A237597 -id:A237597 - cp's OEIS frontend

A237598 a(n) = |{0 < k < prime(n): pi(k*n) is a square}|, where pi(.) is given by A000720.

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

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 0.
(ii) For each n > 9, there is a positive integer k < prime(n)/2 such that pi(k*n) is a triangular number.
See also A237612 for the least k > 0 with pi(k*n) a square.

			a(3) = 1 since pi(3*3) = 2^2 with 3 < prime(3) = 5.
a(6) = 2 since pi(4*6) = 3^2 with 4 < prime(6) = 13, and pi(9*6) =  4^2 with 9 < prime(6) = 13.
a(15) = 1 since pi(28*15) = 9^2 with 28 < prime(15) = 47.
a(62) = 1 since pi(68*62) = 24^2 with 68 < prime(62) = 293.
a(459) = 1 since pi(2544*459) = 301^2 with 2544 < prime(459) = 3253.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2500
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A000217, A000290, A000720, A237578, A237597, A237612, A237614.

sq[n_]:=IntegerQ[Sqrt[PrimePi[n]]]
a[n_]:=Sum[If[sq[k*n],1,0],{k,1,Prime[n]-1}]
Table[a[n],{n,1,70}]

A237612 Least positive integer k such that A000720(k*n) is a square, or 0 if such a number k does not exist.

Original entry on oeis.org

1, 1, 3, 2, 2, 4, 1, 1, 1, 1, 5, 2, 2, 2, 28, 34, 9, 3, 3, 5, 20, 7, 1, 1, 1, 1, 1, 1, 2, 14, 5, 17, 3, 16, 12, 23, 18, 4, 4, 30, 46, 10, 50, 23, 36, 18, 40, 14, 2, 2, 3, 3, 1, 1, 1, 1, 1, 1, 32, 7, 11, 68, 19, 79, 29, 267, 10, 8, 12, 6

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

According to the conjecture in A237598, we should always have 0 < a(n) < prime(n).

			a(3) = 3 since A000720(3*3) = 4 is a square, but neither A000720(1*3) = 2  nor A000720(2*3) = 3 is a square.

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. A000290, A000720, A237578, A237597, A237598, A237614.

sq[n_]:=IntegerQ[Sqrt[PrimePi[n]]]
Do[Do[If[sq[k*n],Print[n," ",k];Goto[aa]],{k,1,Prime[n]-1}];
Print[n," ",0];Label[aa];Continue,{n,1,100}]

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}]

A237615 a(n) = |{0 < k < n: k^2 + k - 1 and pi(k*n) are both prime}|, where pi(.) is given by A000720.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 2, 1, 2, 1, 3, 2, 1, 4, 1, 3, 4, 4, 2, 4, 3, 6, 2, 2, 2, 3, 7, 4, 3, 4, 5, 6, 1, 3, 2, 3, 9, 3, 3, 4, 7, 5, 8, 5, 2, 2, 5, 5, 4, 5, 6, 4, 5, 6, 10, 6, 6, 10, 9, 9, 10, 12, 2, 8, 7, 3, 6, 6, 4, 6

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 5.
(ii) For each n = 4, 5, ..., there is a positive integer k < n with k^2 + k - 1 and pi(k*n) + 1 both prime. Also, for any integer n > 6, there is a positive integer k < n with k^2 + k - 1 and pi(k*n) - 1 both prime.
(iii) For every integer n > 15, there is a positive integer k < n such that pi(k) - 1 and pi(k*n) are both prime.
Note that part (i) is a refinement of the first assertion in the comments in A237578.

			a(8) = 1 since 4^2 + 4 - 1 = 19 and pi(4*8) = 11 are both prime.
a(33) = 1 since 28^2 + 28 - 1 = 811 and pi(28*33) = 157 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, A combinatorial conjecture on primes, a message to Number Theory List, Feb. 9, 2014.

Cf. A000040, A000720, A002327, A045546, A237578, A237597, A237598.

p[k_,n_]:=PrimeQ[k^2+k-1]&&PrimeQ[PrimePi[k*n]]
a[n_]:=Sum[If[p[k,n],1,0],{k,1,n-1}]
Table[a[n],{n,1,70}]

A237643 Least positive integer m such that {A000720(k*n): 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, 2, 3, 8, 8, 12, 13, 14, 27, 25, 32, 25, 16, 23, 94, 41, 46, 67, 38, 60, 77, 55, 84, 46, 88, 79, 85, 113, 82, 155, 114, 141, 178, 132, 124, 176, 155, 96, 135, 176, 146, 148, 126, 125, 183, 191, 185, 194, 166, 261, 378, 230, 278, 203, 199, 161, 293, 286, 175, 274

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

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

Note that a(15) = 94 = 2*prime(15).

