A000720 -id:A000720 - cp's OEIS frontend

A237497 a(n) = |{0 < k <= n/2: pi(k*(n-k)) is prime}|, where pi(.) is given by A000720.

0, 0, 0, 2, 2, 1, 2, 1, 1, 0, 1, 4, 3, 1, 1, 1, 3, 2, 6, 2, 2, 2, 4, 1, 1, 3, 3, 3, 1, 3, 3, 7, 4, 5, 4, 6, 5, 5, 3, 3, 3, 5, 7, 4, 1, 6, 7, 7, 5, 4, 1, 2, 3, 5, 5, 6, 8, 8, 6, 4, 9, 8, 6, 3, 7, 9, 6, 5, 4, 10, 5, 4, 6, 6, 4, 9, 10, 6, 8, 7

Offset: 1

Zhi-Wei Sun, Feb 08 2014

nonn

Conjecture: a(n) > 0 for all n > 10, and a(n) = 1 for no n > 51. Moreover, for any integer n > 10, there is a positive integer k < n with 2*k + 1 and pi(k*(n-k)) both prime.

			a(6) = 1 since 6 = 1 + 5 with pi(1*5) = 3 prime.
a(8) = 1 since 8 = 2 + 6 with pi(2*6) = pi(12) = 5 prime.
a(25) = 1 since 25 = 4 + 21 with pi(4*21) = pi(84) = 23 prime.
a(51) = 1 since 51 = 14 + 37 with pi(14*37) = pi(518) = 97 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A000720, A237284, A237291, A237453, A237496.

p[k_,m_]:=PrimeQ[PrimePi[k*m]]
a[n_]:=Sum[If[p[k,n-k],1,0],{k,1,n/2}]
Table[a[n],{n,1,80}]

A237597 a(n) = |{0 < k < prime(n): n divides pi(k*n)}|, where pi(.) is given by A000720.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 3, 3, 2, 4, 3, 3, 5, 7, 1, 3, 3, 5, 2, 5, 4, 4, 5, 5, 3, 7, 3, 2, 3, 4, 8, 4, 2, 6, 4, 5, 6, 8, 7, 2, 8, 2, 7, 1, 3, 6, 4, 6, 5, 1, 7, 4, 4, 3, 5, 6, 4, 8, 6, 5, 2, 5, 8, 4, 2, 5, 7, 5, 3, 1, 3, 2, 6, 3, 2, 4

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

Conjecture: a(n) > 0 for all n > 0.

See also A237614 for the least k > 0 with pi(k*n) divisible by n.

			a(6) = 1 since pi(11*6) = 3*6 with 11 < prime(6) = 13.
a(19) = 1 since pi(33*19) = 6*19 with 33 < prime(19) = 67.
a(759) = 1 since pi(2559*759) = 191*759 with 2559 < prime(759) = 5783.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2500

Cf. A000040, A000720, A237578, A237598, A237612, A237614.

a[n_]:=Sum[If[Mod[PrimePi[k*n],n]==0,1,0],{k,1,Prime[n]-1}]
Table[a[n],{n,1,80}]

A247604 Least integer m > 0 with pi(m*n) = sigma(m+n), where pi(.) and sigma(.) are given by A000720 and A000203.

Original entry on oeis.org

18, 11, 360, 251, 168, 36, 6, 285, 1185, 792, 29, 11, 245078, 5, 1869, 46074, 573, 42863, 11, 5, 8129, 60806, 1443, 452, 15, 39298437, 386891, 1041920, 1290489, 17630, 35569, 10, 8174777, 3152500, 4291325, 57880072, 55991485, 127358, 93462807, 93314912

Offset: 5

Zhi-Wei Sun, Sep 21 2014

nonn

Conjecture: a(n) exists for every n = 5,6,...

			a(5) = 18 since pi(5*18) = 24 = sigma(5+18).

Zhi-Wei Sun and Hiroaki Yamanouchi, Table of n, a(n) for n = 5..52 (terms a(5)-a(40) from Zhi-Wei Sun)
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Cf. A000203, A000720, A247600, A247601, A247602, A247603, A247672, A247673.

