A000720 -id:A000720 - cp's OEIS frontend

A259789 Least integer k > 1 such that pi(k)pi(kn) is a square, where pi(.) is the prime-counting function given by A000720.

2, 27, 8, 2, 2, 9, 3, 5, 96, 10, 9, 2, 2, 2, 28, 4, 9, 11, 8, 195, 3, 3, 723, 28, 573, 225, 2, 2, 2, 35, 46, 132, 4, 4, 65, 14, 58, 11, 8, 967, 311, 10, 98, 3, 3, 21, 94, 20, 2, 2, 28, 23, 30, 16, 29, 3419, 134, 4, 251, 7

Offset: 1

Zhi-Wei Sun, Jul 05 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, every positive rational number r can be written as m/n, where m and n are positive integers with pi(m)*pi(n) a positive square.

For example, 25/32 = 13102500/16771200 with pi(13102500)*pi(16771200) = 855432*1077512 = 921738245184 = 960072^2, and 49/58 = 1076068567/1273713814 with pi(1076068567)*pi(1273713814) = 54511776*63975626 = 3487424993971776 = 59054424^2.

			a(1) = 2 since pi(2)*pi(2*1) = 1^2.
a(2) = 27 since pi(27)*pi(27*2) = 9*16 = 12^2.
a(8) = 5 since pi(5)*pi(5*8) = 3*12 = 6^2.
a(9) = 96 since pi(96)*pi(96*9) = 24*150 = 60^2.
a(675) = 1465650 since pi(1465650)*pi(1465650*675) = 111747*50331648 = 5624410669056 = 2371584^2.
a(946) = 1922745 since pi(1922745)*pi(1922745*946) = 143599*89749375 = 12887920500625 = 3589975^2.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..60
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
Zhi-Wei Sun, A new theorem on the prime-counting function, Ramanujan J. 42(2017), 59-67. (See also arXiv:1409.5685 [math.NT], 2014.)
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000040, A000290, A000720, A259712.

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[k=1; Label[bb]; k=k+1; If[SQ[PrimePi[k]*PrimePi[k*n]], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue,{n,1,60}]
liks[n_]:=Module[{k=2},While[!IntegerQ[Sqrt[PrimePi[k]PrimePi[k*n]]],k++];k]; Array[liks,60] (* Harvey P. Dale, Jul 12 2024 *)

main(size) = {v=vector(size);for(t=1,size,v[t]=1;until(issquare(primepi(v[t])*primepi(t*v[t])),v[t]++));return(v);} \\ Anders Hellström, Jul 05 2015

A291538 a(n) = PrimePi(n^3) - PrimePi(n)^3, where PrimePi = A000720.

Original entry on oeis.org

0, 3, 1, 10, 3, 20, 4, 33, 65, 104, 92, 144, 111, 184, 260, 348, 313, 422, 370, 495, 635, 786, 728, 904, 1092, 1291, 1498, 1731, 1707, 1961, 1897, 2181, 2486, 2806, 3152, 3490, 3466, 3851, 4267, 4685, 4653, 5111, 5045, 5549, 6066, 6617, 6541, 7124, 7723, 8359, 9007, 9685, 9650, 10383, 11106, 11859, 12669, 13487, 13498, 14374

Offset: 1

Jonathan Sondow, following a suggestion from Altug Alkan, Aug 25 2017

nonn

All terms are positive except a(1) = 0, by the PNT with error term for large n and computation for smaller n. In particular, PrimePi(n^3) > PrimePi(n)^3 for n > 1. Indeed, by A291539 and A291540, PrimePi(n^3) > PrimePi(n) * PrimePi(n^2) > PrimePi(n)^3 for n > 7.
For prime(n)^3 - prime(n^3), see A262199.
For PrimePi(n^2) - PrimePi(n)^2, see A291440.

			a(3) = PrimePi(3^3) - PrimePi(3)^3 = 9 - 2^3 = 1.

Cf. A000720, A123914, A262199, A291440, A291539, A291540, A291541, A291542.

Table[ PrimePi[n^3] - PrimePi[n]^3, {n, 60}]

a(n) = primepi(n^3) - primepi(n)^3; \\ Michel Marcus, Sep 10 2017

a(n) = A000720(n^3) - A000720(n)^3.
a(n) = A291539(n) + A291540(n).
a(n) ~ (n^3 / log(n))*(1/3  - 1/log(n)^2) as n tends to infinity.

A332422 If n = Product (p_j^k_j) then a(n) = Sum ((-1)^(pi(p_j) + 1) * pi(p_j)), where pi = A000720.

