A020639 -id:A020639 - cp's OEIS frontend

A286472 Compound filter (for counting prime gaps): a(1) = 1, a(n) = 2*A032742(n) + (1 if n is composite and spf(A032742(n)) > nextprime(spf(n)), and 0 otherwise). Here spf is the smallest prime factor, A020639.

1, 2, 2, 4, 2, 6, 2, 8, 6, 11, 2, 12, 2, 15, 10, 16, 2, 18, 2, 20, 15, 23, 2, 24, 10, 27, 18, 28, 2, 30, 2, 32, 23, 35, 14, 36, 2, 39, 27, 40, 2, 42, 2, 44, 30, 47, 2, 48, 14, 51, 35, 52, 2, 54, 23, 56, 39, 59, 2, 60, 2, 63, 42, 64, 27, 66, 2, 68, 47, 71, 2, 72, 2, 75, 50, 76, 22, 78, 2

Offset: 1

Antti Karttunen, May 11 2017

nonn

For n > 1, a(n) is odd if and only if n is a composite with its smallest prime factor occurring only once and with a gap of at least one between the smallest and the next smallest prime factor.

For all i, j: a(i) = a(j) => A073490(i) = A073490(j). This follows because A073490(n) can be computed by recursively invoking a(n), without needing any other information.

			For n = 4 = 2*2, the two smallest prime factors (taken with multiplicity) are 2 and 2, and the difference between their indices is 0, thus a(4) = 2*A032742(4) + 0 = 2*(4/2) + 0 = 2.
For n = 6 = 2*3 = prime(1)*prime(2), the difference between the indices of two smallest prime factors is 1 (which is less than required 2), thus a(6) = 2*A032742(6) + 0 = 2*(6/2) + 0 = 6.
For n = 10 = 2*5 = prime(1)*prime(3), the difference between the indices of two smallest prime factors is 2, thus a(10) = 2*A032742(10) + 1 = 2*(10/2) + 1 = 11.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A032742, A285729, A286471, A286473, A286474, A286475, A286476.
Cf. A000040 (primes give the positions of 2's).
Cf. A073490 (one of the matched sequences).

Table[Function[{p, d}, 2 d + If[And[CompositeQ@ n, FactorInteger[d][[1, 1]] > NextPrime[p]], 1, 0] - Boole[n == 1]] @@ {#, n/#} &@ FactorInteger[n][[1, 1]], {n, 98}] (* Michael De Vlieger, May 12 2017 *)

from sympy import primefactors, divisors, nextprime
def ok(n): return 1 if isprime(n)==0 and min(primefactors(divisors(n)[-2])) > nextprime(min(primefactors(n))) else 0
def a(n): return 1 if n==1 else 2*divisors(n)[-2] + ok(n) # Indranil Ghosh, May 12 2017

(define (A286472 n) (if (= 1 n) n (+ (* 2 (A032742 n)) (if (> (A286471 n) 2) 1 0))))

a(n) = 2*A032742(n) + [A286471(n) > 2], a(1) = 1.

A286473 Compound filter (for counting primes of form 4k+1, 4k+2 and 4k+3): a(n) = 4*A032742(n) + (A020639(n) mod 4), a(1) = 1.

Original entry on oeis.org

1, 6, 7, 10, 5, 14, 7, 18, 15, 22, 7, 26, 5, 30, 23, 34, 5, 38, 7, 42, 31, 46, 7, 50, 21, 54, 39, 58, 5, 62, 7, 66, 47, 70, 29, 74, 5, 78, 55, 82, 5, 86, 7, 90, 63, 94, 7, 98, 31, 102, 71, 106, 5, 110, 45, 114, 79, 118, 7, 122, 5, 126, 87, 130, 53, 134, 7, 138, 95, 142, 7, 146, 5, 150, 103, 154, 47, 158, 7, 162, 111, 166, 7, 170, 69, 174, 119, 178, 5, 182, 55

Offset: 1

Antti Karttunen, May 11 2017

nonn

For all i, j: a(i) = a(j) => A079635(i) = A079635(j). This follows because A079635(n) can be computed by recursively invoking a(n), without needing any other information.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A020639, A032742, A285729, A286472, A286474, A286475, A286476.

