A038110 -id:A038110 - cp's OEIS frontend

A284411 Least prime p such that more than half of all integers are divisible by n distinct primes not greater than p.

3, 37, 42719, 5737850066077

Offset: 1

Peter Munn, Mar 26 2017

The proportion of all integers that satisfy the divisibility criterion for p=prime(m) is determined using the proportion that satisfy it over any interval of primorial(m)=A002110(m) integers.
a(4) is from De Koninck, 2009; calculation credited to David Grégoire.
a(5) is about 7.887*10^34 assuming the Riemann Hypothesis, and about 7*10^34 unconditionally (De Koninck and Tenenbaum, 2002). - Amiram Eldar, Dec 05 2024

			Exactly half of the integers are divisible by 2, so a(1)>2. Two-thirds of all integers are divisible by 2 or 3, so a(1) = 3.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, pp. 13, 216 and 368.

Jean-Marie De Koninck and Gérald Tenenbaum, Sur la loi de répartition du k-ième facteur premier d'un entier, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 133, No. 2 (2002), pp. 191-204.
Gérald Tenenbaum, Some of Erdős' unconventional problems in number theory, thirty-four years later, Erdős Centennial, Janos Bolyai Math. Soc., 2013, 651-681. HAL Id: hal-01281530.

Cf. A002110, A077761, A096294, A194156, A281889.

Cf. A038110, A038111, A342479, A342480, A378720, A378721.

a(n) is least p=prime(m) such that 2*Sum_{k=0..n-1} A096294(m,k) < A002110(m).

log(log(a(n))) = n - b + O(1/sqrt(n)), where b = 1/3 + A077761 (De Koninck and Tenenbaum, 2002). - Amiram Eldar, Dec 05 2024

Definition edited by N. J. A. Sloane, Apr 01 2017

A286941 Irregular triangle read by rows: the n-th row corresponds to the totatives of the n-th primorial, A002110(n).

Original entry on oeis.org

1, 1, 5, 1, 7, 11, 13, 17, 19, 23, 29, 1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209

Offset: 1

Jamie Morken and Michael De Vlieger, May 16 2017

Values in row n of a(n) are those of row n of A286942 complement those of row n of A279864.
From Michael De Vlieger, May 18 2017: (Start)
Numbers t < p_n# such that gcd(t, p_n#) = 0, where p_n# = A002110(n).
Numbers in the reduced residue system of A002110(n).
A005867(n) = number of terms of a(n) in row n; local minimum of Euler's totient function.
A048862(n) = number of primes in row n of a(n).
A048863(n) = number of nonprimes in row n of a(n).
Since 1 is coprime to all n, it delimits the rows of a(n).
The prime A000040(n+1) is the second term in row n since it is the smallest prime coprime to A002110(n) by definition of primorial.
The smallest composite in row n is A001248(n+1) = A000040(n+1)^2.
The Kummer numbers A057588(n) = A002110(n) - 1 are the last terms of rows n, since (n - 1) is less than and coprime to all positive n. (End)

			The triangle starts
1;
1, 5;
1, 7, 11, 13, 17, 19, 23, 29;
1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209;

Michael De Vlieger, Table of n, a(n) for n = 1..6299 (rows 1 <= n <= 6 flattened).
Mathoverflow, Lower bound for Euler's totient for almost all integers. - _Michael De Vlieger_, May 18 2017
Eric Weisstein's World of Mathematics, Totative. - _Michael De Vlieger_, May 18 2017

Cf. A002110, A005867, A048862, A057588, A279864, A286941, A286942, A309497, A038110, A058250, A329815.

Cf. A285784 (nonprimes that appear), A335334 (row sums).

Table[Function[P, Select[Range@ P, CoprimeQ[#, P] &]]@ Product[Prime@ i, {i, n}], {n, 4}] // Flatten (* Michael De Vlieger, May 18 2017 *)

row(n) = my(P=factorback(primes(n))); select(x->(gcd(x, P) == 1), [1..P]); \\ Michel Marcus, Jun 02 2020

More terms from Michael De Vlieger, May 18 2017

A053144 Cototient of the n-th primorial number.

Original entry on oeis.org

1, 4, 22, 162, 1830, 24270, 418350, 8040810, 186597510, 5447823150, 169904387730, 6317118448410, 260105476071210, 11228680258518030, 529602053223499410, 28154196550210460730, 1665532558389396767070

Offset: 1

Labos Elemer, Feb 28 2000

nonn

a(n) > A005367(n), a(n) > A002110(n)/2.