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

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

Cf. A000720, A237578, A237597, A237598, A237612, A237614, A237656.

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

A238224 Number of pairs {j, k} with 0 < j < k <= n and k == 1 (mod j) such that pi(jn) divides pi(kn), where pi(.) is given by A000720.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 20 2014

nonn

Conjecture: a(n) > 0 for all n > 1.
This is a refinement of part (i) of the conjecture in A238165.
We have verified the conjecture for n up to 21500.

			a(18) = 1 since 6 == 1 (mod 1), and pi(1*18) = 7 divides pi(6*18) = 28.
a(50) = 1 since 7 == 1 (mod 3), and pi(3*50) = 35 divides pi(7*50) = 70.
a(379) = 1 since 353 == 1 (mod 4), and pi(4*379) = 240 divides pi(353*379) = 12480.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2900
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000720, A237578, A237597, A237598, A238165.

m[k_,j_,n_]:=Mod[PrimePi[k*n],PrimePi[j*n]]==0
a[n_]:=Sum[If[m[j*q+1,j,n],1,0],{j,1,n-1},{q,1,(n-1)/j}]
Table[a[n],{n,1,80}]

A238165 Number of pairs {j, k} with 0 < j < k <= n such that pi(jn) divides pi(kn), where pi(.) is given by A000720.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 1, 5, 5, 5, 3, 5, 12, 5, 5, 7, 3, 2, 12, 7, 8, 9, 9, 6, 6, 11, 9, 12, 9, 15, 12, 12, 13, 7, 16, 12, 18, 15, 16, 11, 8, 8, 13, 15, 20, 13, 7, 15, 13, 7, 18, 7, 18, 15, 11, 15, 15, 12, 15, 17, 6, 18, 17, 16, 11, 15, 9, 18, 15, 13

Offset: 1

Zhi-Wei Sun, Feb 19 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 1.
(ii) For any integer n > 4, the sequence pi(k*n)^(1/k) (k = 1, ..., n) is strictly decreasing.
See also A238224 for a refinement of part (i) of this conjecture.

			a(5) = 3 since pi(1*5) = 3 divides both pi(3*5) = 6 and pi(5*5) = 9, and pi(2*5) = 4 divides pi(4*5) = 8.
a(7) = 1 since pi(1*7) = 4 divides pi(3*7) = 8.

Zhi-Wei Sun, Table of n, a(n) for n = 1..400

Cf. A000720, A237578, A237597, A237598, A237612, A237614, A237615, A238224.

m[k_,j_]:=Mod[PrimePi[k],PrimePi[j]]==0
a[n_]:=Sum[If[m[k*n,j*n],1,0],{k,2,n},{j,1,k-1}]
Do[Print[n," ",a[n]],{n,1,70}]

cp's OEIS Frontend

A237598 a(n) = |{0 < k < prime(n): pi(k*n) is a square}|, where pi(.) is given by A000720.

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A237612 Least positive integer k such that A000720(k*n) is a square, or 0 if such a number k does not exist.

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

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A237615 a(n) = |{0 < k < n: k^2 + k - 1 and pi(k*n) are both prime}|, where pi(.) is given by A000720.

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A237643 Least positive integer m such that {A000720(k*n): k = 1, ..., m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

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A238224 Number of pairs {j, k} with 0 < j < k <= n and k == 1 (mod j) such that pi(j*n) divides pi(k*n), where pi(.) is given by A000720.

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A238165 Number of pairs {j, k} with 0 < j < k <= n such that pi(j*n) divides pi(k*n), where pi(.) is given by A000720.

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A238224 Number of pairs {j, k} with 0 < j < k <= n and k == 1 (mod j) such that pi(jn) divides pi(kn), where pi(.) is given by A000720.

A238165 Number of pairs {j, k} with 0 < j < k <= n such that pi(jn) divides pi(kn), where pi(.) is given by A000720.