A038107 -id:A038107 - cp's OEIS frontend

A263100 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k^2) + pi(m^2/2), where pi(x) denotes the number of primes not exceeding x.

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

Offset: 1

Zhi-Wei Sun, Oct 09 2015

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 28.
(ii) Any integer n > 0 can be written as pi(k^2) + pi((m^2+1)/2) with k and m positive integers.
(iii) Each n = 1,2,3,... can be written as pi(k^2/2) + pi((m^2+1)/2) with k and m positive integers.
See also A262995, A262999, A263001 and A263020 for similar conjectures.

			a(1) = 1 since 1 = 0 + 1 = pi(1^2) + pi(2^2/2).
a(3) = 1 since 3 = 2 + 1 = pi(2^2) + pi(2^2/2).
a(28) = 1 since 28 = 11 + 17 = pi(6^2) + pi(11^2/2).

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

Cf. A000290, A000720, A038107, A262995, A262999, A263001, A263020.

s[n_]:=s[n]=PrimePi[n^2]
t[n_]:=t[n]=PrimePi[n^2/2]
Do[r=0; Do[If[s[k]>n, Goto[bb]]; Do[If[t[j]>n-s[k], Goto[aa]]; If[t[j]==n-s[k], r=r+1]; Continue, {j, 1, n-s[k]+1}]; Label[aa]; Continue, {k, 1, n}];
Label[bb]; Print[n, " ", r]; Continue, {n,1,100}]

A263321 Least positive integer m such that the numbers phi(k)*pi(k^2) (k = 1..n) are pairwise incongruent modulo m.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 16, 19, 19, 19, 29, 29, 29, 37, 37, 59, 59, 59, 59, 59, 59, 59, 59, 101, 101, 101, 133, 133, 133, 133, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 175, 245, 269, 269, 269, 269, 379, 379

Offset: 1

Zhi-Wei Sun, Oct 14 2015

nonn

Part (i) of the conjecture in A263319 implies that a(n) exists for any n > 0.

Conjecture: a(n) <= n^2 for all n > 0, and the only even term is a(7) = 16.

			a(7) = 16 since the 7 numbers phi(1)*pi(1^2) = 0, phi(2)*pi(2^2) = 2, phi(3)*pi(3^2) = 8, phi(4)*pi(4^2) = 12, phi(5)*pi(5^2) = 36, phi(6)*pi(6^2) = 22 and phi(7)*pi(7^2) = 90 are pairwise incongruent modulo 16, but not so modulo any positive integer smaller than 16.

Zhi-Wei Sun, Table of n, a(n) for n = 1..4000
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), 2794-2812.

Cf. A000010, A000290, A000720, A038107, A263317, A263319.

f[n_]:=f[n]=EulerPhi[n]*PrimePi[n^2]
Le[n_,m_]:=Le[m,n]=Length[Union[Table[Mod[f[k],m],{k,1,n}]]]
Do[n=1;m=1;Label[aa];If[Le[n,m]==n,Goto[bb],m=m+1;Goto[aa]];
Label[bb];Print[n," ",m];If[n<50,n=n+1;Goto[aa]]]

A380331 a(n) = number of primes < n^4.

Original entry on oeis.org

0, 0, 6, 22, 54, 114, 210, 357, 564, 847, 1229, 1715, 2334, 3107, 4052, 5191, 6542, 8152, 10022, 12187, 14683, 17531, 20768, 24421, 28546, 33118, 38236, 43934, 50203, 57097, 64683, 72992, 82025, 91932, 102588, 114204, 126726, 140235, 154787, 170426, 187134

Offset: 0

Clark Kimberling, Jan 21 2025

nonn

			a(2) = 6 because there are 6 primes < 16.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Subsequence of A038107, and hence of A000720.

Cf. A000040, A000583.

Mathematica
```
Table[PrimePi[n^4], {n, 0, 60}]
```

a(n)=primepi(n^4) \\ Charles R Greathouse IV, Jan 21 2025

from sympy import primepi
def A380331(n): return primepi(n**4) # Chai Wah Wu, Jan 23 2025

a(n) = A000720(A000583(n)). - Pontus von Brömssen, Jan 21 2025

a(n) ~ (1/4)*n^4/log n. - Charles R Greathouse IV, Jan 23 2025

A060304 Number of primes below n^3 does not exceed n times the number of primes below n^2.

Original entry on oeis.org

0, 0, 3, 6, 15, 19, 37, 47, 69, 82, 113, 139, 180, 216, 244, 300, 381, 423, 486, 553, 638, 726, 820, 887, 1029, 1152, 1256, 1376, 1527, 1659, 1794, 1992, 2156, 2357, 2517, 2739, 2909, 3085, 3365, 3627, 3933, 4200, 4380, 4687, 4960, 5313, 5547, 5917, 6395

Offset: 0

Labos Elemer, Mar 26 2001

nonn

			n=10, 10*pi(100)=250, pi(1000)=168, a(10)=250-168=82.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Segal, On π(x+y)<=π(x)+π(y), Transactions American Mathematical Society, 104 (1962), 523-527.