Limit_{n->oo} a(n)/A002110(n) = 1 because (in the limit) the quotient is the probability that a randomly selected integer contains at least one of the first n primes in its factorization. - Geoffrey Critzer, Apr 08 2010

			In the reduced residue system of q(4) = 2*3*5*7 - 210 the number of coprimes to 210 is 48, while a(4) = 210 - 48 = 162 is the number of values divisible by one of the prime factors of q(4).

Michael De Vlieger, Table of n, a(n) for n = 1..350

Cf. A002110, A051953, A005867, A038110, A038111, A174909, A293558.
Cf. A000040 (prime numbers).
Cf. A161527, A060753.
Column 1 of A281891.

Abs[Table[ Total[Table[(-1)^(k + 1)* Total[Apply[Times, Subsets[Table[Prime[n], {n, 1, m}], {k}], 2]], {k, 0, m - 1}]], {m, 1, 22}]] (* Geoffrey Critzer, Apr 08 2010 *)
Array[# - EulerPhi@ # &@ Product[Prime@ i, {i, #}] &, 17] (* Michael De Vlieger, Feb 17 2019 *)

a(n) = prod(k=1, n, prime(k)) - prod(k=1, n, prime(k)-1); \\ Michel Marcus, Feb 08 2019

a(n) = A051953(A002110(n)) = A002110(n) - A005867(n).
a(n) = a(n-1)*A000040(n) + A005867(n-1). - Bob Selcoe, Feb 21 2016
a(n) = (1/A000040(n+1) - A038110(n+1)/A038111(n+1))*A002110(n+1). - Jamie Morken, Feb 08 2019
a(n) = A161527(n)*A002110(n)/A060753(n+1). - Jamie Morken, May 13 2022

A091440 Smallest number m such that m#/phi(m#) >= n, where m# indicates the primorial (A034386) of m and phi is Euler's totient function.

Original entry on oeis.org

1, 2, 3, 7, 13, 23, 43, 79, 149, 257, 461, 821, 1451, 2549, 4483, 7879, 13859, 24247, 42683, 75037, 131707, 230773, 405401, 710569, 1246379, 2185021, 3831913, 6720059, 11781551, 20657677, 36221753, 63503639, 111333529, 195199289, 342243479, 600036989

Offset: 1

T. D. Noe, Jan 09 2004

nonn

Does the ratio of adjacent terms converge?
It appears that lim_{n->infinity} a(n+1)/a(n) = 1.7532... - Jon E. Schoenfield, Feb 21 2019
For n > 1, a(n) is smallest prime p = prime(k) such that no fewer than (n-1)/n of any p# consecutive integers are divisible by a prime not greater than p. Cf. A053144(k)/A002110(k). - Peter Munn, Apr 29 2017
Also, the smallest prime p such that the sum of the reciprocals of the p-smooth numbers converges to at least n. - Keith F. Lynch, Apr 29 2023
Also, if m is a random integer much larger than the square of a(n), and m is not divisible by any prime less than or equal to a(n), the probability that m is prime is n/log(m). - Keith F. Lynch, Dec 17 2023

			7#/phi(7#) = (2*3*5*7)/(1*2*4*6) = 4.375 >= 4, 5#/phi(5#) = 3.75. Hence a(4) = 7.

Jon E. Schoenfield, Table of n, a(n) for n = 1..44
Eric Weisstein's World of Mathematics, Primorial.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A091439, A000010, A002110, A005579, A034386, A053144, A038110, A060753, A164347.

Mathematica
```
prod=1; i=0; Table[While[prod
				
```

al(lim) = local(mm,n,m); mm=3; n=2; m=1; forprime(x=3,lim, n*=x; m*= (x-1); if (n\m >= mm, print1(x","); mm++)); /* This will generate all terms of this sequence from the 3rd onward, up to lim. The computation slows down for large values because of the size of the internal values. */ \\ Fred Schneider, Aug 13 2009, modified by Franklin T. Adams-Watters, Aug 29 2009

a(n) = prime(A005579(n)) for n >= 4. - Amiram Eldar, Apr 18 2025

More terms from David W. Wilson, Sep 28 2005

Sequence reference in name corrected by Peter Munn, Apr 29 2017

A342479 a(n) is the numerator of the asymptotic density of numbers whose second smallest prime divisor (A119288) is prime(n).

