A057809 -id:A057809 - cp's OEIS frontend

A076243 Remainder when 3rd-order prime ppp(n) = A038580(n) is divided by n.

0, 1, 1, 3, 2, 5, 4, 3, 8, 9, 5, 7, 10, 5, 7, 3, 2, 11, 17, 1, 20, 21, 11, 19, 12, 17, 14, 17, 18, 19, 18, 23, 28, 27, 11, 19, 15, 7, 2, 21, 40, 25, 31, 1, 19, 15, 9, 31, 46, 47, 10, 15, 43, 23, 14, 9, 17, 19, 18, 41, 24, 27, 50, 3, 14, 29, 13, 3, 4, 39, 21, 1, 47, 19, 31, 13, 6, 17

Offset: 1

Labos Elemer, Oct 08 2002

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240, A076241, A076242.

MapIndexed[Mod[#1, First@ #2] &, Nest[Prime, Range@ 79, 3]] (* Michael De Vlieger, Jul 22 2017 *)

a(n) = ppp(n) mod n = A038580(n) mod n.

A087240 First differences of A087235.

Original entry on oeis.org

25, 87, 240, 774, 1960, 5378, 15828, 40420, 110477, 306255, 823267, 2219935, 6035446, 16314684, 44245201, 119849369, 324308909, 879921037, 2385657333, 6467079803, 17541637367, 47581552613, 129104928784, 350330765356, 950772205549, 2580621276899, 7005302330033

Offset: 2

Labos Elemer, Sep 04 2003

nonn

Giovanni Resta, Table of n, a(n) for n = 2..49

Cf. A038623-A038627, A057809, A087235-A087241.

a(n)=A087235(n+1)-A087235(n)

More terms from Giovanni Resta, Sep 01 2018

A256394 Prime values of pi(n) that divide n.

Original entry on oeis.org

2, 3, 11, 67, 71, 439, 1051, 6469, 40087, 100361, 100363, 251737, 251761, 637319, 637327, 4124459, 10553513, 10553551, 27067277, 69709733, 179993171, 465769817, 3140421769, 8179002109, 8179002133, 55762149029, 55762149071, 382465573489, 1003652347081

Offset: 1

Jonathan Sondow, Apr 13 2015

nonn

a(n) is the largest prime factor of n, since pi(n) ~ n / log n.

			pi(6) = 3 is prime, and 3 divides 6, so 3 is a member.

Giovanni Resta, Table of n, a(n) for n = 1..49
K. Gaitanas, An explicit formula for the prime counting function, arXiv:1311.1398 [math.NT], 2013.
K. Gaitanas, An Explicit Formula for the Prime Counting Function Which is Valid Infinitely Often, Amer. Math. Monthly, 122 (2015), 283.
S. W. Golomb, On the Ratio of N to pi(N), American Mathematical Monthly, 69 (1962), 36-37.
R. T. Harger and W. L. Hightower, An Interesting Property of x/pi(x), College Math. J., 40 (2009), 213-214.
Eric Weisstein's World of Mathematics, Prime Counting Function

Cf. A000720, A006530, A038623-A038627, A057809, A057810, A071394, A087235-A087241.

c = 0; lpf[n_] := If[ PrimeQ[n], c++; n, Transpose[ FactorInteger[n]][[1, -1]]]; Do[ If[lpf[n] == c, Print[ PrimePi[n]]], {n, 2, 10^7}]
PrimePi[Select[Select[Range[2,10^6],IntegerQ[#/PrimePi[#]]&],PrimeQ[PrimePi[#]]&]] (* Ivan N. Ianakiev, Apr 15 2015 *)
Select[Table[{PrimePi[n],n},{n,10^6}],PrimeQ[#[[1]]]&&Divisible[#[[2]],#[[1]]]&][[All,1]] (* The program generates the first 9 terms of the sequence. To generate more, increase the constant for n. *) (* Harvey P. Dale, Feb 08 2022 *)

for(n=1,10^6,if(isprime(p=primepi(n))&&!(n%primepi(n)),print1(p,", "))) \\ Derek Orr, Apr 14 2015

a(n) = A000720(A071394(n)) = A006530(A071394(n)).