Cf. A060199, A060208, A000720.

Table[n*PrimePi[n^2]-PrimePi[n^3], {n, 1, 100}]

a(n) = n*pi(n*n) - pi(n*n*n). - Jonathan Sondow, Feb 17 2014

a(n) = n*A038107(n) - A038098(n). - Michel Marcus, Feb 17 2014

A163516 a(n) = floor( Sum_{x=2..n} x/log(x) ).

Original entry on oeis.org

0, 2, 5, 8, 11, 14, 18, 22, 26, 30, 35, 40, 45, 50, 56, 61, 67, 74, 80, 87, 94, 101, 108, 116, 123, 131, 140, 148, 157, 165, 175, 184, 193, 203, 213, 223, 233, 243, 254, 265, 276, 287, 299, 310, 322, 334, 346, 359, 371, 384, 397, 410, 424, 437, 451, 465, 479, 493

Offset: 1

Cino Hilliard, Jul 30 2009

nonn

a(n) closely approximates the number of primes < n^2, that is, A038107(n) = Pi(n^2).
In fact, the sum is as good as Li(n^2). For n = 10^9,
  a(n)    = 24739954333817884.
  Pi(n^2) = 24739954287740860 = A006880(18).
  Li(n^2) = 24739954309690415 = A057754(18) = A089896(18).
  R(n^2)  = 24739954284239494 = A057793(18).
Now x/(log(x)-1) is a much better approximation of Pi(x) than x/log(x).
  10^18/(log(10^18)-1) = 24723998785919976 and
  10^18/log(10^18)     = 24127471216847323.
Ironically though, a(n) = Sum_{x=2..n} x/(log(x)-1) is far from Pi(n^2), see A058290.

			For n = 10, floor(Sum_{x=2..n} x/log(x)) = 30, the 10th term.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Table[Floor[Sum[j/Log[j], {j, 2, n}]], {n,1,50}] (* G. C. Greubel, Jul 27 2017 *)
Join[{0},Floor[Accumulate[Table[x/Log[x],{x,2,60}]]]] (* Harvey P. Dale, May 22 2021 *)

nthsum(n) = for(j=1,n,print1(floor(sum(x=2,j,x/log(x)))","));

a(10^n) = A163521(n).

Offset corrected, definition detailed, 7 references to other sequences added by R. J. Mathar, Aug 29 2009

A213649 Smallest k such that there exists a square between prime(n) and prime(n+k).

Original entry on oeis.org

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

Offset: 1

Michel Lagneau, Jun 17 2012

nonn

a(A038107(n)) = 1 for n >= 2.

a(n) is of the form {S1} union {S2} union ... union {Sk} union ... where a subset Sk is of the form {xk, xk - 1, xk - 2, …, 1 }. We obtain a subsequence Max {Sn} = {xn} = {2, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 5, 5, 4, 6, 7, 5, 6, 6, 7, 7, 7, 6, 9, …}.

			a(7)=3 because prime(7) = 17, prime(7+3) = 29 and  17 < 25 < 29 where 25 is square.

Michel Lagneau, Table of n, a(n) for n = 1..5000

Cf. A038107, A014085.

with(numtheory):for n from 1 to 100 do:ii:=0:for k from 1 to 100 while(ii=0) do:p1:=ithprime(n):p2:=ithprime(n+k):i:=0:for m from p1+1 to p2-1 do:c:=sqrt(m):if c=floor(c) then i:=i+1:else fi:od: if i<>0 then ii:=1:printf(`%d, `,k):else fi:od:od:

A307533 Primes p such that p+2 has exactly two distinct prime factors.

Original entry on oeis.org

13, 19, 31, 37, 43, 53, 61, 67, 73, 83, 89, 97, 109, 113, 127, 131, 139, 151, 157, 173, 181, 199, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 349, 353, 367, 373, 379, 389, 401, 409, 421, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541

Offset: 1

Paolo Galliani, Apr 13 2019

(13,31), (37,73), (157,751), (199,991) are pairs of emirps belonging to this sequence such that the lesser term of the pair is the reverse of the greater. Are there infinitely many such pairs?
Are there infinitely many triples in the sequence like (61,67,73) and (251,257,263), that is, infinitely many a(n) such that a(n+1)=a(n)+6 and a(n+2)=a(n)+12?
The triples found so far are (61,67,73), (251,257,263) and (367,373,379). The first terms of the triples found are 61, 251 and 367, which belong to the sequence A038107.

			61 is in the sequence because 61 + 2 = 63 has exactly two distinct prime factors (3 and 7).

Robert Israel, Table of n, a(n) for n = 1..10000

filter:= proc(n) isprime(n) and nops(numtheory:-factorset(n+2))=2 end proc:
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Jul 28 2019

Select[Range[500], PrimeQ[#] && PrimeNu[# + 2] == 2 &] (* Amiram Eldar, Apr 14 2019 *)

isok(p) = isprime(p) && (omega(p+2) == 2); \\ Michel Marcus, May 02 2019

A370796 Number of primes between (prime(n)+1)^2 and (prime(n+1)-1)^2.

