A247394 -id:A247394 - cp's OEIS frontend

A247511 Number of preprimes of the third kind (A247394) with the smallest prime divisor equals prime(n).

12, 12, 3, 4, 2, 3, 4, 1, 4, 1, 1, 2, 3, 4, 1, 4, 2, 1, 2, 2, 4, 5, 0, 0, 1, 0, 0, 5, 3, 2, 2, 5, 1, 2, 2, 2, 2, 2, 1, 4, 0, 1, 0, 4, 6, 2, 0, 2, 2, 1, 4, 2, 3, 3, 1, 2, 1, 1, 2, 5, 1, 1, 0, 4, 2, 4, 1, 1, 1, 3, 2, 2, 1, 2, 2, 1, 2, 3, 1, 3, 0, 2, 1, 3, 3, 1

Offset: 1

Vladimir Shevelev, Sep 18 2014

nonn

Vladimir Shevelev, A classification of the positive integers over primes

Cf. A000040, A156759, A247393, A247394, A247509, A247510, A247606.

More terms from Peter J. C. Moses, Sep 18 2014

A247509 Number of preprimes (A156759, n>1) such that the smallest prime divisor equals prime(n).

Original entry on oeis.org

3, 3, 2, 4, 2, 3, 2, 2, 3, 2, 4, 3, 2, 2, 3, 3, 2, 4, 3, 2, 3, 2, 2, 4, 3, 2, 3, 2, 2, 5, 2, 3, 2, 4, 2, 3, 3, 2, 3, 3, 2, 5, 2, 3, 2, 3, 5, 3, 2, 2, 3, 2, 3, 3, 3, 3, 2, 4, 3, 2, 2, 5, 3, 2, 2, 3, 2, 4, 2, 2, 2, 3, 3, 3, 2, 2, 3, 2, 2, 3, 2, 4, 2, 3, 2, 2, 4

Offset: 1

Vladimir Shevelev, Sep 18 2014

nonn

			For n=2, using the formula, we have a(2)=pi(25/3)-1=3.

Indranil Ghosh, Table of n, a(n) for n = 1..1000
Vladimir Shevelev, A classification of the positive integers over primes

Cf. A000040, A156759, A247393, A247394.

a[1] = 3;a[n_] := PrimePi[Prime[n + 1]^2 / Prime[n]] - n + 1; Table[a[n], {n, 1, 87}] (* Indranil Ghosh, Mar 09 2017 *)

for (n=1, 87, print1(if(n==1, 3, primepi(prime(n + 1)^2 / prime(n)) - n + 1),", ")) \\ Indranil Ghosh, Mar 09 2017

For n>1, a(n) = pi(prime(n+1)^2/prime(n))-n +1, where pi(x) is the prime counting function (cf. A000720). - Vladimir Shevelev, Sep 28 2014

More terms from Peter J. C. Moses, Sep 18 2014

A247395 The smallest numbers of every class in a classification of positive numbers (see comment).

Original entry on oeis.org

1, 2, 4, 10, 26, 50, 122, 170, 290, 362, 530, 842, 962, 1370, 1682, 1850, 2210, 2810, 3482, 3722, 4490, 5042, 5330, 6242, 6890, 7922, 9410, 10202, 10610, 11450, 11882, 12770, 16130, 17162, 18770, 19322, 22202, 22802, 24650, 26570, 27890, 29930, 32042, 32762

Offset: 0

Vladimir Shevelev, Sep 16 2014

Consider a classification of the positive numbers with classes {1}, A000040 (primes), A156759 (n>=2) (preprimes, or preprimes of the first kind), A247393 (preprimes of the second kind), A247394 (preprimes of the third kind), etc.

Then a(0)=1, a(1)=2; for n>=3, a(n) is the smallest number which is a preprime of the (n-1)st kind.

Indranil Ghosh, Table of n, a(n) for n = 0..1000
Vladimir Shevelev, A classification of the positive integers over primes

Cf. A000040, A156759, A247393, A247394.

Table[If[n>2, Prime[n - 1]^2 + 1, 2^n], {n, 0, 43}] (* Indranil Ghosh, Mar 08 2017 *)

a(n)=if(n>2, (prime(n-1))^2 + 1, 2^n) \\ Charles R Greathouse IV, Sep 17 2014

For n>=3, a(n) = (prime(n-1))^2 + 1.

