A038107 -id:A038107 - cp's OEIS frontend

A237658 Positive integers m with pi(m) and pi(m^2) both prime, where pi(.) is given by A000720.

6, 17, 33, 34, 41, 59, 60, 69, 109, 110, 111, 127, 157, 161, 246, 287, 335, 353, 367, 368, 404, 600, 709, 711, 713, 718, 740, 779, 804, 1153, 1162, 1175, 1437, 1472, 1500, 1526, 1527, 1679, 1729, 1742, 1787, 1826, 2028, 2082, 2104, 2223, 2422, 2616, 2649, 2651

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

The conjecture in A237657 implies that this sequence has infinitely many terms.

For primes in this sequence, see A237659.

			a(1) = 6 since pi(6) = 3 and pi(6^2) = 11 are both prime, but none of pi(1) = 0, pi(2) = 1, pi(3^2) = 4, pi(4^2) = 6 and pi(5^2) = 9 is prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10001 (n = 1..3000 from Zhi-Wei Sun)

Cf. A000040, A000290, A038107, A237595, A237656, A237657, A237659.

p[m_]:=PrimeQ[PrimePi[m]]&&PrimeQ[PrimePi[m^2]]
n=0;Do[If[p[m],n=n+1;Print[n," ",m]],{m,1,1000}]

isok(n) = isprime(primepi(n)) && isprime(primepi(n^2)); \\ Michel Marcus, Apr 28 2018

A262976 Number of ordered ways to write n as 2^x + y^2 + pi(z^2) with x >= 0, y >= 0 and z > 0, where pi(m) denotes the number of primes not exceeding m.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 6, 4, 6, 6, 7, 6, 7, 5, 6, 10, 5, 9, 10, 7, 7, 9, 9, 4, 12, 10, 9, 8, 7, 10, 9, 10, 7, 15, 10, 6, 13, 10, 9, 10, 16, 10, 10, 9, 8, 15, 9, 8, 15, 12, 12, 7, 12, 11, 14, 12, 8, 16, 6, 10, 11, 14, 8, 11, 17, 10, 16, 9, 13, 16, 15, 8, 18, 13, 10, 14, 10, 12, 16, 12, 13, 18, 11, 9, 17, 17, 9, 15, 16, 15, 9, 12, 12, 17, 12, 9, 21, 10, 11

Offset: 1

Zhi-Wei Sun, Oct 05 2015

nonn

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

(ii) Each positive integer can be written as 2^x + pi(y^2) + pi(z^2) with x >= 0, y > 0 and z > 0.

			a(1) = 1 since 1 = 2^0 + 0^2 + pi(1^2).
a(2) = 2 since 2 = 2^0 + 1^2 + pi(1^2) = 2 + 0^2 + pi(1^2).
a(3) = 2 since 3 = 2^0 + 0^2 + pi(2^2) = 2 + 1^2 + pi(1^2).

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..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000079, A000290, A000720, A038107, A262746, A262887.

SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=PrimePi[n^2]
Do[r=0;Do[If[f[x]>=n,Goto[aa]];Do[If[2^y>n-f[x],Goto[bb]];If[SQ[n-f[x]-2^y],r=r+1],{y,0,Log[2,n-f[x]]}];Label[bb];Continue,{x,1,n}];Label[aa];Print[n," ",r];Continue,{n,1,100}]

A263319 a(n) = pi(n^2)*phi(n)/2, where pi(x) denotes the number of primes not exceeding x, and phi(.) is Euler's totient function given by A000010.