More terms from Giovanni Resta, Sep 01 2018

A076241 Remainder when 2nd order prime pp(n)=A006450(n) is divided by n.

Original entry on oeis.org

0, 1, 2, 1, 1, 5, 3, 3, 2, 9, 6, 1, 10, 9, 1, 1, 5, 13, 8, 13, 10, 5, 17, 5, 9, 1, 23, 27, 19, 17, 27, 3, 14, 15, 19, 13, 31, 17, 16, 31, 38, 37, 35, 27, 31, 21, 28, 17, 12, 47, 43, 43, 39, 31, 26, 45, 13, 1, 17, 23, 17, 53, 11, 15, 1, 53, 10, 25, 64, 41, 38, 41, 68, 33, 59, 63, 65

Offset: 1

Labos Elemer, Oct 08 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240-A076243.

[NthPrime(NthPrime(n)) mod(n): n in [1..100]]; // Vincenzo Librandi, Jul 10 2017

Table[Mod[Prime[Prime[n]], n], {n, 100}] (* Vincenzo Librandi, Jul 10 2017 *)

a(n) = prime(prime(n)) % n; \\ Michel Marcus, Jul 09 2017

a(n) = A006450(n) mod n.

A076242 Remainder when 3rd order prime A038580(n) is divided by n-th prime=A000040(n).

Original entry on oeis.org

1, 2, 1, 3, 6, 10, 5, 8, 17, 19, 27, 31, 38, 35, 28, 39, 17, 17, 10, 38, 68, 63, 13, 55, 48, 4, 74, 100, 37, 29, 47, 121, 115, 136, 105, 28, 128, 109, 159, 90, 114, 31, 151, 4, 86, 108, 81, 147, 149, 189, 185, 119, 231, 166, 88, 238, 197, 233, 64, 186, 258, 111, 128, 260

Offset: 1

Labos Elemer, Oct 08 2002

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240-A076243.

Table[Mod[Prime[Prime[Prime[n]]],Prime[n]],{n,70}] (* Harvey P. Dale, Sep 28 2013 *)

a(n) = Mod[A038580(n), A000040(n)]

A087267 a(n) = gcd(n, pi(n)) where pi is A000720.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 1, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 4, 1, 2, 1, 3, 1, 1, 9, 1, 1, 10, 1, 1, 11, 1, 1, 1, 1, 2, 3, 4, 1, 1, 1, 2, 1, 2, 1, 3, 1, 5, 3, 1, 1, 2, 1, 8, 1, 2, 1, 1, 1, 2, 9, 2, 1, 6, 1, 1, 1, 1, 1, 4, 1, 1, 3, 1, 7, 3, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 4, 3, 2, 1, 24, 1, 1, 1, 25, 1, 2, 1

Offset: 1

Labos Elemer, Sep 16 2003

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A007053, A000720, A087266, A057809.

Mathematica
```
Table[GCD[w, PrimePi[w]], {w, 1, 256}]
```

a(n) = gcd(n, primepi(n)); \\ Michel Marcus, Apr 22 2018

A087269 Nonprime solutions to gcd(x, pi(x)) = gcd(x, A000720(x)) = 1.

Original entry on oeis.org

1, 9, 12, 18, 21, 25, 26, 28, 32, 34, 35, 36, 42, 45, 49, 52, 55, 57, 60, 65, 68, 69, 70, 74, 76, 81, 84, 85, 86, 87, 88, 91, 95, 98, 99, 104, 106, 110, 111, 112, 119, 121, 128, 129, 130, 133, 135, 141, 143, 145, 147, 155, 158, 159, 160, 161, 162, 165, 170, 172, 177

Offset: 1

Labos Elemer, Sep 16 2003

What is the density of this sequence? - David A. Corneth, Oct 21 2019

			There are 37 primes below the nonprime 162, so pi(162) = 37 and as gcd(162, pi(162)) = gcd(162, 37) = 1, 162 is in the sequence. - _David A. Corneth_, Oct 21 2019

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000720, A007053, A057809, A087266, A087267, A087268.

t=Table[GCD[w, PrimePi[w]], {w, 1, 1000}]; f=Flatten[Position[t, 1]]; cf=Part[f, Flatten[Position[PrimeQ[f], False]]]

first(n) = {n = max(n, 2); my(q = 2, i = 1, t = 1, res = vector(n)); res[1] = 1; forprime(p = 3, oo, for(j = q + 1, p - 1, if(gcd(t, j) == 1, i++; if(i <= n, res[i] = j; , return(res); ) ) ); t++; q = p ) } \\ David A. Corneth, Oct 21 2019

A058011 Consider the sequence {b(m)} of nonprimes; sequence gives values of m where gcd{m, b(m)} increases.