Do[m=1;Label[aa];If[PrimePi[n*m]==DivisorSigma[1,m+n],Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];
Label[bb];Continue,{n,5,40}]

a(41)-a(44) from Hiroaki Yamanouchi, Oct 04 2014

A047886 Triangle read by rows: T(n,k) = pi(n+k) - pi(n) - pi(k), where pi() = A000720 (n >= 0, 0 <= k <= n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, -1, -1, -2, 0, 1, 0, -1, -1, -1, -1, 0, 0, -1, -2, -1, -2, -1, -2, 0, 0, -1, -1, -1, -1, -1, -2, -2, 0, 0, 0, -1, 0, -1, -1, -2, -1, -1, 0, 1, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, -1, -1, -2, -1, -2, -1, -1, -1, -2, 0, 1, 0, -1

Offset: 0

N. J. A. Sloane

T(n,0)=0; for n > 0: T(n,1)=A010051(n); T(n,n)=-A060208(n). - Reinhard Zumkeller, Apr 15 2008

A212210-A212213 are the preferred versions of this array.

			Triangle begins
  0;
  0,  1;
  0,  1,  0;
  0,  0,  0, -1;
  0,  1,  0,  0,  0;
  0,  0,  0, -1, -1, -2;
  ...

Cf. A047885, A212210-A212213.

Flatten[Table[PrimePi[n+k]-PrimePi[n]-PrimePi[k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Feb 22 2012 *)

More terms from James Sellers, Dec 22 1999

A099802 Bisection of A000720.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 9, 10, 11, 11, 11, 12, 12, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 19, 19, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 30, 30, 30, 30, 30, 31, 31, 32, 32, 32, 33, 34, 34, 34, 34, 34

Offset: 1

N. J. A. Sloane, Nov 19 2004

Maximal number of primes possible in a string of 2n consecutive numbers. - Lekraj Beedassy, Dec 04 2004
a(n) = A139325(n,1) + 1. - Reinhard Zumkeller, Apr 14 2008
Or the number of primes <= 2n. - Juri-Stepan Gerasimov, Oct 29 2009

Cf. A099081.

with(numtheory):seq(pi(2*n),n=1..86); # Emeric Deutsch, Apr 12 2005

a(n)=primepi(2*n) \\ Charles R Greathouse IV, Sep 02 2015

[prime_pi(2*n) for n in range(1, 75)] # Zerinvary Lajos, Jun 06 2009

a(n) = A000720(n) + A035250(n) - A010051(n). - Reinhard Zumkeller, Jul 05 2010

a(n) = A000720(2*n). - Wesley Ivan Hurt, Jun 16 2013

More terms from Emeric Deutsch, Apr 12 2005

A103357 Numbers n such that n and pi(n) (A000720) are palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 262, 323, 393, 525, 535, 555, 666, 818, 878, 949, 2002, 3773, 5775, 6116, 13031, 19591, 39093, 41414, 47374, 59295, 63236, 81918, 94549, 95759, 252252, 394493, 594495, 662266, 674476, 686686, 698896, 764467

Offset: 1

Zak Seidov, Feb 02 2005

Giovanni Resta, Table of n, a(n) for n = 1..196 (terms < 10^16)

Corresponding palindromic pi(n) in A103358.

Cf. A046941, A046942, A103358, A103402, A103403.

NextPalindrome[n_] := Block[ {l = Floor[ Log[10, n] + 1], idn = IntegerDigits[n]}, If[ Union[ idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[ idn, Ceiling[l/2]]]] FromDigits[ Take[ idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[ idn, Ceiling[l/2]], Reverse[ Take[ idn, Floor[l/2]]] ]], idfhn = FromDigits[ Take[ idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[ idfhn], Drop[ Reverse[ IntegerDigits[ idfhn]], Mod[l, 2]]]] ]]]];
p = 0; a = {}; Do[p = NextPalindrome[ p]; q = IntegerDigits[ PrimePi[ p]]; If[ Reverse[q] == q, Print[{p, FromDigits[q]}]; AppendTo[a, p]], {n, 10^4}]; a (* Robert G. Wilson v, Feb 03 2005 *)

a(n) = P_A103358(n).

