.

1, 1, 1, 3, 1, 3, 1, 5, 5, 3, 5, 3, 1, 5, 5, 3, 5, 3, 1, 5, 5, 3, 5, 3, 1, 5, 5, 3, 5, 3, 1, 7, 7, 3, 7, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 1, 7, 7, 3, 7, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 1, 7, 7, 3, 7, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3

Offset: 0

Antti Karttunen, Apr 01 2022

The terms larger than one are given by the k-th prime (A000040), where k is the position of the second least significant nonzero digit in the primorial base expansion of n, counted from the right. See the example.

			For n = 13, its primorial base representation (see A049345) is "201" as 13 = 2*A002110(2) + 1*A002110(0). The one-based index of the second least significant nonzero digit ("2"), when counted from the right, is 3, therefore a(13) = A000040(3) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..60060
Index entries for sequences related to primorial base.

Cf. A060735 (gives the positions of ones after the initial one at a(0)=1).

Cf. A000040, A002110, A049345, A053669, A119288, A276086, A351567.

a[n_] := Module[{k = n, p = 2, s = {}, r, i}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; i = Position[s, ?(# > 0 &)] // Flatten; If[Length[i] < 2, 1, Prime[i[[2]]]]]; Array[a, 100, 0] (* _Amiram Eldar, Mar 13 2024 *)

A119288(n) = if(1>=omega(n), 1, (factor(n))[2,1]);
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351566(n) = A119288(A276086(n));

a(n) = A119288(A276086(n)).
For all n, a(n) > A351567(n).
If a(n) > 1, then a(n) > A053669(n).

A119315 Numbers with composite numbers as third divisors.

Original entry on oeis.org

4, 8, 9, 16, 20, 25, 27, 28, 32, 40, 44, 49, 52, 56, 64, 68, 76, 80, 81, 88, 92, 99, 100, 104, 112, 116, 117, 121, 124, 125, 128, 136, 140, 148, 152, 153, 160, 164, 169, 171, 172, 176, 184, 188, 196, 200, 207, 208, 212, 220, 224, 232, 236, 243, 244, 248, 256, 260

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

m is a term iff A067029(m) > 1 and (A001221(m) = 1 or A020639(m)^2 <= A119288(m)).
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 3, 23, 221, 2194, 21895, 219307, 2193435, 21937419, 219396872, 2193979781, ... . Apparently, the asymptotic density of this sequence exists and equals 0.219... . - Amiram Eldar, Jul 02 2022
Numbers k such that A292269(k) is composite, which must then be a square of prime (A001248) by necessity. - Antti Karttunen, Jul 02 2022

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

Complement of A119316.
A025475, A092259, and A355445 are subsequences.
Cf. A000005, A001221, A001248, A002808, A020639, A027750, A067029, A292269, A355453 (characteristic function).
Cf. also A355455.

Select[Range[300],CompositeQ[Divisors[#][[3]]]&]//Quiet (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 03 2021 *)
Select[Range[260], (f = FactorInteger[#])[[1, 2]] > 1 && (Length[f] == 1 || f[[1, 1]]^2 < f[[2, 1]]) &] (* Amiram Eldar, Jul 02 2022 *)

A355453(n) = ((n>1) && !isprime(n) && !isprime(divisors(n)[3]));
isA119315(n) = A355453(n); \\ Antti Karttunen, Jul 02 2022

A364702 Numbers k in A361098 that are not divisible by A007947(k)^2.

Original entry on oeis.org

48, 50, 54, 75, 80, 96, 98, 112, 135, 147, 160, 162, 189, 192, 224, 240, 242, 245, 250, 252, 270, 294, 300, 320, 336, 338, 350, 352, 360, 363, 375, 378, 384, 396, 405, 416, 448, 450, 468, 480, 486, 490, 504, 507, 525, 528, 540, 550, 560, 567, 578, 588, 594, 600

Offset: 1

Michael De Vlieger, Aug 03 2023

nonn

Subset of A126706, the set of numbers k neither prime powers nor squarefree, i.e., k such that A001222(k) > A001221(k) > 1.
Let p = A119288(k) be the second smallest prime factor of k. Let q = A053669(k) be the smallest prime that does not divide k. Let r = rad(k) = A007947(k) be the squarefree kernel of k. Define sequence S = A361098 = {k : Omega(k) > omega(k) > 1, q*r < k, p*r <= k} = A361098.
Sequence T = A286708 represents numbers in A001694 that are not prime powers. Numbers k in T are such that k = m*r^2, m >= 1, by definition. Since we may rewrite q*r < k instead as q*r < m*r^2, it is clear since omega(r) > 1, that q < r. Further, we may rewrite p*r <= k instead as p*r <= m*r^2, and since p | r, p < r as omega(r) > 1, we see that S contains T.
This sequence gives k that are in S but not in T.