Original entry on oeis.org

1, 4, 6, 8, 27, 30, 33, 96, 100, 120, 330, 335, 340, 350, 355, 360, 1008, 1080, 1092, 1116, 1122, 1128, 1134, 3059, 3066, 3073, 3080, 3087, 3094, 8408, 8424, 8440, 8456, 8464, 8472, 23526, 23535, 24300, 64540, 64580, 64610, 64620, 64650, 64690, 64700

Offset: 0

Robert G. Wilson v, Nov 13 2000

nonn

			a(4) = 8 because eight is the fourth nonprime number and GCD(4,8) > GCD(3,6).

Cf. A058012. Apart from initial term same as A057809.

r=0; Do[a = GCD[n, n - PrimePi[n] ]; If[a > r, r = a; Print[n]], {n, 1, 10^5} ]

A087268 Solutions to gcd(x, pi(x)) = 1, where pi is A000720.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 12, 13, 17, 18, 19, 21, 23, 25, 26, 28, 29, 31, 32, 34, 35, 36, 37, 41, 42, 43, 45, 47, 49, 52, 53, 55, 57, 59, 60, 61, 65, 67, 68, 69, 70, 71, 73, 74, 76, 79, 81, 83, 84, 85, 86, 87, 88, 89, 91, 95, 97, 98, 99, 101, 103, 104, 106, 107, 109, 110, 111

Offset: 1

Labos Elemer, Sep 16 2003

nonn

			All primes are included.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A007053, A000720, A087266, A087267, A057809.

Flatten[Position[Table[GCD[w, PrimePi[w]], {w, 1, 256}], 1]]

isok(n) = gcd(n, primepi(n)) == 1; \\ Michel Marcus, Apr 22 2018

A065136 Numbers n such that n = pi(n)*k + 1 for some k.

Original entry on oeis.org

3, 9, 11, 13, 28, 34, 37, 43, 121, 336, 341, 351, 356, 361, 1081, 1087, 1135, 3060, 3074, 3081, 3088, 3095, 8409, 8425, 8441, 8457, 8465, 8473, 23527, 23536, 24301, 64541, 64581, 64591, 64601, 64611, 64651, 64661, 64691, 64701, 64711, 64721

Offset: 1

Labos Elemer, Oct 15 2001

nonn

Solutions to Mod[n,PrimePi[n]] = 1, i.e. A065134(n) = 1.

			n=28: Pi(28)=9 and 28=3*Pi(28)+1, so 28 is here; n=27 is present in A057809. A large proportion of A057809(m)+1 numbers (but not all of them) arise in this sequence. Numbers from A057809 arise in clusters [see grouping around 8450, 64650, 480900 etc.]

Giovanni Resta, Table of n, a(n) for n = 1..1172

A000720, A065134, A057809.

cp's OEIS Frontend

A076243 Remainder when 3rd-order prime ppp(n) = A038580(n) is divided by n.

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A087240 First differences of A087235.

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A256394 Prime values of pi(n) that divide n.

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A076241 Remainder when 2nd order prime pp(n)=A006450(n) is divided by n.

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A076242 Remainder when 3rd order prime A038580(n) is divided by n-th prime=A000040(n).

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A087267 a(n) = gcd(n, pi(n)) where pi is A000720.

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A087269 Nonprime solutions to gcd(x, pi(x)) = gcd(x, A000720(x)) = 1.

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A058011 Consider the sequence {b(m)} of nonprimes; sequence gives values of m where gcd{m, b(m)} increases.

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A087268 Solutions to gcd(x, pi(x)) = 1, where pi is A000720.