More terms from Peter J. C. Moses, Sep 16 2014

A247396 Number of even numbers in classes of classification of the positive numbers defined in comment in A247395.

Original entry on oeis.org

0, 1, 3, 8, 12, 36, 24, 60, 36, 84, 156, 60, 204, 156, 84, 180, 300, 336, 120, 384, 276, 144, 456, 324, 516, 744, 396, 204, 420, 216, 444, 1680, 516, 804, 276, 1440, 300, 924, 960, 660, 1020, 1056, 360, 1860, 384, 780, 396, 2460, 2604, 900, 456, 924, 1416, 480

Offset: 0

Vladimir Shevelev, Sep 16 2014

nonn

In the classification every class contains no more than a finite number of numbers with a given least prime divisor.

			a(6) = (prime(6)^2 - prime(5)^2)/2 = (13^2 - 11^2)/2 = 24. - _Indranil Ghosh_, Mar 08 2017

Indranil Ghosh, Table of n, a(n) for n = 0..1000
Vladimir Shevelev, A classification of the positive integers over primes

Cf. A000040, A156759, A247393, A247394, A247395.

A247396:=n->(ithprime(n)^2 - ithprime(n-1)^2)/2: 0,1,3,seq(A247396(n), n=3..100); # Wesley Ivan Hurt, Apr 18 2017

a[0] = 0; a[1] = 1; a[2] = 3; a[n_] := (Prime[n]^2 - Prime[n - 1]^2) / 2; Table[a[n], {n, 0, 53}] (* Indranil Ghosh, Mar 08 2017 *)

for(n=0, 53, print1(if(n>2, (prime(n)^2 - prime(n - 1)^2)/2, if(n<2, n, 3)),", ")) \\ Indranil Ghosh, Mar 08 2017

For n>=3, a(n) = (prime(n)^2 - prime(n-1)^2)/2.

More terms from Peter J. C. Moses, Sep 17 2014

A247510 Number of preprimes of the second kind (A247393) with the smallest prime divisor equals prime(n).

Original entry on oeis.org

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

Offset: 1

Vladimir Shevelev, Sep 18 2014

nonn

Vladimir Shevelev, A classification of the positive integers over primes

Cf. A000040, A156759, A247393, A247394, A247509, A247511, A247606.

More terms from Peter J. C. Moses, Sep 18 2014

A247606 Number of non-semiprimes among "preprimes" of the n-th kind (defined in comment in A247395).

Original entry on oeis.org

1, 7, 15, 36, 31, 62, 59, 111, 161, 113, 224, 175, 155, 258, 370, 358, 240, 436, 346, 297, 557, 504, 691, 806, 477, 367, 554, 489, 938, 1743, 786, 959, 725, 1526, 669, 1215, 1207, 1022, 1359, 1286, 958, 1947, 773, 1206, 1328, 3078, 2740, 1165, 915, 1459, 1787

Offset: 1

Vladimir Shevelev, Sep 22 2014

nonn

One can prove that non-semiprimes we can find among preprimes of the n-th kind only with the smallest prime divisor 2,3,...,prime(n), where n=1 corresponds to A156759, n=2 corresponds to A247393, n=3 corresponds to A247394, etc. For example, for n=1, only among even numbers of A156759; for n=2 - only among even numbers and numbers with the smallest prime divisor 3 of A247393, etc. Thus, for every n>=1, among preprimes of the n-th kind almost all numbers are semiprimes.

Vladimir Shevelev, A classification of the positive integers over primes

Cf. A156759, A247393, A247394, A247395, A247396, A247509, A247510, A247511.

More terms from Peter J. C. Moses, Sep 22 2014

A247834 Maximal non-semiprime number which is a "preprime" of the n-th kind (defined in comment in A247395).

Original entry on oeis.org

8, 45, 125, 343, 325, 833, 1331, 1573, 2197, 2057, 3211, 3289, 4913, 4901, 6859, 6647, 8303, 10051, 10469, 11191, 12167, 15341, 16399, 17081, 18259, 22103, 24389, 26071, 29791, 27347, 31117, 35557, 36163, 36859, 39401, 42439, 50653, 50933, 52111, 56129, 56699

Offset: 1

Vladimir Shevelev, Sep 24 2014

nonn

Conjecture: the sequence contains all cubes of primes, except for 3^3 (cf. A030078).
Prime(n)^3 is in the sequence iff the interval [prime(n)^(3/2), prime(n)*sqrt(prime(n+1))] contains a prime.
A simple algorithm for finding the position k=k(n) for which a(k) = prime(n)^3 is given in A247835 (see formula and example there).
Conjecture: every term has the form a(n)= p*q*r, where p<=q<=r are primes.