			Let B = A126706.
B(1) = 12 is not in the sequence since 3*6 > 12.
B(2) = 18 is not in the sequence, since, though 3*6 = 18, 5*6 > 18.
B(6) = S(1) = 36 is not in the sequence since, though 3*6 < 36 and 5*6 < 36, rad(36)^2 = 6^2 | 36, hence B(6) = T(1).
B(10) = S(2) = a(1) = 48 is in the sequence since rad(48) = 6, and 6^2 does not divide 48.
B(11) = S(3) = a(2) = 50 is in the sequence since rad(50) = 10, and 10^2 does not divide 50, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A001694, A007947, A053669, A119288, A126706, A286708, A361098.

nn = 2^10; a053669[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p]; s = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]; Reap[Do[n = s[[j]]; If[And[#1*a053669[n] < n, #1*#2 <= n, ! Divisible[n, #1^2]] & @@ {Times @@ #, #[[2]]} &@ FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}] ][[-1, -1]]

This sequence is A361098 \ A286708.

A372720 a(n) = A000005(n) - A008479(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 3, 1, 4, 3, 3, 1, 4, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 4, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 4, 1, 2, 3, 4, 1, 1, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 4, 1, 3, 3, 4, 3, 7, 1, 5, 1, 3, 1, 10, 3, 3, 3

Offset: 1

Michael De Vlieger, May 13 2024

sign

A095960(50) = 3, a(50) = 2.

a(162) = -2 is the first negative term.

			Table of a(n), b(n) = A000005(n), and c(n) = A008479(n) for n <= 12:
  n  b(n) c(n) a(n)
 ------------------
  1    1    1    0
  2    2    1    1
  3    2    1    1
  4    3    2    1
  5    2    1    1
  6    4    1    3
  7    2    1    1
  8    4    3    1
  9    3    2    1
 10    4    1    3
 11    2    1    1
 12    6    2    4
a(12) = 4 since 12 has 6 divisors {1, 2, 3, 4, 6, 12}, and row 12 of A369609 has 2 terms {6, 12}.
a(18) = 3 since 18 has 6 divisors {1, 2, 3, 6, 9, 18}, and row 18 of A369609 has 3 terms {6, 12, 18}.
a(50) = 2 since 50 has 6 divisors {1, 2, 5, 10, 25, 50}, and row 50 of A369609 has 4 terms {10, 20, 40, 50}
a(162) = -2 since 162 has 10 divisors {1,2,3,6,9,18,27,54,81,162} but row 162 of A369609 has 12 terms {6,12,18,24,36,48,54,72,96,108,144,162}.
a(500) = 0 since 500 has as many divisors {1,2,4,5,10,20,25,50,100,125,250,500} as terms in row 500 of A369609 {10,20,40,50,80,100,160,200,250,320,400,500}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A003557, A008479, A095960, A119288, A162306, A183093, A303554, A355432, A360767, A369609.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], _?(Divisible[r, rad[#]] &)], {n, 120}]

a(n) = my(f=factor(n)[, 1], s); forvec(v=vector(#f, i, [1, logint(n, f[i])]), if(prod(i=1, #f, f[i]^v[i])<=n, s++)); numdiv(n) - s; \\ after A008479 \\ Michel Marcus, Jun 03 2024

a(n) = A095960(n) for n in A303554, i.e., for squarefree n or prime powers n.
a(n) = A095960(n) for n in A360767, i.e., for nonsquarefree composite n such that omega(n) > 1 and A003557(n) < A119288(n), since A008479(n) is the number of terms k in row n of A010846 such that k <= A003557(n).
a(n) = A183093(n) - A355432(n).