Original entry on oeis.org

2, 0, 0, 7, 0, 10, 0, 14, 32, 0, 38, 23, 0, 24, 51, 53, 0, 62, 30, 0, 71, 33, 76, 124, 44, 0, 42, 0, 51, 301, 48, 114, 0, 233, 0, 122, 126, 59, 135, 133, 0, 283, 0, 66, 0, 386, 396, 77, 0, 86, 173, 0, 349, 177, 187, 198, 0, 199, 100, 0, 412, 636, 113, 0, 114, 668, 224, 463, 0, 119, 236, 359

Offset: 1

Rafik Khalfi, Mar 02 2024

nonn

If (prime(n),prime(n+1)) is a twin prime pair, then a(n)=0.

			For n=1, (prime(1+1)-1)^2 = 4, (prime(1)+1)^2 = 9 and we have two primes between 4 and 9, so a(1)=2.

Cf. A050216.

A370796:= proc (n)
local count, a, b, p:
count := 0:
a := (ithprime(n)+1)^2:
b := (ithprime(n+1)-1)^2:
p := n:
while ithprime(p) <= b do if a <= ithprime(p) then count := count+1 end if:
p := p+1 end do:
return count end proc:
A370796(1) := 2:
map(A370796, [$1 .. 100]);

Table[Abs[ PrimePi[(Prime[n+1]-1)^2]- PrimePi[(Prime[n]+1)^2]],{n,72}] (* James C. McMahon, Mar 02 2024 *)

from sympy import primepi, prime, nextprime
def A370796(n): return -primepi(((p:=prime(n))+1)**2)+primepi((nextprime(p)-1)**2) if n>1 else 2 # Chai Wah Wu, Mar 27 2024

a(n) = A038107(A000040(n+1)-1) - A038107(A000040(n)+1) for all n > 1;

a(n) = A038107(A000040(n)+1) - A038107(A000040(n+1)-1) for n=1.

A380332 a(n) = number of primes between n^2 and n^4.

Original entry on oeis.org

0, 0, 4, 18, 48, 105, 199, 342, 546, 825, 1204, 1685, 2300, 3068, 4008, 5143, 6488, 8091, 9956, 12115, 14605, 17446, 20676, 24322, 28441, 33004, 38114, 43805, 50066, 56951, 64529, 72830, 81853, 91751, 102397, 114004, 126516, 140016, 154559, 170186, 186883, 204880, 224009, 244527, 266283, 289506, 314148, 340292, 368114, 397407

Offset: 0

Clark Kimberling, Jan 26 2025

nonn

p(2) = 4 because there are 4 primes between 4 and 16.

Cf. A000040, A000720, A038107, A079648, A380331.

Table[PrimePi[n^4] - PrimePi[n^2], {n, 0, 60}]

a(n) = primepi(n^4) - primepi(n^2); \\ Michel Marcus, Jan 27 2025

from sympy import primepi
def A380332(n): return -primepi(m:=n**2)+primepi(m**2) # Chai Wah Wu, Jan 27 2025

a(n) = PrimePi(n^4) - PrimePi(n^2).

A132635 Number of primes, 0's, and 1's in [0, n^2).

Original entry on oeis.org

0, 1, 4, 6, 8, 11, 13, 17, 20, 24, 27, 32, 36, 41, 46, 50, 56, 63, 68, 74, 80, 87, 94, 101, 107, 116, 124, 131, 139, 148, 156, 164, 174, 183, 193, 202, 212, 221, 230, 242, 253, 265, 276, 285, 297, 308, 321, 331, 344, 359, 369, 380, 395, 411, 423, 436, 447, 459, 476

Offset: 0

Roger L. Bagula and Gary W. Adamson, Aug 26 2008

Number of p-adic cyclotomic type levels in A_n-1 or SU(n).

			SU(5) at n^2-1=24 is exactly inside SU(9) at n^2-1=80.

Table[Sum[If[m == 0 || m == 1, 1, If[PrimeQ[m], 1, 0]], {m, 0,n^2 - 1}], {n, 0, 30}]

a(n)=if(n<2,n,primepi(n^2)+2) \\ Charles R Greathouse IV, Nov 07 2011

a(n) = pi(n^2) + 2 = A038107(n) + 2, n > 1. [Charles R Greathouse IV, Nov 07 2011]

cp's OEIS Frontend

A263100 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k^2) + pi(m^2/2), where pi(x) denotes the number of primes not exceeding x.

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A263321 Least positive integer m such that the numbers phi(k)*pi(k^2) (k = 1..n) are pairwise incongruent modulo m.

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A380331 a(n) = number of primes < n^4.

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A060304 Number of primes below n^3 does not exceed n times the number of primes below n^2.

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A163516 a(n) = floor( Sum_{x=2..n} x/log(x) ).

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A213649 Smallest k such that there exists a square between prime(n) and prime(n+k).

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A307533 Primes p such that p+2 has exactly two distinct prime factors.

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A370796 Number of primes between (prime(n)+1)^2 and (prime(n+1)-1)^2.

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