Original entry on oeis.org

0, 1, 4, 6, 18, 11, 45, 36, 66, 50, 150, 68, 234, 132, 192, 216, 488, 198, 648, 312, 510, 460, 1089, 420, 1140, 732, 1161, 822, 2044, 616, 2430, 1376, 1810, 1528, 2400, 1260, 3942, 2052, 2880, 2008, 5260, 1644, 5943, 2950, 3672, 3509, 7567, 2736, 7497, 3670

Offset: 1

Zhi-Wei Sun, Oct 14 2015

nonn

Conjecture: (i) All the terms of this sequence are pairwise distinct.
(ii) All the numbers phi(n)*pi(n*(n-1)) (n = 1,2,3,...) are pairwise distinct.
(iii) All the numbers phi(n^2)*pi(n^2) = n*phi(n)*pi(n^2) (n = 1,2,3,...) are pairwise distinct.
We have checked this conjecture via Mathematica. For example, we have verified that a(n) (n = 1..4*10^5) are indeed pairwise distinct.
See also A263325 for a similar conjecture.

			a(1) = 0 since pi(1^2)*phi(1)/2 = 0*1/2 = 0.
a(2) = 1 since pi(2^2)*phi(2)/2 = 2*1/2 = 1.
a(3) = 4 since pi(3^2)*phi(3)/2 = 4*2/2 = 4.

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

Cf. A000010, A000290, A000720, A002618, A038107, A263317, A263321, A263325.

[#PrimesUpTo(n^2)*EulerPhi(n)/2: n in [1..80]]; // Vincenzo Librandi, Oct 15 2015

a[n_]:=a[n]=PrimePi[n^2]*EulerPhi[n]/2
Do[Print[n," ",a[n]],{n,1,50}]

a(n) = primepi(n^2)*eulerphi(n)/2; \\ Michel Marcus, Oct 15 2015

A194189 Number of primes between the n-th triangular number and the n-th square.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 6, 7, 8, 9, 12, 13, 15, 17, 18, 22, 25, 27, 30, 32, 35, 38, 41, 43, 48, 52, 55, 58, 62, 64, 68, 73, 79, 83, 86, 89, 93, 97, 103, 110, 114, 120, 123, 129, 132, 139, 141, 149, 157, 162, 162, 173, 183, 186, 192, 195, 198, 207, 213, 222, 229

Offset: 1

Reinhard Zumkeller, Nov 01 2011

nonn

a(n) = A038107(n) - A111208(n).

			a(10) = #{59,61,67,71,73,79,83,89,97} = 9;
a(11) = #{67,71,73,79,83,89,97,101,103,107,109,113} = 12;
a(12) = #{79,83,89,97,101,103,107,109,113,127,131,137,139} = 13.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A010051, A000217, A000290.

a194189 n = sum $ map a010051 [n*(n+1) `div` 2 + 1 .. n^2 - 1]

Table[PrimePi[n^2] - PrimePi[n*(n+1)/2], {n, 100}] (* T. D. Noe, Nov 01 2011 *)
PrimePi[#[[2]]]-PrimePi[#[[1]]]&/@Module[{nn=70},Thread[{Accumulate[ Range[ nn]],Range[nn]^2}]] (* Harvey P. Dale, Aug 21 2019 *)

A237659 Primes p with pi(p) and pi(p^2) both prime, where pi(.) is given by A000720.

Original entry on oeis.org

17, 41, 59, 109, 127, 157, 353, 367, 709, 1153, 1787, 3319, 3407, 3911, 5851, 6037, 6217, 6469, 8389, 9103, 9319, 10663, 13709, 14107, 14591, 15683, 18433, 19463, 19577, 20107, 21727, 23209, 27809, 29383, 32797, 35023, 36251, 36599, 38351, 39239

Offset: 1

Zhi-Wei Sun, Feb 11 2014

nonn

This is a subsequence of A237658.

Conjecture: The sequence has infinitely many terms.

			a(1) = 17 with pi(17) = 7 and pi(17^2) = 61 both prime.
a(2) = 41 with pi(41) = 13 and pi(41^2) = 263 both prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..600 from Zhi-Wei Sun)

Cf. A000040, A000290, A000720, A006450, A038107, A237656, A237657, A237658.

p[m_]:=PrimeQ[PrimePi[m^2]]
n=0;Do[If[p[Prime[Prime[k]]],n=n+1;Print[n," ",Prime[Prime[k]]]],{k,1,1000}]
Select[Prime[Range[4500]],AllTrue[{PrimePi[#],PrimePi[#^2]},PrimeQ]&] (* Harvey P. Dale, May 10 2025 *)

A263325 a(n) = sigma(n)*pi(n^2), where sigma(n) is the sum of all (positive) divisors of n, and pi(x) is the number of primes not exceeding x.

