A032742 -id:A032742 - 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).

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).

A300236 Möbius transform of A032742 (the largest proper divisor of n).

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 2, 2, 4, 0, 2, 0, 6, 4, 4, 0, 4, 0, 4, 6, 10, 0, 4, 4, 12, 6, 6, 0, 4, 0, 8, 10, 16, 6, 6, 0, 18, 12, 8, 0, 6, 0, 10, 8, 22, 0, 8, 6, 16, 16, 12, 0, 12, 10, 12, 18, 28, 0, 8, 0, 30, 12, 16, 12, 10, 0, 16, 22, 18, 0, 12, 0, 36, 16, 18, 10, 12, 0, 16, 18, 40, 0, 12, 16, 42, 28, 20, 0, 16, 12, 22, 30, 46, 18, 16, 0, 36

Offset: 1

Antti Karttunen, Mar 10 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A008683, A032742, A300239.

Table[DivisorSum[n, # MoebiusMu[n/#]/FactorInteger[#][[1, 1]] &], {n, 98}] (* Michael De Vlieger, Mar 10 2018 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A300236(n) = sumdiv(n,d,moebius(n/d)*A032742(d));

a(n) = Sum_{d|n} A008683(n/d)*A032742(d).

A318508 a(n) = A032742(n) AND A001065(n)-A032742(n), where AND is bitwise-and (A004198) and A001065 = sum of proper divisors and A032742 = the largest proper divisor of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 3, 4, 0, 0, 8, 0, 8, 4, 3, 0, 8, 1, 1, 0, 14, 0, 11, 0, 0, 0, 1, 6, 0, 0, 3, 4, 20, 0, 1, 0, 18, 2, 3, 0, 16, 1, 16, 0, 16, 0, 3, 2, 4, 0, 1, 0, 14, 0, 3, 20, 0, 4, 33, 0, 0, 4, 35, 0, 4, 0, 1, 24, 2, 8, 35, 0, 0, 9, 1, 0, 34, 0, 3, 4, 32, 0, 33, 8, 14, 4, 3, 2, 32, 0, 16, 0, 2, 0, 51, 0, 52, 32

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A001065, A004198, A032742, A318505, A318506, A318507.

Cf. also A318457, A318517.

A001065(n) = (sigma(n)-n);
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318505(n) = if(1==n,0,A001065(n)-A032742(n));
A318508(n) = bitand(A318505(n),A032742(n));

a(n) = A004198(A032742(n), A318505(n)).

For n > 1, a(n) = A001065(n) - A318506(n) = (A001065(n) - A318507(n))/2.

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.]

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).

A318506 a(n) = A032742(n) OR A001065(n)-A032742(n), where OR is bitwise-or (A003986) and A001065 = sum of proper divisors and A032742 = the largest proper divisor of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 7, 3, 7, 1, 14, 1, 7, 5, 15, 1, 13, 1, 14, 7, 11, 1, 28, 5, 15, 13, 14, 1, 31, 1, 31, 15, 19, 7, 55, 1, 19, 13, 30, 1, 53, 1, 22, 31, 23, 1, 60, 7, 27, 21, 30, 1, 63, 15, 60, 23, 31, 1, 94, 1, 31, 21, 63, 15, 45, 1, 58, 23, 39, 1, 119, 1, 39, 25, 62, 11, 55, 1, 106, 31, 43, 1, 106, 23, 43, 29, 60, 1, 111, 13, 62, 31

Offset: 1

Antti Karttunen, Aug 28 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A001065, A003986, A032742, A318505, A318507, A318508.

Cf. also A318456, A318516.

A001065(n) = (sigma(n)-n);
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318505(n) = if(1==n,0,A001065(n)-A032742(n));
A318506(n) = bitor(A318505(n),A032742(n));

a(n) = A003986(A032742(n), A318505(n)).

For n > 1, a(n) = A001065(n) - A318508(n).

A318507 a(n) = A032742(n) XOR A001065(n)-A032742(n), where XOR is bitwise-or (A003987) and A001065 = sum of proper divisors and A032742 = the largest proper divisor of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 0, 1, 7, 2, 6, 1, 12, 1, 4, 1, 15, 1, 5, 1, 6, 3, 8, 1, 20, 4, 14, 13, 0, 1, 20, 1, 31, 15, 18, 1, 55, 1, 16, 9, 10, 1, 52, 1, 4, 29, 20, 1, 44, 6, 11, 21, 14, 1, 60, 13, 56, 23, 30, 1, 80, 1, 28, 1, 63, 11, 12, 1, 58, 19, 4, 1, 115, 1, 38, 1, 60, 3, 20, 1, 106, 22, 42, 1, 72, 23, 40, 25, 28, 1, 78, 5, 48, 27, 44, 21, 92, 1

Offset: 1

Antti Karttunen, Aug 28 2018

Note that here zeros occur only on even perfect numbers (even terms of A000396), in contrast to A318457, which would be zero also for any hypothetical odd perfect number. - Antti Karttunen, Aug 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A000396, A001065, A003987, A032742, A318505, A318506, A318508.

Cf. also A106409, A318457, A318462, A318517.

A001065(n) = (sigma(n)-n);
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A318505(n) = if(1==n,0,A001065(n)-A032742(n));
A318507(n) = bitxor(A318505(n),A032742(n));

a(n) = A003987(A032742(n), A318505(n)).

For n > 1, a(n) = A001065(n) - 2*A318508(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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

A300236 Möbius transform of A032742 (the largest proper divisor of n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A318508 a(n) = A032742(n) AND A001065(n)-A032742(n), where AND is bitwise-and (A004198) and A001065 = sum of proper divisors and A032742 = the largest proper divisor of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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.

Views