More terms from Robert G. Wilson v, Feb 03 2005

A237453 Number of primes p < n with p*n + pi(p) prime, where pi(.) is given by A000720.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 3, 2, 1, 2, 1, 2, 3, 3, 2, 3, 1, 1, 1, 3, 2, 4, 3, 3, 3, 2, 1, 2, 1, 1, 3, 3, 1, 2, 3, 3, 3, 4, 3, 3, 2, 2, 6, 4, 3, 5, 3, 2, 3, 2, 4, 4, 3, 1, 3, 5, 2, 5, 3, 1, 2, 3, 2, 4, 2, 3, 2

Offset: 1

Zhi-Wei Sun, Feb 08 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 4, and a(n) = 1 for no n > 144. Moreover, for any positive integer n, there is a prime p < sqrt(2*n)*log(5n) with p*n + pi(p) prime.
(ii) For each integer n > 8, there is a prime p <= n + 1 with (p-1)*n - pi(p-1) prime.
(iii) For every n = 1, 2, 3, ... there is a positive integer k < 3*sqrt(n) with k*n + prime(k) prime.
(iv) For each n > 13, there is a positive integer k < n with k*n + prime(n-k) prime.
We have verified that a(n) > 0 for all n = 5, ..., 10^8.

			a(3) = 1 since 2 and 2*3 + pi(2) = 6 + 1 = 7 are both prime.
a(10) = 1 since 5 and 5*10 + pi(5) = 50 + 3 = 53 are both prime.
a(107) = 1 since 89 and 89*107 + pi(89) = 9523 + 24 = 9547 are both prime.
a(144) = 1 since 59 and 59*144 + pi(59) = 8496 + 17 = 8513 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A000720, A233296, A237284, A237291, A237496, A237497.

a[n_]:=Sum[If[PrimeQ[Prime[k]*n+k],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

vector(100, n, sum(k=1, primepi(n-1), isprime(prime(k)*n+k))) \\ Colin Barker, Feb 08 2014

A237768 Number of primes p < n with pi(n-p) a Sophie Germain prime, where pi(.) is given by A000720.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 3, 2, 2, 2, 1, 2, 3, 2, 3, 3, 2, 2, 1, 1, 3, 3, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 5, 5, 4, 4, 4, 3, 4, 4, 4, 4, 3, 3, 4, 4, 4, 4, 3, 3, 4, 4, 3, 3, 2, 2, 3, 3, 1, 1, 3, 3, 5, 5, 2, 2, 1, 1, 3, 3, 4, 4, 3, 3, 4, 4, 4, 4, 1

Offset: 1

Zhi-Wei Sun, Feb 13 2014

nonn

Conjecture: a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 12, 20, 21, 26, 27, 30, 31, 32, 60, 61, 68, 69, 80, 81.
This is stronger than part (i) of the conjecture in A237705.
We have verified that a(n) > 0 for all n = 5, ..., 2*10^7.

			a(5) = 1 since 2, pi(5-2) = pi(3) = 2 and 2*2 + 1 = 5 are all prime.
a(12) = 1 since 7, pi(12-7) = pi(5) = 3 and 2*3 + 1 = 7 are all prime.
a(81) = 1 since 47, pi(81-47) = pi(34) = 11 and 2*11 + 1 = 23 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A000720, A005384, A237284, A237705, A237706.

sg[n_]:=PrimeQ[n]&&PrimeQ[2n+1]
a[n_]:=Sum[If[sg[PrimePi[n-Prime[k]]],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

