A286473 -id:A286473 - cp's OEIS frontend

A286386 Compound filter: a(n) = 2*A286473(n) + (1 if n is a square, 0 otherwise).

3, 12, 14, 21, 10, 28, 14, 36, 31, 44, 14, 52, 10, 60, 46, 69, 10, 76, 14, 84, 62, 92, 14, 100, 43, 108, 78, 116, 10, 124, 14, 132, 94, 140, 58, 149, 10, 156, 110, 164, 10, 172, 14, 180, 126, 188, 14, 196, 63, 204, 142, 212, 10, 220, 90, 228, 158, 236, 14, 244, 10, 252, 174, 261, 106, 268, 14, 276, 190, 284, 14, 292, 10, 300, 206, 308, 94, 316, 14, 324, 223

Offset: 1

Antti Karttunen, May 13 2017

nonn

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

Cf. A000290 (gives the positions off odd terms), A010052, A286473, A286366, A286388.

from sympy import sqrt, divisors, primefactors
import math
def a010052(n): return 1 if n<1 else int(math.floor(sqrt(n))) - int(math.floor(sqrt(n - 1)))
def a286473(n): return 1 if n==1 else 4*divisors(n)[-2] + (min(primefactors(n))%4)
def a(n): return 2*a286473(n) + a010052(n) # Indranil Ghosh, May 14 2017

(define (A286386 n) (+ (* 2 (A286473 n)) (A010052 n)))

a(n) = 2*A286473(n) + A010052(n).

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.

Original entry on oeis.org

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.

A286474 Compound filter: a(n) = 4*A032742(n) + (n mod 4), a(1) = 1.

Original entry on oeis.org

1, 6, 7, 8, 5, 14, 7, 16, 13, 22, 7, 24, 5, 30, 23, 32, 5, 38, 7, 40, 29, 46, 7, 48, 21, 54, 39, 56, 5, 62, 7, 64, 45, 70, 31, 72, 5, 78, 55, 80, 5, 86, 7, 88, 61, 94, 7, 96, 29, 102, 71, 104, 5, 110, 47, 112, 77, 118, 7, 120, 5, 126, 87, 128, 53, 134, 7, 136, 93, 142, 7, 144, 5, 150, 103, 152, 45, 158, 7, 160, 109, 166, 7, 168, 69, 174, 119, 176, 5, 182, 55

Offset: 1

Antti Karttunen, May 11 2017

nonn

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

Cf. A032742, A285729, A286472, A286473, A286475, A286476.

Table[If[n == 1, 1, 4 (Divisors[n][[-2]]) + Mod[n, 4]], {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] + n%4 # Indranil Ghosh, May 12 2017

(define (A286474 n) (if (= 1 n) n (+ (* 4 (A032742 n)) (modulo n 4))))

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

A286476 Compound filter: a(n) = 6*A032742(n) + (n mod 6), a(1) = 1.

Original entry on oeis.org

1, 8, 9, 16, 11, 18, 7, 26, 21, 34, 11, 36, 7, 44, 33, 52, 11, 54, 7, 62, 45, 70, 11, 72, 31, 80, 57, 88, 11, 90, 7, 98, 69, 106, 47, 108, 7, 116, 81, 124, 11, 126, 7, 134, 93, 142, 11, 144, 43, 152, 105, 160, 11, 162, 67, 170, 117, 178, 11, 180, 7, 188, 129, 196, 83, 198, 7, 206, 141, 214, 11, 216, 7, 224, 153, 232, 71, 234, 7, 242, 165, 250, 11, 252, 103

Offset: 1

Antti Karttunen, May 11 2017

nonn

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

Cf. A032742, A285729, A286472, A286473, A286474, A286475.

With[{k = 6}, Table[If[n == 1, 1, k (Divisors[n][[-2]]) + Mod[n, k]], {n, 85}]] (* Michael De Vlieger, May 12 2017 *)

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

(define (A286476 n) (if (= 1 n) n (+ (* 6 (A032742 n)) (modulo n 6))))

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

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).

A319714 Filter sequence combining the largest proper divisor of n (A032742) with n's residue modulo 4 (A010873).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 3, 7, 8, 9, 3, 10, 5, 11, 12, 13, 5, 14, 3, 15, 16, 17, 3, 18, 19, 20, 21, 22, 5, 23, 3, 24, 25, 26, 27, 28, 5, 29, 30, 31, 5, 32, 3, 33, 34, 35, 3, 36, 16, 37, 38, 39, 5, 40, 41, 42, 43, 44, 3, 45, 5, 46, 47, 48, 49, 50, 3, 51, 52, 53, 3, 54, 5, 55, 56, 57, 25, 58, 3, 59, 60, 61, 3, 62, 63, 64, 65, 66, 5, 67, 30, 68, 69, 70, 71, 72, 5, 73

Offset: 1

Antti Karttunen, Sep 26 2018

nonn

Restricted growth sequence transform of A286474, or equally, of A286473.

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

Cf. A010873, A032742, A286473, A286474, A319704.

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]));
A286474(n) = if(1==n,n,(4*A032742(n) + (n % 4)));
v319714 = rgs_transform(vector(up_to,n,A286474(n)));
A319714(n) = v319714[n];

cp's OEIS Frontend

A286386 Compound filter: a(n) = 2*A286473(n) + (1 if n is a square, 0 otherwise).

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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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A286474 Compound filter: a(n) = 4*A032742(n) + (n mod 4), a(1) = 1.

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A286476 Compound filter: a(n) = 6*A032742(n) + (n mod 6), a(1) = 1.

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Mathematica

Python

Scheme

Formula

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.

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A319714 Filter sequence combining the largest proper divisor of n (A032742) with n's residue modulo 4 (A010873).

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