Cf. A001511, A007814, A065339, A079635, A083025 (some of the matched sequences).

With[{k = 4}, Table[Function[{p, d}, k d + Mod[p, k] - k Boole[n == 1]] @@ {#, n/#} &@ FactorInteger[n][[1, 1]], {n, 91}]] (* Michael De Vlieger, May 12 2017 *)

from sympy import divisors, primefactors
def a(n): return 1 if n==1 else 4*divisors(n)[-2] + (min(primefactors(n))%4) # Indranil Ghosh, May 12 2017

(define (A286473 n) (if (= 1 n) n (+ (* 4 (A032742 n)) (modulo (A020639 n) 4))))

a(1) = 1, for n > 1, a(n) = 4*A032742(n) + (A020639(n) mod 4).

A115561 a(n) = lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 5, 1, 1, 1, 2, 1, 1, 3, 7, 1, 5, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 7, 1, 11, 5, 1, 1, 2, 1, 5, 1, 13, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 7, 2, 1, 11, 1, 17, 1, 7, 1, 2, 1, 1, 5, 19, 1, 13, 1, 2, 3, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 23, 1, 1, 1, 2, 1, 7, 11, 5, 1

Offset: 1

Reinhard Zumkeller, Mar 10 2006

nonn

From Peter Munn, Jul 14 2019: (Start)
a(n) = 1 if and only if n is 1 or a prime or semiprime. Otherwise a(n) is the 3rd factor when n is written as a product of primes in nondecreasing order. For example, 60 = 2*2*3*5, so a(60) = 3.
Although values equal to 1 are predominant at low indices, their asymptotic density is 0, whereas for values equal to prime(k) for k > 0 the asymptotic density is positive, namely A281890(k,3)/A002110(k)^3. For all sufficiently large n the median value of a(1), a(2), ... a(n) is A281889(3) = 433.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Least Prime Factor

Cf. A001222, A002110, A014673, A020639, A027746, A032742, A054576, A117358, A281889, A281890.

f[n_] := FactorInteger[n][[1, 1]]; Table[f[#/f@ #] &[n/f@ n], {n, 101}] (* Michael De Vlieger, Aug 14 2017 *)

a020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1) \\ after M. F. Hasler in A020639
a(n) = a020639((n/a020639(n))/a020639(n/a020639(n))) \\ Felix Fröhlich, Jul 15 2019

from sympy import divisors, primefactors
def a032752(n): return 1 if n==1 else divisors(n)[-2]
def a020639(n): return 1 if n==1 else primefactors(n)[0]
def a(n): return a020639(a032752(a032752(n)))
print([a(n) for n in range(1, 102)]) # Indranil Ghosh, Aug 12 2017

a(n) = A020639(A054576(n)).

If A001222(n) >= 3, a(n) = A027746(n,3), otherwise a(n) = 1. - Peter Munn, Jul 13 2019

A242057 Even numbers n for which lpf_3(n-1) < lpf_3(n-3), where lpf_3(n) = lpf(n/3^t) (cf. A020639) such that 3^t (t>=0) is the maximal power of 3 which divides n.

