A038607 -id:A038607 - cp's OEIS frontend

A038623 Smallest prime p such that p/pi(p)>=n.

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

Offset: 1

Jud McCranie

nonn

			pi(37)=12 and a(3)=37 is the smallest prime >= 3*12.

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

Cf. A038624-A038627.
Essentially the same as A062743,A038607.
a(n) = prime(A038624(n)).

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

k=n=1; forprime(p=2,, while(p/k>=n, print1(p", "); n++); k++) \\ Charles R Greathouse IV, Oct 15 2016

a(n) = exp(n + 1 + o(1)). - Charles R Greathouse IV, Oct 15 2016

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

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

A100478 Pentanacci pi function: a(1)=a(2)=a(3)=a(4)=a(5)=1; for n>5, a(n) = pi(Sum_{j=1..5} a(n-j)) where pi = A000720.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 4, 4, 6, 7, 9, 10, 11, 14, 15, 17, 19, 21, 23, 24, 27, 30, 30, 32, 34, 36, 37, 39, 40, 42, 44, 46, 47, 47, 48, 50, 51, 53, 53, 54, 55, 56, 58, 58, 60, 61, 62, 62, 62, 63, 63, 64, 65, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66

Offset: 1

Jonathan Vos Post, Nov 22 2004

Starting with other values of a(1), a(2), a(3), a(4), a(5) what behaviors are possible? Does the sequence always stick at a single integer after some point, or can it go into a loop, or is there a third pattern?

a(n) is equal to 66 for 54 <= n <= 10^7. - G. C. Greubel, Apr 06 2023

			a(6) = pi(a(1)+a(2)+a(3)+a(4)+a(5)) = pi(1+1+1+1+1) = pi(5) = 3.
a(7) = pi(a(2)+a(3)+a(4)+a(5)+a(6)) = pi(1+1+1+1+3) = pi(7) = 4.
a(8) = pi(a(3)+a(4)+a(5)+a(6)+a(7)) = pi(1+1+1+3+4) = pi(10) = 4.
a(9) = pi(a(4)+a(5)+a(6)+a(7)+a(8)) = pi(1+1+3+4+4) = pi(13) = 6.
a(10) = pi(a(5)+a(6)+a(7)+a(8)+a(9)) = pi(1+3+4+4+6) = pi(18) = 7.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Andrew Booker, The Nth Prime Page.
I. Flores, k-Generalized Fibonacci numbers, Fib. Quart., 5 (1967), 258-266.
V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.
Eric Weisstein's World of Mathematics, Prime Counting Function

Cf. A001591, A038607.

a[n_]:= a[n]= If[n<6,1,PrimePi[Sum[a[n-j], {j,5}]]];
Table[a[n], {n,80}] (* Robert G. Wilson v, Dec 03 2004 *)

@CachedFunction
def a(n): # a = A100478
    if (n<6): return 1
    else: return prime_pi(sum(a(n-j) for j in range(1,6)))
[a(n) for n in range(1, 81)] # G. C. Greubel, Apr 06 2023

a(n) = pi(a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5)) with a(1) = a(2) = a(3) = a(4) = a(5) = 1.

Edited and extended by Robert G. Wilson v, Dec 03 2004

A245071 a(n) = 12n - prime(n).

Original entry on oeis.org

10, 21, 31, 41, 49, 59, 67, 77, 85, 91, 101, 107, 115, 125, 133, 139, 145, 155, 161, 169, 179, 185, 193, 199, 203, 211, 221, 229, 239, 247, 245, 253, 259, 269, 271, 281, 287, 293, 301, 307, 313, 323, 325, 335, 343, 353, 353, 353, 361, 371, 379, 385, 395, 397, 403, 409, 415, 425

Offset: 1

Freimut Marschner, Jul 21 2014

Prime(n) > n for n > 0. Let prime(n) = k*n with k as an even integer constant, for example, k = 12; then a(n) = k*n - prime(n) is a sequence of odd integers that are positive as long as k*n > prime(n). This is the case up to a(40072) = 11. If k*n < prime(n) then a(n) < 0, a(40073) = -5 up to a(40083) = -5. From a(40084) = 5 up to a(40121) = 5, a(n) > 0 again, but a(n) < 0 for n >= 40122. For k = 12 the table shows this result compared with floor(prime(n)/n) and (prime(n) mod n) <= (prime(n+1) mod (n+1)) for n >= 1. Observations:
(1) If k > floor(prime(n)/n) then a(n) is positive.
(2) If k <= floor(prime(n)/n) and (prime(n) mod n) < (prime(n+1) mod (n+1)) and n > 1 then a(n) is negative.
(3) If k <= floor(prime(n)/n) and (prime(n) mod n) > (prime(n+1) mod (n+1)) then a(n) is positive.
.
n     prime(n) floor(prime(n)/n) (prime(n) mod n)  a(n)
40072 480853        12                 5            11
40073 480881        12                23            -5
40083 481001        11             40079            -5
40084 481003        11             40074             5
40121 481447        12                 5             5
40122 481469        12                13            -5

			a(3) = 12*3 - prime(3) = 36 - 5 = 31.

Freimut Marschner, Table of n, a(n) for n = 1..100000

A000040 (prime(n)), A038605 (floor(prime(n)/n)), A004648 (prime(n) mod n), A038606 (Least k such that k-th prime > n * k), A038607 (the smallest prime number k such that k > n*pi(k)), A102281 (the largest number m such that m = pi(n*m)).

Table[12n - Prime[n], {n, 60}] (* Alonso del Arte, Jul 27 2014 *)

PARI
```
vector(133, n, 12*n-prime(n) )
```

a(n) = 12*n - prime(n).

A111360 Integers k such that sigma(k) + prime(k) is divisible by k.

Original entry on oeis.org

1, 2, 3, 9, 21, 129, 5663, 40087, 184971, 246901, 251737, 1610143, 3098384123, 19819945093, 21323898091, 123112069843, 130057547087

Offset: 1

Ray G. Opao, Nov 07 2005

a(18) > 10^13, if it exists. - Giovanni Resta, Jan 05 2020

			The divisors of 21 are 1,3,7,21 and the 21st prime is 73. 1+3+7+21+73 = 105, which is divisible by 21.

Cf. A000203, A038607.

Select[Range[10^8], Mod[Prime[ # ] + Plus @@ Divisors[ # ], # ] == 0 &] (* Ray Chandler, Jan 24 2006 *)

p=2; for(n=1,100000000, if( (sigma(n)+p) % n == 0, print(n) ) ; p=nextprime(p+1) ; ) \\ R. J. Mathar, Feb 11 2008

a(13)-a(15) from Donovan Johnson, Apr 22 2008
New name from Michel Marcus, Dec 10 2019
a(16)-a(17) from Giovanni Resta, Dec 12 2019

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A038623 Smallest prime p such that p/pi(p)>=n.

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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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A100478 Pentanacci pi function: a(1)=a(2)=a(3)=a(4)=a(5)=1; for n>5, a(n) = pi(Sum_{j=1..5} a(n-j)) where pi = A000720.

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A245071 a(n) = 12n - prime(n).

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A111360 Integers k such that sigma(k) + prime(k) is divisible by k.

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