Original entry on oeis.org

0, 6, 16, 42, 54, 132, 120, 270, 286, 450, 360, 952, 546, 1056, 1152, 1674, 1098, 2574, 1440, 3276, 2720, 3312, 2376, 6300, 3534, 5124, 5160, 7672, 4380, 11088, 5184, 10836, 8688, 10314, 9600, 19110, 8322, 13680, 13440, 22590, 11046, 26304, 12452, 24780, 23868, 22968, 15792, 42408, 20349, 34131

Offset: 1

Zhi-Wei Sun, Oct 14 2015

nonn

Conjecture: (i) All the terms of this sequence are pairwise distinct.
(ii) All the numbers sigma(n)*pi(n*(n+1)) (n = 1,2,3,...) are pairwise distinct.
(iii) All the numbers n*sigma(n)*pi(n^2) (n = 1,2,3,...) are pairwise distinct, and all the numbers sigma(n^2)*pi(n^2) (n = 1,2,3,...) are also pairwise distinct.
(iv) All the numbers n*phi(n)*sigma(n^2) = phi(n^2)*sigma(n^2) (n = 1,2,3,...) are pairwise distinct, where phi(.) is Euler's totient function.
We have verified that the terms a(n) (n = 1..4*10^5) are indeed pairwise distinct.
See also A263319 for a similar conjecture.

			a(1) = 0 since sigma(1)*pi(1^2) = 1*0 = 0.
a(2) = 6 since sigma(2)*pi(2^2) = 3*2 = 6.

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

Cf. A000010, A000203, A000290, A000720, A002618, A038107, A065764, A263317, A263319.

[#PrimesUpTo(n^2)*SumOfDivisors(n): n in [1..80]]; // Vincenzo Librandi, Oct 15 2015

a[n_]:=a[n]=DivisorSigma[1,n]*PrimePi[n^2]
Do[Print[n," ",a[n]],{n,1,50}]

a(n) = sigma(n)*primepi(n^2); \\ Michel Marcus, Oct 15 2015

A139328 Sums of rows of the triangle in A139325.

Original entry on oeis.org

0, 3, 6, 10, 14, 19, 24, 30, 36, 45, 52, 60, 67, 76, 86, 96, 105, 117, 127, 138, 151, 162, 176, 189, 203, 216, 230, 246, 262, 277, 292, 308, 325, 343, 362, 376, 398, 417, 435, 451, 473, 491, 515, 535, 557, 579, 599, 622, 646, 668, 691, 712, 737, 764, 788, 815

Offset: 1

Reinhard Zumkeller, Apr 14 2008

nonn

a(n) = Sum_{k=1..n} A139325(n,k).

			a(4) = #{3,5,7}+#{11,13}+#{17,19,23}+#{29,31} = 3+2+3+2 = 10:
..1 ...3 ...5 ...7 ... primes in first row = {3,5,7},
..9 ..11 ..13 ..15 ... primes in 2nd row = {11,13},
.17 ..19 ..21 ..23 ... primes in 3rd row = {17,19,23},
.25 ..27 ..29 ..31 ... primes in 4th row = {29,31}.

Cf. A000720, A038107, A139325.

a(n) = A000720(2*n^2 - 1) - 1.

A161182 Successive differences between positions of squares in list of nonprimes.

Original entry on oeis.org

1, 1, 3, 5, 6, 9, 9, 12, 13, 16, 16, 19, 20, 22, 25, 25, 26, 30, 31, 33, 34, 36, 38, 41, 40, 43, 46, 47, 48, 51, 53, 53, 56, 57, 60, 61, 64, 66, 65, 68, 69, 72, 76, 75, 78, 78, 83, 82, 82, 89, 90, 88, 89, 95, 96, 100, 101, 98, 104, 103, 105, 110, 108, 112, 115, 115, 118, 120

Offset: 1

Daniel Tisdale, Jun 05 2009

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A078435, sequence of positions of squares in sequence of nonprimes.