A381639 Denominators of Sum_{i=1..omega(n)-1} p_{i}/p_{i+1}, where omega(n) = A001221(n) and p_1 < p_2 < ... p_omega(n) are the distinct prime factors of n; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 5, 1, 1, 3, 1, 5, 7, 11, 1, 3, 1, 13, 1, 7, 1, 15, 1, 1, 11, 17, 7, 3, 1, 19, 13, 5, 1, 21, 1, 11, 5, 23, 1, 3, 1, 5, 17, 13, 1, 3, 11, 7, 19, 29, 1, 15, 1, 31, 7, 1, 13, 33, 1, 17, 23, 35, 1, 3, 1, 37, 5, 19, 11, 39, 1

Offset: 1

Amiram Eldar, Mar 03 2025

First differs from A119288 at n = 30.
First differs from {A226040(n-1)} at n = 35.
Also denominators of the fractions whose numerators are A381641.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and Jean-Louis Nicolas, Grandes valeurs de fonctions liées aux diviseurs premiers consécutifs d'un entier, in: Jean-Marie de Koninck and Claude Levesque (eds.), Théorie des nombres / Number Theory, Proceedings of the International Number Theory Conference held at Université Laval, July 5-18, 1987, De Gruyter, 1989; alternative link.

Cf. A001221, A119288, A226040, A381638 (numerators), A381640, A381641.

a[n_] := Module[{p = FactorInteger[n][[;; , 1]]}, Denominator[Total[Most[p]/Rest[p]]]]; Array[a, 100]

a(n) = {my(p = factor(n)[,1]); denominator(sum(i = 1, #p-1, p[i]/p[i+1]));}

A367455 Numbers not divisible by 6 that are neither squarefree nor prime powers.

Original entry on oeis.org

20, 28, 40, 44, 45, 50, 52, 56, 63, 68, 75, 76, 80, 88, 92, 98, 99, 100, 104, 112, 116, 117, 124, 135, 136, 140, 147, 148, 152, 153, 160, 164, 171, 172, 175, 176, 184, 188, 189, 196, 200, 207, 208, 212, 220, 224, 225, 232, 236, 242, 244, 245, 248, 250, 260, 261

Offset: 1

Michael De Vlieger, Jan 15 2024

A364997 is a proper subset.

The asymptotic density of this sequence is 1/6 - 1/(2*Pi^2). - Amiram Eldar, Jan 20 2024

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A047253, A053669, A119288, A126706, A364997, A365710, A367018.

Select[Range[261], And[Nor[SquareFreeQ[#], PrimePowerQ[#]], Mod[#, 6] != 0] &]

Intersection of A047253 and A126706.
Let p = A119288(k) and q = A053669(k) for k in A126706. Various definitions of this sequence:
{a(n)} = { k : Omega(k) > omega(k) > 1, p > q }.
{a(n)} = { k : Omega(k) > omega(k) > 1, k mod 6 != 0 }.
{a(n)} = { k = mx : x in A367018, rad(m) | x, m > 1. }.

A089992 Second prime divisor of numbers that are not powers of primes (A024619).

Original entry on oeis.org

3, 5, 3, 7, 5, 3, 5, 7, 11, 3, 13, 7, 3, 11, 17, 7, 3, 19, 13, 5, 3, 11, 5, 23, 3, 5, 17, 13, 3, 11, 7, 19, 29, 3, 31, 7, 13, 3, 17, 23, 5, 3, 37, 5, 19, 11, 3, 5, 41, 3, 17, 43, 29, 11, 3, 13, 23, 31, 47, 19, 3, 7, 11, 5, 3, 13, 5, 53, 3, 5, 37, 7, 3, 23, 29, 13, 59, 17, 3, 61, 41, 31, 3, 43

Offset: 1

Cino Hilliard, Jan 14 2004

nonn

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

Cf. A024619, A119288, A089993.

Select[Table[If[Length[(f = FactorInteger[n])] > 1, f[[2, 1]], 1], {n, 1, 150}], # > 1 &]

f(n) = a=factor(n);v=a[,1];ln=length(v);if(ln>1,return(v[2]));
g(m) = for(x=2,m,if(f(x)>0,print1(f(x)",")));

a(n) = A119288(A024619(n)). - Amiram Eldar, Apr 12 2021

Offset corrected by Amiram Eldar, Apr 12 2021

A119314 Complement of A119313.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 47, 49, 52, 53, 56, 59, 61, 64, 67, 68, 71, 73, 76, 79, 80, 81, 83, 88, 89, 92, 97, 99, 100, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121, 124, 125, 127, 128, 131, 136, 137

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

m is a term iff A001221(m) <= 1 or (A067029(m) > 1 and A020639(m)^2 <= A119288(m)).