Original entry on oeis.org

0, 1, -2, 1, 3, -1, -4, 1, -2, 4, 5, -1, -6, -3, 1, 1, 7, -1, -8, 4, -6, 6, 9, -1, 3, -5, -2, -3, -10, 2, 11, 1, 3, 8, -1, -1, -12, -7, -8, 4, 13, -5, -14, 6, 1, 10, 15, -1, -4, 4, 5, -5, -16, -1, 8, -3, -10, -9, 17, 2, -18, 12, -6, 1, -3, 4, 19, 8, 7, 0, -20, -1

Offset: 1

Ilya Gutkovskiy, Feb 12 2020

sign

Sum of odd indices of distinct prime factors of n minus the sum of even indices of distinct prime factors of n.

			a(66) = a(2 * 3 * 11) = a(prime(1) * prime(2) * prime(5)) = 1 - 2 + 5 = 4.

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A002129, A071321, A112798, A066328, A195017, A316524, A325699, A332423.

a[n_] := Plus @@ ((-1)^(PrimePi[#[[1]]] + 1) PrimePi[#[[1]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 72}]
nmax = 72; CoefficientList[Series[Sum[(-1)^(k + 1) k x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^prime(k) / (1 - x^prime(k)).

A335294 a(n) = pi(n) - pi(Sum_{k=1..n-1} a(k)) with a(1) = 1, where pi() is the prime counting function A000720.

Original entry on oeis.org

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

Offset: 1

Altug Alkan, Jun 01 2020

Alkan, Booker, & Luca prove that every nonnegative integer appears infinitely often.

Robert Israel, Table of n, a(n) for n = 1..10000
Altug Alkan, Andrew R. Booker, and Florian Luca, On a recursively defined sequence involving the prime counting function, arXiv:2006.08013 [math.NT], 2020.

Cf. A000720, A335337 (when k appears), A334714 (partial sums).

A[1]:= 1: S:= 1:
for n from 2 to 100 do
  A[n]:= numtheory:-pi(n) - numtheory:-pi(S);
  S:= S + A[n];
od:
seq(A[n],n=1..100); # Robert Israel, Jun 01 2020

a[1] = 1; a[n_] := a[n] = PrimePi[n] - PrimePi[Sum[a[k], {k, 1, n-1}]]; Array[a, 100] (* Amiram Eldar, Jun 01 2020 *)
upto[nn_] := Reap[Block[{i=0, is=0, sp=2, p=2, s=1}, Sow@ 1; Do[ If[n == p, i++; p = NextPrime@ p]; Sow[i - is]; s += i - is; While[ s >= sp, is++; sp = NextPrime@ sp], {n, 2, nn}]]] [[2, 1]]; upto[97] (* Giovanni Resta, Jun 02 2020 *)

a=vector(10^2); a[1] = 1; for(n=2, #a, a[n] = primepi(n) - primepi(sum(k=1, n-1, a[k]))); a

first(n) = {my(res = vector(n), pp = 0, s = 1, ps=0); primepivec = vector(n); forprime(p = 2, n, primepivec[p] = 1; ); for(i = 2, n, primepivec[i] += primepivec[i-1] ); res[1] = 1; for(i = 2, n, if(isprime(i), pp++); res[i] = pp - ps; s+=(pp-ps); ps = primepivec[s]; ); res } \\ David A. Corneth, Jun 01 2020

from sympy import primepi
A = [1]
S = 1
for n in range(1, 101):
    A += [primepi(n+1) - primepi(S), ]
    S += A[n]
print(A) # Indranil Ghosh, Jun 21 2020, after Maple

A073798 pi(n) is a power of 2, where pi(n) = A000720(n) is the number of primes <= n.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 10, 19, 20, 21, 22, 53, 54, 55, 56, 57, 58, 131, 132, 133, 134, 135, 136, 311, 312, 719, 720, 721, 722, 723, 724, 725, 726, 1619, 1620, 3671, 3672, 8161, 8162, 8163, 8164, 8165, 8166, 17863, 17864, 17865, 17866, 17867, 17868, 17869, 17870

Offset: 1

Labos Elemer, Aug 14 2002

nonn

The numbers occur in blocks of consecutive integers: 2, 3-4, 7-10, 19-22, ...; the n-th block starts at the 2^n-th prime (A033844) and ends just before the (2^n + 1)-th prime (A051439).

			10 is in the sequence since pi(10)=4=2^2.