Vladimir Shevelev , A classification of the positive integers over primes

Cf. A030078, A156759, A247393, A247394, A247395, A247396, A247509, A247510, A247511, A247606, A247835, A247867.

More terms from Peter J. C. Moses, Sep 24 2014

A247835 Indices of prime(n)^3 in A247834, or a(n)=0 if prime(n)^3 is not in A247834.

Original entry on oeis.org

1, 0, 3, 4, 7, 9, 13, 15, 21, 27, 29, 37, 43, 47, 52, 61, 71, 74, 83, 89, 94, 105, 111, 123, 138, 145, 149, 158, 161, 168, 196, 208, 220, 226, 246, 248, 261, 276, 287, 299, 316, 319, 340, 345, 358, 364, 392, 422, 432, 436, 447, 464, 470, 496, 512, 530, 544, 549

Offset: 1

Vladimir Shevelev, Sep 24 2014

nonn

Conjecture: all a(n)>0, except for n=2.

			Using the formula, let us find the position in A247834, in which should be 17^3, if 17^3 belongs to A247834. Since 17 = prime(7), then we have a(7) = pi(17^(3/2)) - 6 = pi(70) - 6 = 13. Indeed, A247834(13) = 4913 = 17^3.

Cf. A030078, A156759, A247393, A247394, A247395, A247396, A247509, A247510, A247511, A247606, A247834, A247867.

a(n) = my(p=prime(n)); if(nextprime(ceil(p*sqrt(p))) > p*sqrt(prime(n+1)), 0, primepi(prime(n)^(3/2)) - n + 1); \\ Jinyuan Wang, Feb 17 2021

If prime(n)^3 is in A247834, then a(n) = pi(prime(n)^(3/2)) - n + 1, where pi(x) is the prime counting function (A000720).

More terms from Jinyuan Wang, Feb 17 2021

A247977 If n = 1 or prime, then a(n) = 0; otherwise, if n is a preprime of k-th kind, then a(n) = k.

Original entry on oeis.org

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

Offset: 1

Vladimir Shevelev, Sep 28 2014

Preprimes of k-th kind are defined in comment in A247395.

			If n = 15, then, by the formula, we have a(15) = 2 - 2 + 1 = 1.

Indranil Ghosh, Table of n, a(n) for n = 1..10000
Vladimir Shevelev , A classification of the positive integers over primes

Cf. A000040, A156759, A247393, A247394, A247395, A247396, A247509, A247510, A247511, A247606, A247834, A247835.

Table[If[n==1 || PrimeQ[n], 0, PrimePi[Sqrt[n]] - PrimePi[FactorInteger[n][[1, 1]]] + 1], {n, 1, 125}] (* Indranil Ghosh, Mar 08 2017 *)

for(n=1, 125, print1(if(n==1 || isprime(n), 0, primepi(sqrt(n)) - primepi(vecmin(factor(n)[, 1])) + 1),", ")) \\ Indranil Ghosh, Mar 08 2017

If n is a composite number, then a(n) = pi(sqrt(n)) - pi(lpf(n)) + 1, where pi(x) is prime counting function (cf. A000720), lpf = least prime factor (A020639).

cp's OEIS Frontend

A247511 Number of preprimes of the third kind (A247394) with the smallest prime divisor equals prime(n).

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A247509 Number of preprimes (A156759, n>1) such that the smallest prime divisor equals prime(n).

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A247395 The smallest numbers of every class in a classification of positive numbers (see comment).

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A247396 Number of even numbers in classes of classification of the positive numbers defined in comment in A247395.

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A247510 Number of preprimes of the second kind (A247393) with the smallest prime divisor equals prime(n).

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A247606 Number of non-semiprimes among "preprimes" of the n-th kind (defined in comment in A247395).

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A247834 Maximal non-semiprime number which is a "preprime" of the n-th kind (defined in comment in A247395).

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A247835 Indices of prime(n)^3 in A247834, or a(n)=0 if prime(n)^3 is not in A247834.

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