Original entry on oeis.org

10, 16, 22, 26, 28, 34, 36, 40, 46, 50, 56, 64, 66, 70, 76, 82, 86, 92, 96, 100, 106, 112, 116, 120, 126, 130, 134, 136, 142, 144, 146, 154, 156, 160, 162, 166, 170, 176, 184, 186, 190, 196, 202, 204, 206, 210, 214, 216, 222, 226, 232, 236, 244, 254, 256, 260

Offset: 1

Vladimir Shevelev, Aug 13 2014

nonn

An analog of A245024.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A245024, A243937, A020639.

lpf[n_]:=lpf[n]=First[First[FactorInteger[n]]];
lpf3[n_]:=lpf3[n]=If[#==1,1,lpf[#]]&[n/3^IntegerExponent[n,3]]
Select[Range[4,300,2],lpf3[#-1]Peter J. C. Moses, Aug 13 2014 *)

lpf3(n)=m=n/3^valuation(n, 3); if(m>1, factor(m)[1,1], 1)
select(n->lpf3(n-1)Jens Kruse Andersen, Aug 19 2014

More terms from Peter J. C. Moses, Aug 13 2014

A249148 a(1) = 1, after which, if a(n-1) = 1, a(n) = 1 + the total number of 1's that have occurred in the sequence so far, otherwise a(n) = the total number of times the least prime dividing a(n-1) [i.e., A020639(a(n-1))] occurs as a divisor (counted with multiplicity for each term) in the previous terms from a(1) up to and including a(n-1).

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 3, 2, 4, 6, 7, 1, 5, 1, 6, 8, 11, 1, 7, 2, 12, 14, 15, 6, 16, 20, 22, 23, 1, 8, 26, 27, 10, 28, 30, 31, 1, 9, 13, 2, 32, 37, 1, 10, 38, 39, 14, 40, 43, 1, 11, 3, 15, 16, 47, 1, 12, 49, 7, 8, 52, 54, 55, 9, 22, 56, 59, 1, 13, 5, 10, 60, 62, 63, 25, 14, 64, 70, 71, 1, 14, 72, 75, 28, 77, 15, 29, 1, 15, 30, 78, 79, 1, 16, 83

Offset: 1

Antti Karttunen, Oct 24 2014

nonn

Inspired by A248034.

After a(1), it is very likely that 1's occur only just after primes, although they do not necessarily occur after every prime. For example, 13 is the first prime whose initial occurrence is not followed by 1.

			a(1) = 1 by definition.
For n = 2, we see that a(1) = 1, which is the only 1 that has occurred in the sequence so far, and thus a(2) = 1+1 = 2.
For n = 3, we see that a(2) = 2, with the least prime dividing it being 2, which has occurred so far only once (namely in a(2)), thus a(3) = 1.
For n = 4, we see that a(3) = 1, and there has occurred two 1's so far (as a(1) and a(3)), thus a(4) = 2+1 = 3.
For n = 5, we see that a(4) = 3, with the least prime dividing it being 3, which has occurred now just once, thus a(5) = 1.
For n = 6, we see that a(5) = 1, and there has occurred three 1's so far (as a(1), a(3) and a(5)), thus a(6) = 3+1 = 4.
For n = 7, we see that a(6) = 4 = 2*2, with its least prime 2 dividing it two times, and also occurring once at a(2), thus a(7) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A020639, A049084, A055396, A061395, A249147, A248034, A249144, A249069, A249070.

A049084(n) = if(isprime(n), primepi(n), 0); \\ This function from Charles R Greathouse IV
A249148_write_bfile(up_to_n) = { my(pfcounts, n, a_n, f, k); pfcounts = vector(up_to_n); a_n = 1; for(n = 1, up_to_n, if((1 == a_n), pfcounts[1]++; a_n = pfcounts[1], f=factor(a_n); for(i=1,#f~,k = A049084(f[i,1])+1; pfcounts[k] += f[i,2]); a_n = pfcounts[A049084(f[1,1])+1]); write("b249148.txt", n, " ", a_n)); };
A249148_write_bfile(10000);
(MIT/GNU Scheme) ;; With memoizing definec-macro from Antti Karttunen's IntSeq-library and factor function from Aubrey Jaffer's SLIB-library.
(definec (A249148 n) (if (= 1 n) 1 (vector-ref (A249148aux_primefactor_counts (- n 1)) (A055396 (A249148 (- n 1))))))
(definec (A249148aux_primefactor_counts n) (cond ((= 1 n) (vector 2)) (else (let* ((a_n (A249148 n)) (copy-of-prevec (vector-copy (A249148aux_primefactor_counts (- n 1)))) (newsize (max (vector-length copy-of-prevec) (+ 1 (A061395 a_n)))) (pf_counts_vec (vector-grow copy-of-prevec newsize))) (let loop ((pf_indices (map A049084 (factor a_n)))) (cond ((null? pf_indices) pf_counts_vec) (else (vector-set! pf_counts_vec (car pf_indices) (+ 1 (or (vector-ref pf_counts_vec (car pf_indices)) 0))) (loop (cdr pf_indices)))))))))