Chai Wah Wu, Table of n, a(n) for n = 1..1103 (n = 1..665 from Ivan Neretin)

Cf. A000079, A000720, A015910, A073797, A073799.

pow2[n_] := n==1||(n>1&&IntegerQ[n/2]&&pow2[n/2]); Select[Range[20000], pow2[PrimePi[ # ]]&]
Flatten@Table[Range[p = Prime[2^k], NextPrime[p] - 1], {k, 0, 11}] (* Ivan Neretin, Jan 21 2017 *)

isok(n) = my(pi = primepi(n)); (pi==1) || (pi==2) || (ispower(primepi(n),,&k) && (k==2)); \\ Michel Marcus, Jan 23 2017

Edited by Dean Hickerson, Aug 15 2002

A132090 a(n) = pi(pi(n)), where pi = A000720.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Sep 03 2007

nonn

a(n) = a(n-1) + 1 if n is in A006450, otherwise a(n) = a(n-1). - Robert Israel, Jan 20 2016

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000720, A006450, A000040, A000720, A137588.

a132090 = a000720 . a000720  -- Reinhard Zumkeller, Jun 23 2015

map(numtheory:-pi@@2,[$1..100]); # Robert Israel, Jan 20 2016

PrimePi@ PrimePi@ Range@ 105 (* Robert G. Wilson v, Jan 20 2016 *)

a(n)=primepi(primepi(n)) \\ Charles R Greathouse IV, Aug 12 2014

a(n) ~ n/log^2 n. - Charles R Greathouse IV, Aug 12 2014

A237496 Number of ordered ways to write n = k + m (0 < k <= m) with pi(k) + pi(m) - 2 prime, where pi(.) is given by A000720.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 4, 4, 3, 2, 3, 3, 3, 5, 3, 1, 4, 5, 5, 7, 4, 1, 2, 1, 1, 1, 1, 1, 3, 6, 7, 8, 8, 8, 8, 8, 9, 11, 11, 11, 11, 9, 7, 7, 4, 1, 2, 1, 2, 3, 5, 7, 10, 14, 14, 14, 10, 6, 10, 14, 16, 19, 16, 13, 12, 11, 10, 7, 6, 5, 3, 3, 4, 3, 6, 9, 13, 17, 18

Offset: 1

Zhi-Wei Sun, Feb 08 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 5.

(ii) Any integer n > 23 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m) prime. Also, each integer n > 25 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m) - 1 prime.

			a(6) = 1 since 6 = 3 + 3 with pi(3) + pi(3) - 2 = 2 + 2 - 2 = 2 prime.
a(17) = 1 since 17 = 2 + 15 with pi(2) + pi(15) - 2 = 1 + 6 - 2 = 5 prime.
a(99) = 1 since 99 = 1 + 98 with pi(1) + pi(98) - 2 = 0 + 25 - 2 = 23 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000

Cf. A000040, A000720, A232465, A237284, A237291, A237453, A237497.

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

A237614 Least positive integer k with A000720(k*n) divisible by n, or 0 if such a number k does not exist.

Original entry on oeis.org

1, 2, 2, 2, 6, 11, 11, 7, 3, 3, 3, 8, 13, 13, 8, 14, 14, 14, 33, 33, 9, 15, 9, 4, 4, 42, 22, 22, 43, 4, 36, 99, 10, 10, 10, 10, 38, 38, 38, 38, 31, 24, 17, 17, 17, 62, 24, 194, 55, 80, 11, 40, 11, 11, 11, 11, 11, 57, 11, 11, 33, 18, 18, 83, 164, 5, 5, 5, 156, 5

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

According to the conjecture in A237597, we should always have 0 < a(n) < prime(n).

			a(3) = 2 since pi(2*3) = 3 is divisible by 3, but pi(1*3) = 2 is not.

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

Cf. A000720, A237578, A237597, A237598, A237612.

Do[Do[If[Mod[PrimePi[k*n],n]==0,Print[n," ",k];Goto[aa]],{k,1,Prime[n]-1}];
Print[n," ",0];Label[aa];Continue,{n,1,70}]

A237656 Least positive integer m such that {A000720(k^2): k = 1, ..., m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

Original entry on oeis.org

1, 5, 3, 6, 8, 10, 18, 17, 30, 41, 28, 43, 29, 33, 43, 27, 66, 47, 98, 105, 155, 114, 113, 100, 49, 62, 118, 146, 85, 125, 80, 117, 74, 101, 167, 137, 168, 282, 176, 287, 129, 178, 151, 140, 163, 139, 262, 267, 277, 234, 285, 188, 203, 163, 192, 239, 188, 241, 252, 252

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

Conjecture: a(n) is always positive. Moreover, a(n) < 2*prime(n+1) - 2 for all n > 0.

