A062742 -id:A062742 - cp's OEIS frontend

A038624 Values of pi(x) where x exceeds n * pi(x).

1, 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

Offset: 1

Jud McCranie

nonn

"Exceeds" can be interpreted as ">" or ">=" since the corresponding primes are never multiples of their indices. - R. J. Mathar, Jun 08 2008
Equivalently, a(n) = minimal k such that prime(k)/k >= n. - Enoch Haga, Oct 19 2007
a(n) = A062742(n) = A038606(n) for n >= 3. - Jaroslav Krizek, Dec 13 2009

			x exceeds 3*pi(x) when pi(x)=12, so a(3)=12

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A038623-A038627.
Essentially the same as A062742.
Cf. A038606 (variant).

Join[{k = 1}, Table[While[Prime[k]/k < n, k++]; k, {n, 2, 18}]] (* Jayanta Basu, Jul 10 2013 *)

a(24)-a(28) from Robert G. Wilson v, Sep 26 2005
Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar.
a(29)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A038606 Least k such that k-th prime > n * k.

Original entry on oeis.org

1, 5, 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, 382465573483, 1003652347100

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net) 1998 Jul

nonn

Log(a(n)) =~ -1.295 + 0.964312n. - Robert G. Wilson v, Jan 25 2002
Numbers n such that prime(n) (mod n) begins the next cycle of terms in A004648. Generally prime(i) (mod i) exceeds prime(i-1) (mod i-1) but there are numerous times where for a short run prime(i) (mod i) is minimally less than its predecessor. Here n is substantially less. See Labos's graph.
A090973(a(n)) = n+1. [From Reinhard Zumkeller, Aug 16 2009]
With offset 2: Index j of prime p(j) such that ceiling[p(j)/j]=n is first satisfied. a(n) = A062742(n) = A038624(n) for n >= 3. [From Jaroslav Krizek, Dec 13 2009]

Giovanni Resta, Table of n, a(n) for n = 1..50
Labos, E. Graph of first 50000 terms of A004648
Andrew R. Booker, The Nth Prime Page

Cf. A038607, A004648, A038625.

A038606 := proc(n)
    for k from 1 do
        if ithprime(k)> n*k then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Aug 24 2013

k = 1; Do[ While[ Floor[ Prime[k]/k] < n, k++ ]; Print[k]; k++, {n, 1, 30} ]

k=1;n=1;forprime(p=3,4e9,if(p/n++>k,print1(n", ");k++)) \\ Charles R Greathouse IV, Sep 06 2011

a(n) = pi(A038607(n)) = A000720(A038607(n)).

Edited by Robert G. Wilson v, Jan 25 2002
a(21)=179992909 corrected by Ray Chandler, Dec 01 2004
a(29)-a(30) from Charles R Greathouse IV, Sep 06 2011
a(31)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A062743 Smallest prime prime(m) such that floor(prime(m)/m) = n.

Original entry on oeis.org

3, 2, 37, 127, 347, 1087, 3109, 8419, 24317, 64553, 175211, 480881, 1304707, 3523901, 9558533, 25874843, 70115473, 189961529, 514272533, 1394193607, 3779851091, 10246935679, 27788566133, 75370121191, 204475052401

Offset: 1

Labos Elemer, Jul 12 2001

nonn

a(n+1)/a(n) -> e as n -> infinity, as do the m's.

Giovanni Resta, Table of n, a(n) for n = 1..50 (first 29 terms from Robert G. Wilson v)

Essentially the same as A038623.

Cf. A038607, A038625.

Do[ k = 1; While[ Floor[ Prime[m]/ m] != n, m++ ]; Print[Prime[k] ], {n, 1, 27} ]

A062742(n) = pi(a(n)).

More terms from Robert G. Wilson v, Jul 13 2001
a(27) from Farideh Firoozbakht, Sep 12 2005
Corrected by T. D. Noe, Nov 14 2006
a(30)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Sep 01 2018

A062357 a(n) = np(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).

Original entry on oeis.org

-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

Offset: 1

Labos Elemer, Jul 13 2001

A sequence based on the solution of the equation: 1+(1+n)*prime(n)/x-n*prime(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

			n = 10: a(10) = 10*31-11*29 = 310-319 = -9;
n = 54: a(54) = 54*257-55*251 = 13878-13805 = 73;
n = 55: a(55) = 55*263-56*257 = 14465-14392 = 73; consecutive terms are often equal to each other.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A001223, A062742, A062743.

[n*NthPrime(n + 1) - (n + 1)*NthPrime(n): n in [1..75]]; // Vincenzo Librandi, Jun 29 2018

seq(n*ithprime(n+1)-(n+1)*ithprime(n),n=1..80); # Muniru A Asiru, Jun 29 2018

Table[(Prime[w+1]-Prime[w])*w-Prime[w], {w, 1, 1024}]

a(n)={n*prime(n + 1) - (n + 1)*prime(n)} \\ Harry J. Smith, Aug 06 2009

a(n) = n*A000040(n+1) - (n+1)*A000040(n) = n*A001223(n) - A000040(n).

A072916 Number of m such that floor(prime(m)/m) = n.

Original entry on oeis.org

3, 8, 19, 41, 117, 254, 616, 1642, 3766, 9461, 24183, 60252, 151368, 385600, 979844, 2507393, 6428977, 16513542, 42642649, 110283280, 285776799, 742428731, 1932223170, 5038580446, 13159683245, 34423463648, 90173540312

Offset: 1

Zak Seidov Aug 11 2002

nonn

			Only m = 2,3,4 give [p(m)/m] = 1, so a(1) = 3.
There are 8 values of m giving floor(prime(m)/m) = 2, namely m = 1,5,6,7,8,9,10,11, so a(2) = 8.

Cf. A062742, A102281.

a(n_) := Length[Cases[Table[Floor[Prime[m]/m], {m, 1, 1000000}], n]]

a(16)-a(27) from Farideh Firoozbakht, Sep 13 2005

Typo corrected by David W. Wilson, Oct 22 2005

A090974 Duplicate of A038606.

Original entry on oeis.org

0, 1, 5, 12, 31, 69, 181, 443, 1052, 2701, 6455, 15928, 40073, 100362, 251707, 637235

Offset: 0

dead

With offset 1: Index j of prime p(j) such that ceiling[p(j)/j]=n is first satisfied for n >= 2. a(n) = A038606(n) = A062742(n) = A038624(n) for n >= 3. [From Jaroslav Krizek, Dec 13 2009]

cp's OEIS Frontend

A038624 Values of pi(x) where x exceeds n * pi(x).

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A038606 Least k such that k-th prime > n * k.

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A062743 Smallest prime prime(m) such that floor(prime(m)/m) = n.

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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).

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Magma

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A072916 Number of m such that floor(prime(m)/m) = n.

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A090974 Duplicate of A038606.

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A062357 a(n) = np(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).