A249744 a(n) = 0 if n is 1 or a prime, otherwise, when n = A020639(n) * A032742(n), a(n) = the largest m < n such that A020639(m) = A020639(n), where A020639(n) and A032742(n) are the smallest prime and the largest proper divisor dividing n.

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 0, 6, 3, 8, 0, 10, 0, 12, 9, 14, 0, 16, 0, 18, 15, 20, 0, 22, 5, 24, 21, 26, 0, 28, 0, 30, 27, 32, 25, 34, 0, 36, 33, 38, 0, 40, 0, 42, 39, 44, 0, 46, 7, 48, 45, 50, 0, 52, 35, 54, 51, 56, 0, 58, 0, 60, 57, 62, 55, 64, 0, 66, 63, 68, 0, 70, 0, 72, 69, 74, 49, 76, 0, 78, 75, 80, 0, 82, 65, 84, 81, 86, 0, 88, 77, 90, 87, 92, 85, 94, 0, 96, 93, 98, 0, 100

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

For all composite numbers, a(n) tells what is the previous number processed by the sieve of Eratosthenes, i.e., number which is immediately left of n on the same row where n is in arrays like A083140, A083221.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Can be used to compute A078898.

Cf. A000040, A001248, A020639, A083140, A083221, A249738, A250469, A250470.

a[1] = 0; a[?PrimeQ] = 0; a[n] := For[p = FactorInteger[n][[1, 1]]; m = n - p, True, m = m - p, If[FactorInteger[m][[1, 1]] == p, Return[m]]]; Array[a, 102] (* Jean-François Alcover, Mar 08 2016 *)

(define (A249744 n) (* (A020639 n) (A249738 n)))

a(n) = A020639(n) * A249738(n).
Other identities. For all n >= 1 it holds:
a(2n) = 2n-2.
a(A001248(n)) = A000040(n). [I.e., a(p^2) = p for primes p.]

A270436 a(1) = 1, for n > 1, a(n) = A020639(n)^A065621(A067029(n)) * a(A028234(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 128, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 384, 25, 26, 2187, 28, 29, 30, 31, 8192, 33, 34, 35, 36, 37, 38, 39, 640, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 4374, 55, 896, 57, 58, 59, 60, 61, 62, 63, 16384, 65, 66, 67, 68, 69, 70, 71, 1152, 73, 74, 75

Offset: 1

Antti Karttunen, May 27 2016

Antti Karttunen, Table of n, a(n) for n = 1..12167

Cf. A020639, A028234, A065621, A067029.
Cf. A270428 (same sequence sorted into ascending order).
Cf. also A270418, A270419, A270437 and permutation A273671.

f[p_, e_] := p^BitXor[e - 1, 2*e - 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 13 2023 *)

Multiplicative with a(p^e) = p^A065621(e).
a(1) = 1, for n > 1, a(n) = A020639(n)^A065621(A067029(n)) * a(A028234(n)).
Other identities. For all n >= 1:
A270418(a(n)) = n, A270419(a(n)) = 1.

A276347 Numbers n for which A020639(n) = A272565(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 78, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 92, 93, 94, 95

Offset: 1

Antti Karttunen, Sep 11 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A020639, A272565.
Complement: A276437.
Subsequence of A276568.
Subsequences: A047229 (after the initial zero), A192503.

A286475 Compound filter (for counting primes of form 6k+1, 6k+2, 6k+3 and 6k+5): a(n) = 6*A032742(n) + (A020639(n) mod 6), a(1) = 1.

Original entry on oeis.org

1, 8, 9, 14, 11, 20, 7, 26, 21, 32, 11, 38, 7, 44, 33, 50, 11, 56, 7, 62, 45, 68, 11, 74, 35, 80, 57, 86, 11, 92, 7, 98, 69, 104, 47, 110, 7, 116, 81, 122, 11, 128, 7, 134, 93, 140, 11, 146, 43, 152, 105, 158, 11, 164, 71, 170, 117, 176, 11, 182, 7, 188, 129, 194, 83, 200, 7, 206, 141, 212, 11, 218, 7, 224, 153, 230, 67, 236, 7, 242, 165, 248, 11, 254, 107

Offset: 1

Antti Karttunen, May 11 2017

nonn

			For n = 55 = 5*11, a(n) = 6*A032742(55) + (5 modulo 6) = 6*11 + 5 = 71.
For n = 121 = 11*11, a(n) = 6*A032742(121) + (11 modulo 6) = 6*11 + 1 = 71.
For n = 91 = 7*13, a(n) = 6*A032742(91) + (7 modulo 6) = 6*13 + 1 = 79.
For n = 169 = 13*13, a(n) = 6*A032742(169) + (13 modulo 6) = 6*13 + 1 = 79.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A020639, A032742, A285729, A286472, A286473, A286474, A286476.

With[{k = 6}, Table[Function[{p, d}, k d + Mod[p, k] - k Boole[n == 1]] @@ {#, n/#} &@ FactorInteger[n][[1, 1]], {n, 85}]] (* Michael De Vlieger, May 12 2017 *)

from sympy import divisors, primefactors
def a(n): return 1 if n==1 else 6*divisors(n)[-2] +(min(primefactors(n))%6) # Indranil Ghosh, May 12 2017

(define (A286475 n) (if (= 1 n) n (+ (* 6 (A032742 n)) (modulo (A020639 n) 6))))

a(1) = 1, for n > 1, a(n) = 6*A032742(n) + (A020639(n) mod 6).

A319717 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)), and a single bit A319710(n) telling whether the smallest prime factor is unitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 18, 5, 19, 20, 21, 22, 23, 5, 24, 7, 25, 26, 27, 28, 29, 7, 30, 31, 32, 5, 33, 7, 34, 35, 36, 5, 37, 38, 39, 40, 41, 5, 42, 43, 44, 45, 46, 5, 47, 7, 48, 49, 50, 51, 52, 7, 53, 54, 55, 5, 56, 7, 57, 58, 59, 60, 61, 7, 62, 63, 64, 5, 65, 66, 67, 68, 69, 5, 70, 71, 72, 73, 74, 75, 76, 7, 77, 78, 79, 5, 80, 7, 81, 82, 83, 5, 84, 7, 85, 86, 87, 5, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Antti Karttunen, Oct 04 2018

nonn

Restricted growth sequence transform of triple [A010875(A020639(n)), A032742(n), A319710(n)] (with a separate value allotted for a(1)), or equally, of ordered pair [A319716(n), A319710(n)].
In addition to A319716, this filter sequence also records in the value of a(n) also the fact whether the smallest prime factor of n is unitary or not. This information is enough to determine the modulo 6 residues of all the divisors of n, thus sequences like A002324 are essentially functions of this sequence. Moreover, a lot of other information is immediately (and unavoidably) present, for example the exact prime signature of n, including also the relative order of exponents.
Any such filtering sequence can be perceived also in terms of what information it leaves out from a(n) that would be needed to reconstruct whole n from each a(n). If the whole n could be reconstructed from a(n) each time, then sequence a would be injective, and would be useless for filtering, because then it would match with any sequence. In this filter, what is left out is only the exact identity of the smallest prime factor, although its residue class mod 6 is retained. However, when the smallest prime factor is 2 or 3, this can be seen from that residue value, so for any number x in A047229, both A020639(x) and A032742(x) are known, and as x = A020639(x)*A032742(x), it means such numbers must occur in their own singleton equivalence classes.
Likewise, for any n in A283050, even if not divisible by 2 or 3, when we have A319710(n) stored in the triple as 1, this immediately gives away the exact identity of the smallest prime factor, which is equal to A014673(n) = A020639(A032742(n)) in these cases.
Thus there is a substantial subset of N (containing at least the union of A047229 and A283050) which is actually in the "blind sector" of this filter, "where anything goes", as this sequence obtains only unique values in that subdomain.
There is a related filter sequence A319996, which operates by "cleaving n from its high end" (by storing the residue class of the largest prime factor, A006530, instead of the smallest, together with n/A006530(n)), which has its own blind spots, but fortunately, they do not fully coincide with the blind spots of this filter. Naturally, any sequence like A002324 should match both to this sequence and A319996.
For all i, j:
  a(i) = a(j) => A002324(i) = A002324(j),
  a(i) = a(j) => A067029(i) = A067029(j),
  a(i) = a(j) => A071178(i) = A071178(j),
  a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A319716(i) = A319716(j) => A319690(i) = A319690(j).

			For n = 65 = 5*13 and 143 = 11*13, the smallest prime factor is of the form 6k+5,  doesn't occur more than once in the factorization, and the largest proper divisor is the same number (13) in both cases, thus a(65) = a(143) (= 51, a running count value allotted by rgs-transform for this equivalence class).
For n = 1805 (5*19^2), 3971 (11*19^2), 6137 (17*19^2), it's like above, but the largest proper divisor is in all three cases 361 = 19^2, thus a(1805) = a(3971) = a(6137) (= 1405).
Note that such nontrivial equivalence classes may only contain numbers that are 5-rough, A007310, with no prime factors 2 or 3, and also, they may not contain numbers from A283050. See the comments section.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A101296, A319707, A319716, A319996.
Cf. also A320004 (analogous sequence for modulo 4 residues).
Differs from A319707 for the first time at n=143, where a(143) = 51, differs from A319716 for the first time at n=121, where a(121) = 95.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A286476(n) = if(1==n,n,(6*A032742(n) + (n % 6)));
A319710(n) = ((n>1)&&(factor(n)[1,2]>1));
v319717 = rgs_transform(vector(up_to,n,[A286476(n),A319710(n)]));
A319717(n) = v319717[n];

cp's OEIS Frontend

A286472 Compound filter (for counting prime gaps): a(1) = 1, a(n) = 2*A032742(n) + (1 if n is composite and spf(A032742(n)) > nextprime(spf(n)), and 0 otherwise). Here spf is the smallest prime factor, A020639.

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A286473 Compound filter (for counting primes of form 4k+1, 4k+2 and 4k+3): a(n) = 4*A032742(n) + (A020639(n) mod 4), a(1) = 1.

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A115561 a(n) = lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.

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A242057 Even numbers n for which lpf_3(n-1) < lpf_3(n-3), where lpf_3(n) = lpf(n/3^t) (cf. A020639) such that 3^t (t>=0) is the maximal power of 3 which divides n.

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A249744 a(n) = 0 if n is 1 or a prime, otherwise, when n = A020639(n) * A032742(n), a(n) = the largest m < n such that A020639(m) = A020639(n), where A020639(n) and A032742(n) are the smallest prime and the largest proper divisor dividing n.

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A270436 a(1) = 1, for n > 1, a(n) = A020639(n)^A065621(A067029(n)) * a(A028234(n)).

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A276347 Numbers n for which A020639(n) = A272565(n).

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