A156759 -id:A156759 - cp's OEIS frontend

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

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

A247393 Numbers n such that the second maximal prime <= sqrt(n) is the least prime divisor of n.

Original entry on oeis.org

10, 12, 14, 16, 18, 20, 22, 24, 27, 33, 39, 45, 55, 65, 85, 95, 115, 133, 161, 187, 209, 253, 299, 391, 493, 527, 551, 589, 703, 779, 817, 851, 943, 1073, 1189, 1247, 1363, 1457, 1643, 1739, 1927, 2173, 2279, 2537, 2623, 2867, 3149, 3337, 3431, 3551, 3953

Offset: 1

Vladimir Shevelev, Sep 16 2014

nonn

These numbers we call "preprimes" of the second kind in contrast to A156759 for n>=2, for which the maximal prime <= sqrt(n) is the least prime divisor of n. Terms of A156759 (n>=2) we call "preprimes" (cf. comment there).

			a(1)=10. Indeed, in interval [2,sqrt(10)] we have two primes: 2 and 3. Maximal from them 3, the second maximal is 2, and 2=lpf(10).

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

Cf. A156759.

Select[Range[4000], Prime[PrimePi[Sqrt[#]]-1] == FactorInteger[#][[1,1]] &] (* Indranil Ghosh, Mar 08 2017 *)

select(n->prime(primepi(sqrtint(n))-1)==factor(n)[1, 1], vector(10^4, x, x+8)) \\ Jens Kruse Andersen, Sep 17 2014

lpf(a(n)) = prime(pi(sqrt(a(n)))-1), where pi(n) = A000720(n).

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

a(52..10000) from Jens Kruse Andersen, Sep 17 2014

A247394 Numbers n for which the third maximal prime <= sqrt(n) is the least prime divisor of n.

Original entry on oeis.org

26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 111, 117, 125, 145, 155, 203, 217, 259, 287, 319, 341, 377, 403, 481, 629, 697, 731, 799, 893, 989, 1081, 1219, 1357, 1537, 1829, 1961, 2183, 2419, 2501, 2747, 2881, 3053

Offset: 1

Vladimir Shevelev, Sep 16 2014

nonn

These numbers we call "preprimes" of the third kind in contrast to A156759 for n>=2, for which the maximal prime <= sqrt(n) is the least prime divisor of n; and to A247393 for which the second maximal prime <= sqrt(n) is the least prime divisor of n.

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

Cf. A156759, A247393.

Select[Range[4000], Prime[PrimePi[Sqrt[#]]-2] == FactorInteger[#][[1,1]] &] (* Indranil Ghosh, Mar 08 2017 *)

select(n->prime(primepi(sqrtint(n))-2)==factor(n)[1, 1], vector(10^4, x, x+24)) \\ Jens Kruse Andersen, Sep 17 2014

lpf(a(n)) = prime(pi(isqrt(a(n)))-2), with pi(n) = A000720(n), lpf(n) = A020639(n) and isqrt(n) = A000196(n).

Terms up to a(54) from Peter J. C. Moses, Sep 16 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

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

Original entry on oeis.org

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

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

A158846 Primes which are removed with the algorithm of A156284, starting the selection with the interval (2^4, 2^5).

Original entry on oeis.org

19, 29, 41, 47, 53, 59, 61, 97, 149, 167, 173, 233, 239, 251, 271, 283, 313, 331, 349, 373, 409, 433, 439, 499, 509, 521, 557, 563, 593, 641, 677, 743, 761, 797, 827, 887, 911, 941, 953, 1013, 1019, 1021, 1039, 1051, 1129, 1171, 1237, 1279, 1291

Offset: 1

Vladimir Shevelev, Mar 28 2009

nonn

We iteratively scan integer intervals (2^(m-1)..2^m), first the one with m=5, then m=6, m=7, etc., and start with the set S={3,5,7,11,...} of all odd primes. For each prime p = 2^m-k, 2^(m-1) < p < 2^m, p is removed from S if k is in S. Basically, all the upper primes of primes pairs are removed when the prime pair sums to a power of 2 which are larger than 2^4. The sequence shows all p that are removed from S at any stage m.

Powers 2^m, m >= 5, are not expressible as sums of two primes which are not in the sequence.

Cf. A156284, A158756, A156759.

A158846 := proc()
        local mmax,prrem,m,prm,pi,p,q ;
        mmax := 12 ; prrem := {} ;
        for m from 5 to mmax do
                prm := {} ;
                for pi from 1 do
                        k := ithprime(pi) ;
                        p := 2^m-k ;
                        if p <= 2^(m-1) then  break; end if;
                        if isprime(p) and not k in prrem then prm := prm union {p} ;
                        end if ;
                end do:
                prrem := prrem union prm ;
        end do: print( sort(prrem)) ; return ;
end proc:
A158846() ; # R. J. Mathar, Dec 07 2010

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

cp's OEIS Frontend

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

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A247393 Numbers n such that the second maximal prime <= sqrt(n) is the least prime divisor of n.

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A247394 Numbers n for which the third maximal prime <= sqrt(n) is the least prime divisor of 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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A247511 Number of preprimes of the third kind (A247394) 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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