A062743 -id:A062743

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A038623

Smallest prime p such that p/pi(p)>=n.

+10
16

2, 2, 37, 127, 347, 1087, 3109, 8419, 24317, 64553, 175211, 480881, 1304707, 3523901, 9558533, 25874843, 70115473, 189961529, 514272533, 1394193607, 3779851091, 10246935679, 27788566133, 75370121191, 204475052401, 554805820711, 1505578023841, 4086199302113 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	LINKS	`Giovanni Resta, Table of n, a(n) for n = 1..50`
	FORMULA	`a(n) = exp(n + 1 + o(1)). - Charles R Greathouse IV, Oct 15 2016`
	EXAMPLE	`pi(37)=12 and a(3)=37 is the smallest prime >= 3*12.`
	MATHEMATICA	`Prime[Join[{k = 1}, Table[While[Prime[k]/k < n, k++]; k, {n, 2, 18}]]] (* Jayanta Basu, Jul 10 2013 *)`
	PROG	`(PARI) k=n=1; forprime(p=2, , while(p/k>=n, print1(p", "); n++); k++) \\ Charles R Greathouse IV, Oct 15 2016`
	CROSSREFS	`Cf. A038623-A038627.` `Essentially the same as A062743.` `a(n) = prime(A038624(n)).`
	KEYWORD	`nonn`
	AUTHOR	`Jud McCranie`
	EXTENSIONS	`Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar` `a(24)-a(28) from David W. Wilson, Apr 25 2017` `a(29)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018`
	STATUS	`approved`

A062742

Index j of prime p(j) such that floor(p(j)/j) = n is first satisfied.

+10
7

2, 1, 12, 31, 69, 181, 443, 1052, 2701, 6455, 15928, 40073, 100362, 251707, 637235, 1617175, 4124437, 10553415, 27066974, 69709680, 179992909, 465769803, 1208198526, 3140421716, 8179002096, 21338685407, 55762149030, 145935689361 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	LINKS	`Giovanni Resta, Table of n, a(n) for n = 1..50`
	FORMULA	`a(n) = Min_{j\| floor(p(j)/j) = n}. Note that neither p(j)/j nor floor(p(j)/j) is monotonic.` `a(n) = pi(A062743(n)).` `a(n) = A038606(n) = A038624(n) for n >= 3. - Jaroslav Krizek, Dec 13 2009`
	EXAMPLE	`The q(j)=p(j)/j quotient when the value 14 first appears: {j=251706, p(j)=3523841, q(j)=13.9998291} {251707, 3523901, 14.0000119} {251708, 3523903, 13.9999642} {251709, 3523921, 13.9999801} {251710, 3523957, 14.0000675} {251711, 3523963, 14.0000357}`
	PROG	`(PARI) {a062742(m)=local(n, j); for(n=1, m, j=1; while(floor(prime(j)/j)!=n, j++); print1(j, ", "))} a062742(10^7)`
	CROSSREFS	`Essentially the same as A038624.` `Cf. A038606. - R. J. Mathar, Jan 30 2009`
	KEYWORD	`nonn`
	AUTHOR	`Labos Elemer, Jul 12 2001`
	EXTENSIONS	`More terms from Jason Earls, May 15 2002` `a(17)-a(28) from Farideh Firoozbakht and Robert G. Wilson v, Sep 13 2005` `a(29)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018`
	STATUS	`approved`

A062357

a(n) = n*p(n+1)-(n+1)*p(n) = n*d(n)-p(n), where p(n) is the n-th prime and d(n) is the n-th prime-difference, A001223(n).

+10
2

-1, 1, 1, 9, -1, 11, -3, 13, 31, -9, 35, 11, -15, 13, 43, 43, -25, 47, 9, -31, 53, 9, 55, 103, 3, -49, 5, -51, 7, 307, -3, 61, -71, 201, -79, 65, 65, -11, 67, 67, -97, 239, -105, -17, -107, 353, 353, -31, -129, -29, 73, -135, 289, 73, 73, 73, -155, 77, -41, -161, 327, 575, -55, -183, -53, 607, 71, 343, -209, -69, 73, 217 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,4`
	COMMENTS	`A sequence based on the solution of the equation: 1+(1+n)prime(n)/x-nprime(n+1)/x=0 for x. This is an irrational rotation-like sequence: the sequence is similar to a Beatty sequence. - Roger L. Bagula, Jun 06 2002`
	LINKS	`Harry J. Smith, Table of n, a(n) for n = 1..1000`
	FORMULA	`a(n) = nA000040(n+1) - (n+1)A000040(n) = n*A001223(n) - A000040(n).`
	EXAMPLE	`n = 10: a(10) = 1031-1129 = 310-319 = -9;` `n = 54: a(54) = 54257-55251 = 13878-13805 = 73;` `n = 55: a(55) = 55263-56257 = 14465-14392 = 73; consecutive terms are often equal to each other.`
	MAPLE	`seq(nithprime(n+1)-(n+1)ithprime(n), n=1..80); # Muniru A Asiru, Jun 29 2018`
	MATHEMATICA	`Table[(Prime[w+1]-Prime[w])*w-Prime[w], {w, 1, 1024}]`
	PROG	`(PARI) a(n)={nprime(n + 1) - (n + 1)prime(n)} \\ Harry J. Smith, Aug 06 2009` `(Magma) [nNthPrime(n + 1) - (n + 1)NthPrime(n): n in [1..75]]; // Vincenzo Librandi, Jun 29 2018`
	CROSSREFS	`Cf. A000040, A001223, A062742, A062743.`
	KEYWORD	`sign,easy`
	AUTHOR	`Labos Elemer, Jul 13 2001`
	STATUS	`approved`

A364635

a(n) is the largest prime p such that p/PrimePi(p) < n.

+10
0

7, 31, 113, 359, 1129, 3089, 8467, 24281, 64717, 175141, 481447, 1304713, 3524621, 9560081, 25874773, 70119967, 189969349, 514282961, 1394199299, 3779856617, 10246936393, 27788573801, 75370126379, 204475055189, 554805820519, 1505578026059, 4086199303001, 11091501632977 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`2,1`
	COMMENTS	`Sequence begins at a(2) because there exists no prime p such that p/PrimePi(p) < 1.`
	LINKS	`Table of n, a(n) for n=2..29.`
	FORMULA	`a(n) = prime(A102281(n)). - Michel Marcus, Sep 10 2023`
	EXAMPLE	`a(4) = 113 because 113/PrimePi(113) = 113/30 = 3.766... but p/PrimePi(p) >= 4 for all primes > 113.`
	CROSSREFS	`Cf. A000720, A038625, A062743, A102281.`
	KEYWORD	`nonn`
	AUTHOR	`Jon E. Schoenfield, Sep 09 2023`
	STATUS	`approved`

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