A038107 := proc(n) numtheory[pi]( n^2) ; end: A078435 := proc(n) n^2-A038107(n) ; end: A161182 := proc(n) A078435(n)-A078435(n-1) ; end: seq(A161182(n),n=1..100) ; # R. J. Mathar, Jun 22 2009

With[{nn=5000},Differences[Flatten[Position[Complement[Range[0,nn], Prime[ Range[ PrimePi[nn]]]],?(IntegerQ[Sqrt[#]]&)]]]] (* _Harvey P. Dale, Mar 03 2014 *)

a(n) = A078435(n) - A078435(n-1). - R. J. Mathar, Jun 22 2009

Corrected and extended by R. J. Mathar, Jun 22 2009

A220506 Number of primes <= n-th quarter-square.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 15, 16, 18, 20, 22, 24, 25, 29, 30, 32, 34, 36, 39, 42, 44, 46, 48, 52, 54, 58, 61, 62, 66, 68, 72, 75, 78, 81, 85, 89, 92, 96, 99, 101, 105, 109, 114, 118, 122, 126, 129, 133, 137, 141, 146, 150, 154, 158, 162, 167, 172, 177, 181, 187, 191, 195, 200

Offset: 1

Omar E. Pol, Feb 05 2013

nonn

Partial sums of A220492. A bisection is A038107, n >= 1.

Cf. A000040, A000720, A002620, A014085, A111208.

Table[PrimePi[n^2/4], {n, 75}] (* Alonso del Arte, Feb 05 2013 *)

a(n) = A000720(A002620(n)), n >= 1.

A237687 Primes p with pi(p), pi(pi(p)) and pi(p^2) all prime, where pi(.) is given by A000720.

Original entry on oeis.org

59, 127, 709, 1153, 1787, 9319, 13709, 19577, 32797, 35023, 39239, 40819, 53353, 62921, 75269, 90023, 161159, 191551, 218233, 228451, 235891, 238339, 239087, 272999, 289213, 291619, 339601, 439357, 500741, 513683

Offset: 1

Zhi-Wei Sun, Feb 11 2014

nonn

This is a subsequence of A237659.

Conjecture: The sequence has infinitely many terms.

			a(1) = 59 with 59, pi(59) = 17, pi(pi(59)) = pi(17) = 7 and pi(59^2) = 487 all prime.

Chai Wah Wu, Table of n, a(n) for n = 1..1000 (n = 1..150 from Zhi-Wei Sun)

Cf. A000040, A000290, A000720, A006450, A038107, A038580, A237656, A237657, A237658, A237659.

p[m_]:=PrimeQ[PrimePi[m^2]]
n=0;Do[If[p[Prime[Prime[Prime[k]]]],n=n+1;Print[n," ",Prime[Prime[Prime[k]]]]],{k,1,1000}]

cp's OEIS Frontend

A237658 Positive integers m with pi(m) and pi(m^2) both prime, where pi(.) is given by A000720.

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A262976 Number of ordered ways to write n as 2^x + y^2 + pi(z^2) with x >= 0, y >= 0 and z > 0, where pi(m) denotes the number of primes not exceeding m.

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A263319 a(n) = pi(n^2)*phi(n)/2, where pi(x) denotes the number of primes not exceeding x, and phi(.) is Euler's totient function given by A000010.

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A194189 Number of primes between the n-th triangular number and the n-th square.

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A237659 Primes p with pi(p) and pi(p^2) both prime, where pi(.) is given by A000720.

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A263325 a(n) = sigma(n)*pi(n^2), where sigma(n) is the sum of all (positive) divisors of n, and pi(x) is the number of primes not exceeding x.

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Magma

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A139328 Sums of rows of the triangle in A139325.

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A161182 Successive differences between positions of squares in list of nonprimes.

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