A286455 -id:A286455 - cp's OEIS frontend

A286470 a(n) = maximal gap between indices of successive primes in the prime factorization of n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 1, 0, 0, 1, 0, 2, 2, 4, 0, 1, 0, 5, 0, 3, 0, 1, 0, 0, 3, 6, 1, 1, 0, 7, 4, 2, 0, 2, 0, 4, 1, 8, 0, 1, 0, 2, 5, 5, 0, 1, 2, 3, 6, 9, 0, 1, 0, 10, 2, 0, 3, 3, 0, 6, 7, 2, 0, 1, 0, 11, 1, 7, 1, 4, 0, 2, 0, 12, 0, 2, 4, 13, 8, 4, 0, 1, 2, 8, 9, 14, 5, 1, 0, 3, 3, 2, 0, 5, 0, 5, 1, 15, 0, 1, 0, 2, 10, 3, 0, 6, 6, 9, 4, 16, 3, 1

Offset: 1

Antti Karttunen, May 13 2017

nonn

			For n = 70 = 2*5*7 = prime(1)*prime(3)*prime(4), the largest index difference occurs between prime(1) and prime(3), thus a(70) = 3-1 = 2.

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

Cf. A001221, A032742, A055396, A073490, A243055, A286455, A286471, A286472.
Cf. A286469 (version which considers the index of the smallest prime as the initial gap).
Cf. A000961 (positions of zeros).
Differs from A242411 for the first time at n=70, where a(70) = 2, while A242411(70) = 1.

Table[If[Or[n == 1, PrimeNu@ n == 1], 0, Max@ Differences@ PrimePi[FactorInteger[n][[All, 1]]]], {n, 120}] (* Michael De Vlieger, May 16 2017 *)

from sympy import primepi, isprime, primefactors, divisors
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def x(n): return 1 if n==1 else divisors(n)[-2]
def a(n): return 0 if n==1 or len(primefactors(n))==1 else max(a055396(x(n)) - a055396(n), a(x(n))) # Indranil Ghosh, May 17 2017

(define (A286470 n) (cond ((or (= 1 n) (= 1 (A001221 n))) 0) (else (max (- (A055396 (A032742 n)) (A055396 n)) (A286470 (A032742 n))))))

a(1) = 0, for n > 1, if A001221(n) = 1 [when n is a prime power], a(n) = 0, otherwise a(n) = max((A055396(A032742(n))-A055396(n)), a(A032742(n))).

For all n >= 1, a(n) <= A243055(n).

Definition corrected by Zak Seidov, May 16 2017

A286454 Compound filter (prime signature & prime signature of conjugated prime factorization): a(n) = P(A101296(n), A286621(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 8, 9, 12, 32, 23, 20, 13, 49, 38, 51, 47, 82, 49, 35, 68, 51, 80, 72, 124, 140, 122, 74, 18, 175, 26, 111, 155, 334, 192, 65, 257, 280, 82, 116, 255, 329, 355, 99, 327, 570, 380, 177, 72, 469, 437, 132, 31, 72, 532, 216, 498, 74, 257, 144, 599, 634, 597, 448, 632, 745, 159, 119, 784, 1044, 782, 331, 907, 570, 863, 186, 905, 1039, 72, 384, 140, 1335, 1037

Offset: 1

Antti Karttunen, May 14 2017

nonn

Here, instead of A046523 and A278221 we use as the components of a(n) their rgs-versions A101296 and A286621 because of the latter sequence's moderate growth rates.

For all i, j: a(i) = a(j) => A286356(i) = A286356(j).

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

Cf. A000027, A046523, A101296, A122111, A278221, A285334, A286621, A286356, A286455, A286456.

(define (A286454 n) (* (/ 1 2) (+ (expt (+ (A101296 n) (A286621 n)) 2) (- (A101296 n)) (- (* 3 (A286621 n))) 2)))

a(n) = (1/2)*(2 + ((A101296(n)+A286621(n))^2) - A101296(n) - 3*A286621(n)).

A286456 Compound filter: a(n) = P(A056239(n), A243503(n)), with a(1) = 0, where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 1, 5, 5, 13, 18, 25, 13, 19, 40, 41, 40, 61, 71, 41, 25, 85, 41, 113, 83, 72, 111, 145, 71, 42, 160, 42, 142, 181, 84, 221, 41, 112, 218, 73, 72, 265, 285, 161, 142, 313, 143, 365, 217, 85, 361, 421, 111, 74, 85, 219, 308, 481, 73, 113, 238, 286, 446, 545, 143, 613, 540, 144, 61, 162, 218, 685, 415, 362, 144, 761, 112, 841, 643, 86, 538, 114, 309, 925, 217

Offset: 1

Antti Karttunen, May 14 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192 (computed by using the b-files of A056239 and A243503)
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A056239, A243503, A278520, A286454, A286455.

(define (A286456 n) (if (= 1 n) 0 (* (/ 1 2) (+ (expt (+ (A056239 n) (A243503 n)) 2) (- (A056239 n)) (- (* 3 (A243503 n))) 2))))

a(1) = 0 and for n > 1, a(n) = (1/2)*(2 + ((A056239(n)+A243503(n))^2) - A056239(n) - 3*A243503(n)).

A328469 Lexicographically earliest infinite sequence such that a(i) = a(j) => A020639(i) = A020639(j) and A046523(i) = A046523(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 6, 10, 11, 12, 6, 13, 14, 15, 11, 16, 11, 13, 6, 17, 18, 19, 6, 20, 11, 21, 22, 23, 24, 13, 6, 25, 26, 27, 6, 13, 18, 28, 22, 29, 11, 30, 6, 31, 32, 33, 11, 13, 11, 34, 18, 25, 18, 13, 6, 35, 36, 37, 6, 30, 38, 25, 22, 39, 11, 13, 22, 40, 41, 42, 6, 30, 11, 43, 22, 44, 32, 45, 6, 46, 36, 25, 6, 13, 18, 47, 36, 43, 11, 13, 6, 25, 48, 49, 11, 30, 26, 50, 22, 51, 18, 52

Offset: 1

Antti Karttunen, Oct 19 2019

nonn

Restricted growth sequence transform of the ordered pair [A020639(n), A046523(n)], where A020639(n) gives the smallest prime factor of n, while A046523(n) gives the prime signature of n.

For all i, j: a(i) = a(j) => A291761(i) = A291761(j).

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

Cf. A020639, A046523.

Cf. also A101296, A291761, A328470, A328477, A318891, A286455.

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; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
Aux328469(n) = [A020639(n), A046523(n)];
v328469 = rgs_transform(vector(up_to, n, Aux328469(n)));
A328469(n) = v328469[n];

cp's OEIS Frontend

A286470 a(n) = maximal gap between indices of successive primes in the prime factorization of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

Extensions

A286454 Compound filter (prime signature & prime signature of conjugated prime factorization): a(n) = P(A101296(n), A286621(n)), where P(n,k) is sequence A000027 used as a pairing function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A286456 Compound filter: a(n) = P(A056239(n), A243503(n)), with a(1) = 0, where P(n,k) is sequence A000027 used as a pairing function.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A328469 Lexicographically earliest infinite sequence such that a(i) = a(j) => A020639(i) = A020639(j) and A046523(i) = A046523(j) for all i, j.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI