A038111 -id:A038111 - cp's OEIS frontend

A084969 Numbers whose smallest prime factor is 11.

11, 121, 143, 187, 209, 253, 319, 341, 407, 451, 473, 517, 583, 649, 671, 737, 781, 803, 869, 913, 979, 1067, 1111, 1133, 1177, 1199, 1243, 1331, 1397, 1441, 1507, 1529, 1573, 1639, 1661, 1727, 1793, 1837, 1859, 1903, 1969, 1991, 2057, 2101, 2123, 2167, 2189, 2299, 2321

Offset: 1

Robert G. Wilson v, Jun 15 2003

Fifth row of A083140.

Integers k such that gcd(11*k, 210) = 1.

			a(2) = 11*11, a(3) = 11*13.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A084967 (5), A084968 (7), A084970 (13), A332799 (17), A332798 (19), A332797 (23), A008364 (11-rough numbers).

Cf. A008364, A038110, A038111, A083140.

11Select[ Range[210], GCD[ #, 2*3*5*7] == 1 & ]
Select[11*Range[0,200],GCD[#,210]==1&] (* Harvey P. Dale, Dec 23 2013 *)

is(n)=gcd(n,2310)==11 \\ Charles R Greathouse IV, Nov 19 2014

G.f.: 11*x*(x^48 +10*x^47 +2*x^46 +4*x^45 +2*x^44 +4*x^43 +6*x^42 +2*x^41 +6*x^40 +4*x^39 +2*x^38 +4*x^37 +6*x^36 +6*x^35 +2*x^34 +6*x^33 +4*x^32 +2*x^31 +6*x^30 +4*x^29 +6*x^28 +8*x^27 +4*x^26 +2*x^25 +4*x^24 +2*x^23 +4*x^22 +8*x^21 +6*x^20 +4*x^19 +6*x^18 +2*x^17 +4*x^16 +6*x^15 +2*x^14 +6*x^13 +6*x^12 +4*x^11 +2*x^10 +4*x^9 +6*x^8 +2*x^7 +6*x^6 +4*x^5 +2*x^4 +4*x^3 +2*x^2 +10*x +1) / (x^49 -x^48 -x +1). - Colin Barker, Feb 22 2013
a(n) = a(n-48) + 2310 = a(n-1) + a(n-48) - a(n-49). - Charles R Greathouse IV, Nov 19 2014
Lim_{n->infinity} a(n)/n = A038111(5)/A038110(5) = 385/8 = 48.125. - Vladimir Shevelev, Jan 20 2015
a(n) = 11*A008364(n).

a(47)-a(49) from Georg Fischer, Nov 07 2019

New name from Frank Ellermann, Feb 25 2020

A088821 a(n) is the sum of smallest prime factors of numbers from 1 to n.

Original entry on oeis.org

0, 2, 5, 7, 12, 14, 21, 23, 26, 28, 39, 41, 54, 56, 59, 61, 78, 80, 99, 101, 104, 106, 129, 131, 136, 138, 141, 143, 172, 174, 205, 207, 210, 212, 217, 219, 256, 258, 261, 263, 304, 306, 349, 351, 354, 356, 403, 405, 412, 414, 417, 419, 472, 474, 479, 481, 484

Offset: 1

Labos Elemer, Oct 22 2003

nonn

M. Kalecki, On certain sums extended over primes or prime factors, Prace Mat, Vol. 8 (1963), pp. 121-127.
J. Sandor, D. S. Mitrinovic, B. Crstici, Handbook of Number Theory I, Volume 1, Springer, 2005, Chapter IV, p. 121.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Project Euler, Problem 521 - Smallest prime factor

Cf. A020639, A046669, A088822, A088823, A088824, A088825.

P:=List(List([2..60],n->Factors(n)),i->i[1]);;
a:=Concatenation([0],List([1..Length(P)],i->Sum([1..i],k->P[k]))); # Muniru A Asiru, Nov 29 2018

Prepend[Accumulate[Rest[Table[FactorInteger[i][[1,1]],{i,60}]]],0] (* Harvey P. Dale, Jan 09 2011 *)

a(n) = sum(k=2, n, factor(k)[1,1]); \\ Michel Marcus, May 15 2017

a(n) ~ n^2/(2 log n) [Kalecki]. - Thomas Ordowski, Nov 29 2018

a(n) = Sum_{prime p} n(p)*p, where n(p) is the number of integers in [1,n] with smallest prime factor spf(.) = A020639(.) = p, decreasing from n(2) = floor(n/2) to n(p) = 1 for p >= sqrt(n), possibly earlier, and n(p) = 0 for p > n. One has n(p) ~ D(p)*n where D(p) = (Product_{primes q < p} 1-1/q)/p = A038110/A038111 is the density of numbers having p as smallest prime factor. - M. F. Hasler, Dec 05 2018

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

Original entry on oeis.org

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

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

A309497 Irregular triangle read by rows: T(n,k) = A060753(n)k-A038110(n)A286941(n,k).

Original entry on oeis.org

0, 1, 2, 1, 11, 2, 1, 8, 7, 14, 13, 4, 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37, 24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11, -2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8

Offset: 0

Jamie Morken, Aug 05 2019

The sequence is Primorial rows of A308121.
Row n has length A005867(n).
Row n > 1 average value = A060753(n)/2.
Row n > 1 has sum = A002110(n-1)*A038110(n)/2.
First value on row(n) = A161527(n-1).
Last value on row(n) = A038110(n) for n > 2.
For n > 1, A060753(n) = Max(row) + Min(row).
For values x and y on row n > 1 at positions a and b on the row:
x + y = A060753(n), where a = A005867(n-1) - (b-1).
For n > 2 the penultimate value on row A002110(n) is given by
  A038110(n)*A000040(n)-A060753(n).
Related identity:
  A038110(n)/A038111(n)*(Prime(n)^2) - (A038110(n)/A038111(n)*((A038110(n)*Prime(n) - A060753(n))*Prime(n)/A038110(n))) = 1.

			The triangle starts:
row1: 0;
row2: 1;
row3: 2, 1;
row4: 11, 2, 1, 8, 7, 14, 13, 4;
row5: 27, -18, 1, 4, 23, 26, 13, 32, 19, 22, 41, 44, 31, 18, 37, 24, 27, 46, 33, 36, 23, -6, -3, 16, 19, 38, 41, 12, -1, 2, -11, 8, 11, -2, 17, 4, -9, -6, 13, 16, 3, 22, 9, 12, 31, 34, 53, 8;

Michael De Vlieger, Table of n, a(n) for n = 0..6299 (rows n = 0..6, flattened)
Eric Weisstein's World of Mathematics, Totative

Cf. A058250, A005867, A002110, A038110, A038111, A060753, A286941, A058262, A161527, A083140, A308121.

row[0] = 0; row[n_] := -(v = Numerator[Product[1 - 1/Prime[i], {i, 1, n}] / Prime[n]] * Select[Range[(p = Product[Prime[i], {i, 1, n}])], CoprimeQ[p, #] &]) + Denominator[Product[((pr = Prime[i]) - 1)/pr, {i, 1, n}]] * Range[Length[v]]; Table[row[n], {n, 0, 4}] // Flatten (* Amiram Eldar, Aug 10 2019 *)

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]

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.

A378720 a(n) is the numerator of the asymptotic density of numbers whose third smallest prime divisor is prime(n).

Original entry on oeis.org

0, 0, 1, 1, 4, 326, 628, 992, 98304, 125568, 733440, 281163264, 386427322368, 3178249003008, 12454223855616, 6450728943845376, 342348724735967232, 20218431581110665216, 39814891891080560640, 82739188294287768944640, 15336676441718784000, 61298453882755419734016000

Offset: 1

Robert G. Wilson v and Amiram Eldar, Dec 05 2024

The third smallest prime divisor of a number k is the third member in the ordered list of the distinct prime divisors of k. Only numbers in A000977 have a third smallest prime divisor.

The partial sums of the fractions first exceed 1/2 after summing 4467 terms. Therefore, the median value of the distribution of the third prime divisor is prime(4467) = 42719 = A284411(3).