Original entry on oeis.org

0, 1, 1, 1, 46, 44, 288, 33216, 613248, 151296, 391584768, 2383570944, 86830424064, 206470840320, 21270238986240, 987259950858240, 1262040231444480, 3022250536693923840, 3884253754215628800, 1102040800033347993600, 1892288242221318144000, 5616902226049109065728000

Offset: 1

Amiram Eldar, Mar 13 2021

The second smallest prime divisor of a number k is the second member in the ordered list of the distinct prime divisors of k. All the numbers that are not prime powers (A000961) have a second smallest prime divisor.

			The fractions begin with 0, 1/6, 1/10, 1/15, 46/1155, 44/1365, 288/12155, 33216/1616615, 613248/37182145, 151296/11849255, 391584768/33426748355, ...
a(1) = 0 since there are no numbers whose second smallest prime divisor is prime(1) = 2.
a(2)/A342480(2) = 1/6 since the numbers whose second smallest prime divisor is prime(2) = 3 are the positive multiples of 6.
a(3)/A342480(3) = 1/10 since the numbers whose second smallest prime divisor is prime(3) = 5 are the numbers congruent to {10, 15, 20} (mod 30) whose density is 3/30 = 1/10.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 337-341.

Amiram Eldar, Table of n, a(n) for n = 1..376
Paul Erdős and Gérald Tenenbaum, Sur les densités de certaines suites d'entiers, Proc. London Math. Soc. (3), Vol. 59, No. 3 (1989), pp. 417-438; alternative link.

Cf. A000961, A038110, A038111, A119288, A342480 (denominators).

f[n_] := Module[{p = Prime[n], q}, q = Select[Range[p - 1], PrimeQ]; Plus @@ (1/(q - 1))*Times @@ ((q - 1)/q)/p]; Numerator @ Array[f, 30]

a(n)/A342480(n) = (1/prime(n)) * Product_{q prime < prime(n)} (1 - 1/q) * Sum_{q prime < prime(n)} 1/(q-1).

Sum_{n>=1} a(n)/A342480(n) = 1 (since the asymptotic density of numbers without a second smallest prime divisor, i.e., the prime powers, is 0).

A342480 a(n) is the denominator of the asymptotic density of numbers whose second smallest prime divisor (A119288) is prime(n).

Original entry on oeis.org

1, 6, 10, 15, 1155, 1365, 12155, 1616615, 37182145, 11849255, 33426748355, 247357937827, 10141675450907, 25652473199353, 2928046583754721, 155186468939000213, 223317113839049087, 558516101711461766587, 796182527971658263007, 241532826894674874877669, 430046252763689411367557

Offset: 1

Amiram Eldar, Mar 13 2021

See A342479 for details.

Amiram Eldar, Table of n, a(n) for n = 1..376

Cf. A038110, A038111, A119288, A342479 (numerators).

f[n_] := Module[{p = Prime[n], q}, q = Select[Range[p - 1], PrimeQ]; Plus @@ (1/(q - 1))*Times @@ ((q - 1)/q)/p]; Denominator @ Array[f, 30]

A256968 Let b(n) = Product_{i=1..n} p_i/(p_i - 1), p_i = i-th prime; a(n) = minimum k such that b(k) >= n.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 9, 14, 22, 35, 55, 89, 142, 230, 373, 609, 996, 1637, 2698, 4461, 7398, 12301, 20503, 34253, 57348, 96198, 161659, 272124, 458789, 774616, 1309627, 2216968, 3757384, 6375166, 10828012, 18409028, 31326514, 53354259, 90945529, 155142139

Offset: 0

N. J. A. Sloane, Apr 17 2015

nonn

A001611 is similar but different.

Equal to A005579 except for n = 2 and n = 3. The following argument shows that they are equal for n > 3. First note that b(k+1) > b(k). Next, Product_{i=1..k} p_i is 2 times an odd number, i.e., it is not divisible by 4. Similarly since p_i - 1 is even for i > 1, Product_{i=1..k} (p_i - 1) is divisible by 2^(k-1), i.e., it is divisible by 4 for k >= 3. Thus b(k) is not an integer for k >= 3. Since b(3) = 15/4 > 3, this means that a(n) = A005579(n) for n > 3 - Chai Wah Wu, Apr 17 2015

			The sequence b(n) for n >= 0 begins 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, 323323/55296, 676039/110592, 2800733/442368, 86822723/13271040, 3212440751/477757440, 131710070791/19110297600, 5663533044013/802632499200, ... = A060753/A038110. So a(3) = 2.