A237769 Number of primes p < n with pi(n-p) - 1 and pi(n-p) + 1 both prime, where pi(.) is given by A000720.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 2, 2, 3, 3, 3, 4, 3, 4, 4, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 3, 3, 1, 1, 2, 2, 3, 4, 3, 3, 4, 3, 5, 5, 3, 3, 2, 2, 5, 5, 3, 3, 3, 3, 5, 5, 2, 2, 3, 3, 3, 4, 2, 2, 6, 6, 9, 8, 4, 4, 3, 3, 6, 6, 5, 5, 4, 4, 7

Offset: 1

Zhi-Wei Sun, Feb 13 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 8, and a(n) = 1 only for n = 9, 34, 35.
(ii) For any integer n > 4, there is a prime p < n such that 3*pi(n-p) - 1, 3*pi(n-p) + 1 and 3*pi(n-p) + 5 are all prime. Also, for each integer n > 8, there is a prime p < n such that 3*pi(n-p) - 1, 3*pi(n-p) + 1 and 3*pi(n-p) - 5 are all prime.
(iii) For any integer n > 6, there is a prime p < n such that phi(n-p) - 1 and phi(n-p) + 1 are both prime, where phi(.) is Euler's totient function.

			a(9) = 1 since 2, pi(9-2) - 1 = 3 and pi(9-2) + 1 = 5 are all prime.
a(34) = 1 since 19, pi(34-19) - 1 = pi(15) - 1 = 5 and pi(34-19) + 1 = pi(15) + 1 = 7 are all prime.
a(35) = 1 since 19, pi(35-19) - 1 = pi(16) - 1 = 5 and pi(35-19) + 1 = pi(16) + 1 = 7 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000720, A001359, A006512, A014574, A022004, A022005, A237705, A237706, A237768.

TQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
a[n_]:=Sum[If[TQ[PrimePi[n-Prime[k]]],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

A247673 Least integer m > 0 with pi(m*n) = sigma(m) + sigma(n), where pi(.) and sigma(.) are given by A000720 and A000203 respectively.

Original entry on oeis.org

23, 47, 359, 25, 11, 33, 9, 17, 182, 11, 15, 304, 12, 160, 6105, 444, 22676, 408, 5, 60, 8, 17888, 9, 125526, 1616818, 334976, 22584, 19548, 10, 286780, 21540, 6698792, 640720, 2466378, 75999272, 646104, 573678, 801525615, 1116040868, 3565308, 127408112

Offset: 5

Zhi-Wei Sun, Sep 22 2014

nonn

Conjecture: a(n) exists for every n = 5, 6, ... .

			a(5) = 23 since pi(5*23) = 30 = sigma(5) + sigma(23).

Zhi-Wei Sun and Hiroaki Yamanouchi, Table of n, a(n) for n = 5..53 (terms a(5)-a(41) from Zhi-Wei Sun)
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Cf. A000203, A000720, A247600, A247601, A247602, A247603, A247604, A247672.

Do[m=1; Label[aa]; If[PrimePi[m*n]==DivisorSigma[1,m]+DivisorSigma[1,n], Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 5, 41}]

a(42)-a(45) from Hiroaki Yamanouchi, Oct 04 2014

cp's OEIS Frontend

A237497 a(n) = |{0 < k <= n/2: pi(k*(n-k)) is prime}|, where pi(.) is given by A000720.

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A237597 a(n) = |{0 < k < prime(n): n divides pi(k*n)}|, where pi(.) is given by A000720.

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A247604 Least integer m > 0 with pi(m*n) = sigma(m+n), where pi(.) and sigma(.) are given by A000720 and A000203.

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A047886 Triangle read by rows: T(n,k) = pi(n+k) - pi(n) - pi(k), where pi() = A000720 (n >= 0, 0 <= k <= n).

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A099802 Bisection of A000720.

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Maple

PARI

Sage

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A103357 Numbers n such that n and pi(n) (A000720) are palindromic.

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A237453 Number of primes p < n with p*n + pi(p) prime, where pi(.) is given by A000720.

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PARI

A237768 Number of primes p < n with pi(n-p) a Sophie Germain prime, where pi(.) is given by A000720.

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