A247509 -id:A247509 - cp's OEIS frontend

A056608 Least prime factor of the n-th composite number.

2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 5, 2, 3, 2, 2, 2, 3, 2, 5, 2, 2, 3, 2, 2, 2, 3, 2, 2, 7, 2, 3, 2, 2, 5, 2, 3, 2, 2, 2, 3, 2, 5, 2, 2, 3, 2, 2, 2, 3, 2, 7, 2, 2, 3, 2, 2, 5, 2, 3, 2, 2, 7, 2, 3, 2, 5, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 5, 2, 3, 2, 7, 2, 11, 2, 3, 2, 5, 2, 2, 3, 2, 2, 7, 2, 3, 2, 2, 2

Offset: 1

Odimar Fabeny, Aug 07 2000

Record values are seen when n = A120389(m). Conjecture: at each new record the count of the prior record follows A247509. Records seen are 2, 3, 5, 7, 11, ... and when 3, 5, 7, 11 are first seen, there have been 3, 3, 2, and 4 occurrences of 2, 3, 5, and 7. These are A247509(1) through A247509(4). Thus, the count for prime(60) would be A247509(60). - Bill McEachen, Jun 17 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A052369 (largest prime factor of n, where n runs through composite numbers). - Klaus Brockhaus, Jun 23 2009

Cf. A160180.

a056608 = a020639 . a002808  -- Reinhard Zumkeller, Mar 29 2014

[ PrimeDivisors(n)[1]: n in [2..140] | not IsPrime(n) ]; // Klaus Brockhaus, Jun 23 2009

DeleteCases[Table[FactorInteger[n][[1, 1]] * Boole[Not[PrimeQ[n]]], {n, 2, 100}], 0] (* Alonso del Arte, Aug 21 2014 *)
FactorInteger[#][[1,1]]&/@Select[Range[200],CompositeQ] (* Harvey P. Dale, Mar 16 2023 *)

forcomposite(n=1, 1e2, p=factor(n)[1, 1]; print1(p, ", ")) \\ Felix Fröhlich, Aug 03 2014

from sympy import composite, factorint
def A056608(n):
    return min(factorint(composite(n))) # Chai Wah Wu, Jul 22 2019

a(n) = A020639(A002808(n)) = A000040(A118663(n)). - M. F. Hasler, Apr 03 2012

More terms from James Sellers, Aug 25 2000
Definition corrected by Reinhard Zumkeller, Mar 29 2014
Name changed by Alonso del Arte, Aug 21 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

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

A056608 Least prime factor of the n-th composite number.

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

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