Popular Computing (Calabasas, CA), Problem 182 (Suggested by Victor Meally), Annotated and scanned copy of page 10 of Vol. 5 (No. 53, Aug 1977).

Cf. A005579, A060753, A038110, A001611.

from sympy import prime
A256968_list, count, bn, bd = [0,0], 2, 1, 1
for k in range(1,10**4):
    p = prime(k)
    bn *= p
    bd *= p-1
    while bn >= count*bd:
        A256968_list.append(k)
count += 1 # Chai Wah Wu, Apr 17 2015; corrected by Max Alekseyev, Jan 26 2025

More terms from Chai Wah Wu, Apr 17 2015
a(32)-a(33) from Chai Wah Wu, Apr 19 2015
a(0)-a(1) corrected and a(34)-a(39) copied over from A005579 by Max Alekseyev, Jan 26 2025

A061671 Numbers n such that { x +- 2^k : 0 < k < 4 } are primes, where x = 210*n - 105.

Original entry on oeis.org

1, 77, 93, 209, 5197, 7695, 9307, 13442, 13524, 15445, 16192, 28600, 30970, 34228, 36388, 38391, 41625, 50127, 52795, 55546, 69146, 70538, 70642, 70747, 76314, 76642, 90079, 91416, 93496, 94288, 95773, 96415, 101530, 104049, 107559, 118031

Offset: 1

Frank Ellermann, Jun 16 2001

nonn

This sequence does not include the sextet (7,11,13,17,19,23). It is a proper subset of A014561 in a certain sense.

			16057, 16061, 16063, 16067, 16069, 16073 are prime and (16065+105)/210= 77= a(2).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, conjectures following th. 5

Harvey P. Dale, Table of n, a(n) for n = 1..900
F. Ellermann, Illustration for A002110, A005867, A038110, A060753

210 = 7*5*3*2 = A002110(4), cf. A014561.

Select[Range[1, 1000000], Union[PrimeQ[(210*# - 105) + {-8, -4, -2, 2, 4, 8}]] == {True} &]
Select[Range[120000],AllTrue[210#-105+{-8,-4,-2,2,4,8},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 05 2019 *)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 20 2001 and from Frank Ellermann, Nov 26 2001. Mathematica script from Peter Bertok (peter(AT)bertok.com), Nov 27 2001.

A254196 a(n) is the numerator of Product_{i=1..n} (1/(1-1/prime(i))) - 1.

Original entry on oeis.org

1, 2, 11, 27, 61, 809, 13945, 268027, 565447, 2358365, 73551683, 2734683311, 112599773191, 4860900544813, 9968041656757, 40762420985117, 83151858555707, 5085105491885327, 341472595155548909, 24295409051193284539

Offset: 1

Geoffrey Critzer, Jan 26 2015

nonn

The denominators are A038110(n+1).
a(n)/A038110(n+1) = Sum_{k >=2} 1/k where k is a positive integer whose prime factors are among the first n primes. In particular, for n=1,2,3,4,5, a(n)/A038110(n+1) is the sum of the reciprocals of the terms (excepting the first, 1) in A000079, A003586, A051037, A002473, A051038.
Appears to be a duplicate of A161527. - Michel Marcus, Aug 05 2019

			a(1)=1 because 1/2 + 1/4 + 1/8 + 1/16 + ... = 1/1.
a(2)=2 because 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/12 + ... = 2/1.
a(3)=11 because 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/15 + ... = 11/4.
a(4)=27 because Sum_{n>=2} 1/A002473(n) = 27/8.
a(5)=61 because Sum_{n>=2} 1/A051038(n) = 61/16.