Note that a(21) = 155 = 2*prime(22) - 3.

			a(5) = 8 since {A000720(k^2): k = 1, ..., 8} = {0, 2, 4, 6, 9, 11, 15, 18} contains a complete system of residues modulo 5, but {A000720(k^2): k = 1, ..., 7} contains no integer congruent to 3 modulo 5.

Chai Wah Wu, Table of n, a(n) for n = 1..1011 (n = 1..100 from Zhi-Wei Sun)
Zhi-Wei Sun, On a^n+bn modulo m, preprint, arXiv:1312.1166 [math.NT], 2013-2014.

Cf. A000290, A000720, A038107, A237643.

q[m_,n_]:=Length[Union[Table[Mod[PrimePi[k^2],n],{k,1,m}]]]
Do[Do[If[q[m,n]==n,Print[n," ",m];Goto[aa]],{m,n,2*Prime[n+1]-3}];
Print[n," ",0];Label[aa];Continue,{n,1,60}]

A249809 Irregular table read by rows: T(n, k) is the number of times prime p_k has occurred as the smallest prime factor of numbers 1 .. n. (T(1,1) = 0, and for each n > 1, k = 1 .. A000720(n)).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 4, 2, 1, 1, 5, 2, 1, 1, 5, 2, 1, 1, 1, 6, 2, 1, 1, 1, 6, 2, 1, 1, 1, 1, 7, 2, 1, 1, 1, 1, 7, 3, 1, 1, 1, 1, 8, 3, 1, 1, 1, 1, 8, 3, 1, 1, 1, 1, 1, 9, 3, 1, 1, 1, 1, 1, 9, 3, 1, 1, 1, 1, 1, 1, 10, 3, 1, 1, 1, 1, 1, 1, 10, 4, 1, 1, 1, 1, 1, 1, 11, 4, 1, 1, 1, 1, 1, 1, 11, 4, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Nov 06 2014

After the first row {0}, consists of rows of triangular table A249808 with trailing zeros removed.

			Table begins:
       k=1  2  3  4  5  6  7
  n=1:   0;
  n=2:   1;
  n=3:   1, 1;
  n=4:   2, 1;
  n=5:   2, 1, 1;
  n=6:   3, 1, 1;
  n=7:   3, 1, 1, 1;
  n=8:   4, 1, 1, 1;
  n=9:   4, 2, 1, 1;
  n=10:  5, 2, 1, 1;
  n=11:  5, 2, 1, 1, 1;
  n=12:  6, 2, 1, 1, 1;
  n=13:  6, 2, 1, 1, 1, 1;
  n=14:  7, 2, 1, 1, 1, 1;
  n=15:  7, 3, 1, 1, 1, 1;
  n=16:  8, 3, 1, 1, 1, 1;
  n=17:  8, 3, 1, 1, 1, 1, 1;
  ...

Antti Karttunen, Table of n, a(n) for n = 1..10016; the first 294 rows of the table, flattened
Orges Leka, Fractal of prime factorization of integers, lexicographically sorted.
Orges Leka, How to define a fractal from the lexicographic sorting on the prime factorization of natural numbers?

A004526 gives the left edge, A001477 the row sums.

Cf. A000040, A000720, A020639, A249808, A249727, A249728, A055396, A078898.

(define (A249809 n) (A249808bi (A249728 n) (A249727 n))) ;; Code for A249808bi given in A249808.

a(n) = A249808(A249728(n), A249727(n)).

For n > 1, A078898(n) = T(n, A055396(n)).

cp's OEIS Frontend

A259789 Least integer k > 1 such that pi(k)*pi(k*n) is a square, where pi(.) is the prime-counting function given by A000720.

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A291538 a(n) = PrimePi(n^3) - PrimePi(n)^3, where PrimePi = A000720.

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A332422 If n = Product (p_j^k_j) then a(n) = Sum ((-1)^(pi(p_j) + 1) * pi(p_j)), where pi = A000720.

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A335294 a(n) = pi(n) - pi(Sum_{k=1..n-1} a(k)) with a(1) = 1, where pi() is the prime counting function A000720.

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A073798 pi(n) is a power of 2, where pi(n) = A000720(n) is the number of primes <= n.

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A132090 a(n) = pi(pi(n)), where pi = A000720.

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A237496 Number of ordered ways to write n = k + m (0 < k <= m) with pi(k) + pi(m) - 2 prime, where pi(.) is given by A000720.

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A259789 Least integer k > 1 such that pi(k)pi(kn) is a square, where pi(.) is the prime-counting function given by A000720.