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

Union of A119315 and A008578.
(Intersection with A119316) = A008578.
A000961 and A092259 are subsequences.
Cf. A001221, A020639, A067029, A119288, A119313.

Select[Range[140], !CompositeQ[#] || ((f = FactorInteger[#])[[1, 2]] > 1 && (Length[f] == 1 || f[[1, 1]]^2 < f[[2, 1]])) &] (* Amiram Eldar, Jul 02 2022 *)

A119316 Complement of A119315.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 48, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

m is a term iff A067029(m) = 1 or (A001221(m) > 1 and A119288(m) < A020639(m)^2).

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 7, 77, 779, 7806, 78105, 780693, 7806565, 78062581, 780603128, 7806020219, ... . Apparently, the asymptotic density of this sequence exists and equals 0.780... . - Amiram Eldar, Jul 02 2022

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

Union of A119313 and A008578.
(Intersection with A119314) = A008578.
Cf. A001221, A020639, A067029, A119288, A119315.

Select[Range[100], (f = FactorInteger[#])[[1, 2]] == 1 || (Length[f] > 1 && f[[1, 1]]^2 > f[[2, 1]]) &] (* Amiram Eldar, Jul 02 2022 *)

A372972 Numbers k such that A372720(k) is negative.

Original entry on oeis.org

162, 250, 324, 384, 486, 648, 686, 768, 972, 1152, 1250, 1296, 1372, 1458, 1536, 1728, 1875, 1944, 2058, 2250, 2304, 2430, 2500, 2560, 2592, 2662, 2738, 2916, 3000, 3072, 3362, 3402, 3456, 3698, 3750, 3840, 3888, 3993, 4050, 4116, 4374, 4394, 4418, 4500, 4608

Offset: 1

Michael De Vlieger, Jun 02 2024

nonn

Let tau = A000005, let omega = A001221, let f = A008479, and let g = A372720.
For squarefree k, A372720(k) >= 0, since A008479(k) = 1 while tau(k) = 2^omega(k).
For prime power p^m, A372720(p^m) = 1, since A008479(p^m) = m while tau(k) = m+1.
Therefore, apart from a(1) = 1, this sequence is a proper subset of A126706.
In the sequence R = {k = m*s : rad(m) | s, s > 1 in A120944}, there is a smallest term k such that g(k) <= 0 and a largest term k such that g(k) is positive. For instance, in A033845 where s = 6, only {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864} are such that g(k) > 0.
For s > 1, an infinite number of k in R are such that g(k) is negative. For example, with s = 6, all terms k > 864 in A033845 are in this sequence.
Conjecture: proper subset of A361098, hence of A360765 and A360768. This is to say that k = a(n) is such that A003557(k) >= A119288(k), i.e., k/rad(k) >= second smallest prime factor of k, and A003557(k) > A053669(k), where A053669(k) is the smallest prime q that does not divide k.

			a(1) = 162 = 2*3^4, since tau(162) - f(162)
     = (1+1)*(4+1) - card(A369609(162))
     = 10 - 12 = -2.
a(2) = 250 = 2*5^3, since tau(250) - f(250)
     = (1+1)*(3+1) - card(A369609(250))
     = 8 - 9 = -1.
a(3) = 324 = 2^2*3^4, since tau(324) - f(324)
     = (2+1)*(4+1) - card(A369609(324))
     = 15 - 16 = -1, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A007947, A008479, A126706, A372720, A372864.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Position[Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], ?(Divisible[r, rad[#]] &)], {n, 5000}], ?(# < 0 &)][[All, 1]]

cp's OEIS Frontend

A351566 Radix of the second least significant nonzero digit in the primorial base expansion of n, or 1 if there is no such digit.

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A119315 Numbers with composite numbers as third divisors.

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A364702 Numbers k in A361098 that are not divisible by A007947(k)^2.

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A372720 a(n) = A000005(n) - A008479(n).

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A381639 Denominators of Sum_{i=1..omega(n)-1} p_{i}/p_{i+1}, where omega(n) = A001221(n) and p_1 < p_2 < ... p_omega(n) are the distinct prime factors of n; a(1) = 1.

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A367455 Numbers not divisible by 6 that are neither squarefree nor prime powers.

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A089992 Second prime divisor of numbers that are not powers of primes (A024619).

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A119314 Complement of A119313.

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