Robert Israel, Table of n, a(n) for n = 1..422
Eric Weisstein's World of Mathematics, Smooth Number

Cf. A038110, A000079, A003586, A051037, A002473, A051038, A161527.

seq(numer(mul(1/(1-1/ithprime(i)),i=1..n)-1),n=1..20); # Robert Israel, Jan 28 2015

Numerator[Table[Product[1/(1 - 1/p), {p, Prime[Range[n]]}] - 1, {n,1,20}]]
b[0] := 0; b[n_] := b[n - 1] + (1 - b[n - 1]) / Prime[n]
Numerator@ Table[b[n], {n, 1, 20}] (* Fred Daniel Kline, Jun 27 2017 *)

a(n) = numerator(prod(i=1, n, (1/(1-1/prime(i)))) - 1); \\ Michel Marcus, Jun 29 2017

a(n) = A038111(n+1)/prime(n+1)-A038110(n+1). - Robert Israel, Jan 28 2015, corrected Jul 07 2019.

A308121 Irregular triangle read by rows: T(n,k) = A109395(n)k-A076512(n)A038566(n,k).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 3, 4, 2, 1, 1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 1, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 1, 2, 3, 7, 14, 13, 4, 11, 2, 1, 8

Offset: 1

Jamie Morken, May 13 2019

Row n has length A000010(n).
Row n > 1 has sum = n*A076512(n)/2.
First value on row(n) = A076511(n).
Last value on row(n) = A076512(n) for n > 1.
For n > 1, A109395(n) = Max(row) + Min(row).
For values x and y on row n > 1 at positions a and b on the row:
  x + y = A109395(n), where a = A000010(n) - (b-1).
For n > 2 the penultimate value on row A002110(n) is given by
  A038110(n)*A000040(n)-A060753(n).
From Charlie Neder, Jun 05 2019: (Start)
If p is a prime dividing n, then row p*n consists of p copies of row n.
Conjecture: If n is odd, then row 2n can be obtained from row n by interchanging the first and second halves. (End)

			The sequence as an irregular triangle:
  n/k 1, 2, 3, 4, ...
   1: 0
   2: 1
   3: 1, 2
   4: 1, 1
   5: 1, 2, 3, 4
   6: 2, 1
   7: 1, 2, 3, 4, 5, 6
   8: 1, 1, 1, 1
   9: 1, 2, 1, 2, 1, 2
  10: 3, 4, 1, 2
  11: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
  12: 2, 1, 2, 1
  13: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
  14: 4, 5, 6, 1, 2, 3
  15: 7, 14, 13, 4, 11, 2, 1, 8
  ...
  Row sums: 0, 1, 3, 2, 10, 3, 21, 4, 9, 10, 55, 6, 78, 21, 60.
T(14,5) = A109395(14)*5 - A076512(14)*A038566(14,5) = 7*5 - 3*11 = 2.
T(210,2) = A109395(210)*2 - A076512(210)*A038566(210,2) = 35*2 - 8*11 = -18.

Jamie Morken, Table of n, a(n) for n = 1..13413 (Rows n = 1..210 of triangle, flattened)

Cf. A000010, A109395, A038566, A076511, A076512.

Flatten@ Table[With[{a = n/GCD[n, #], b = Numerator[#/n]}, MapIndexed[a First@ #2 - b #1 &, Flatten@ Position[GCD[Table[Mod[k, n], {k, n - 1}], n], 1] /. {} -> {1}]] &@ EulerPhi@ n, {n, 15}] (* Michael De Vlieger, Jun 06 2019 *)

vtot(n) = select(x->(gcd(n, x)==1), vector(n, k, k));
row(n) = my(q = eulerphi(n)/n, v = vtot(n)); vector(#v, k, denominator(q)*k - numerator(q)*v[k]); \\ Michel Marcus, May 14 2019

cp's OEIS Frontend

A284411 Least prime p such that more than half of all integers are divisible by n distinct primes not greater than p.

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A286941 Irregular triangle read by rows: the n-th row corresponds to the totatives of the n-th primorial, A002110(n).

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A053144 Cototient of the n-th primorial number.

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A091440 Smallest number m such that m#/phi(m#) >= n, where m# indicates the primorial (A034386) of m and phi is Euler's totient function.

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A342479 a(n) is the numerator of the asymptotic density of numbers whose second smallest prime divisor (A119288) is prime(n).

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A342480 a(n) is the denominator of the asymptotic density of numbers whose second smallest prime divisor (A119288) is prime(n).

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A256968 Let b(n) = Product_{i=1..n} p_i/(p_i - 1), p_i = i-th prime; a(n) = minimum k such that b(k) >= n.

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A308121 Irregular triangle read by rows: T(n,k) = A109395(n)k-A076512(n)A038566(n,k).