			The fractions begin with 0/1, 0/1, 1/30, 1/30, 4/165, 326/15015, 628/36465, 992/62985, 98304/7436429, 125568/11849255, ..., .
a(1) = a(2) = 0 since there are no numbers whose third prime divisor is 2 or 3.
a(3)/A378721(3) = 1/30 since the numbers whose third prime divisor is 5 are the numbers that are divisible by 2, 3 and 5, and their density if (1/2)*(1/3)*(1/5) = 1/30.
a(4)/A378721(4) = 1/30 since the numbers whose third prime divisor is 7 are the union of the numbers that are divisible by 2, 3 and 7 and not by 5 whose density is (1/2)*(1/3)*(1-1/5)*(1/7) = 2/105, the numbers that are divisible by 2, 5 and 7 and not by 3 whose density is (1/2)*(1-1/3)*(1/5)*(1/7) = 1/105, and the numbers that are divisible by 3, 5 and 7 and not by 2 whose density is (1-1/2)*(1/3)*(1/5)*(1/7) = 1/210, and 2/105 + 1/105 + 1/210 = 1/30.

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..365
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.
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. A000040, A002110, A000977, A038110, A038111, A284411, A342479, A342480, A378721 (denominators).

a[n_] := Block[{p, q = Prime@ Range@ n}, p = Fold[Times, 1, q]; q = Most@ q; Plus @@ Times @@@ Subsets[q -1, {n -3}]/p]; a[1] = 0; Numerator@ Array[a, 22]

a(n) = {my(v = primes(n), q = vecextract(apply(x -> x-1, v),"^-1"), p = vecprod(v), prd = vecprod(q)/p, sm = 0, sb); forsubset([#q, 2], s, sb = vecextract(q, s); sm += 1/vecprod(sb)); numerator(prd * sm);}

a(n)/A378721(n) = (1/prime(n)#) * (Product_{k=1..n-1} (prime(k) - 1)) * Sum_{j=1..n-1, i=1..j-1} 1/((prime(i)-1)*(prime(j)-1)), where prime(n)# = A002110(n) is the n-th primorial number.
Sum_{n>=1} a(n)/A378721(n) = 1.
Sum_{n=1..m} a(n)/A378721(n) > 1/2 for m >= 4467 = primepi(A284411(3)).

A378721 a(n) is the denominator of the asymptotic density of numbers whose third smallest prime divisor is prime(n).

Original entry on oeis.org

1, 1, 30, 30, 165, 15015, 36465, 62985, 7436429, 11849255, 73465381, 33426748355, 50708377254535, 436092044389001, 1863302371480277, 1086305282573001491, 64092011671807087969, 3909612711980232366109, 8449808119441147371913, 18598027670889965365580513, 3543193335582015099413

Offset: 1

Robert G. Wilson v and Amiram Eldar, Dec 05 2024

See A378720 for more details.

Amiram Eldar, Table of n, a(n) for n = 1..365

Cf. A000040, A038110, A038111, A342479, A342480, A378720 (numerators).

a[n_] := Block[{p, q = Prime@ Range@ n}, p = Fold[Times, 1, q]; q = Most@ q; Plus @@ Times @@@ Subsets[q -1, {n -3}]/p]; a[1] = 0; Denominator@ Array[a, 21]

a(n) = {my(v = primes(n), q = vecextract(apply(x -> x-1, v),"^-1"), p = vecprod(v), prd = vecprod(q)/p, sm = 0, sb); forsubset([#q, 2], s, sb = vecextract(q, s); sm += 1/vecprod(sb)); denominator(prd * sm);}

cp's OEIS Frontend

A084969 Numbers whose smallest prime factor is 11.

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A088821 a(n) is the sum of smallest prime factors of numbers from 1 to n.

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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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A053144 Cototient of the n-th primorial number.

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A309497 Irregular triangle read by rows: T(n,k) = A060753(n)*k-A038110(n)*A286941(n,k).

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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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A309497 Irregular triangle read by rows: T(n,k) = A060753(n)k-A038110(n)A286941(n,k).