A287170 a(n) = number of runs of consecutive prime numbers among the prime divisors of n.

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2

Offset: 1

Author

Rémy Sigrist, Jun 04 2017

Keywords

• nonn

Comments

a(n) = 0 iff n = 1.

a(n) = 1 iff n belongs to A073491.

a(p) = 1 for any prime p.

a(A002110(n)) = 1 for any n > 0.

a(n!) = 1 for any n > 1.

a(A066205(n)) = n for any n > 0.

a(n) = a(A007947(n)) for any n > 0.

a(n) = a(A003961(n)) for any n > 0.

a(n*m) <= a(n) + a(m) for any n > 0 and m > 0.

Each number n can be uniquely represented as a product of a(n) distinct terms from A073491; this representation is minimal relative to the number of terms.

Examples

			See illustration of the first terms in the Links section.
The prime indices of 18564 are {1,1,2,4,6,7}, which separate into maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 3; this corresponds to the ordered factorization 18564 = 12 * 7 * 221. - _Gus Wiseman_, Sep 03 2022
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Illustration of the first terms
• Rémy Sigrist, Scatterplot of the ordinal transform of the first 1000000 terms

Crossrefs

Cf. A002110, A003961, A007947, A069010, A087207.

Positions of first appearances are A066205.

These are the row-lengths of A356226 and A356234. Other statistics are:

- length: A287170 (this sequence)

- minimum: A356227

- maximum: A356228

- bisected length: A356229

- standard composition: A356230

- Heinz number: A356231

- positions of first appearances: A356603 or A356232 (sorted)

A001222 counts prime factors, distinct A001221.

A003963 multiplies together the prime indices.

A056239 adds up the prime indices, row sums of A112798.

A073491 lists numbers with gapless prime indices, complement A073492.

Cf. A000005, A053251, A055932, A061395, A073493, A107428, A132747, A137921, A193829, A286470.

Programs

Mathematica

Table[Length[Select[First/@If[n==1,{},FactorInteger[n]],!Divisible[n,NextPrime[#]]&]],{n,30}] (* Gus Wiseman, Sep 03 2022 *)

PARI

a(n) = my (f=factor(n)); if (#f~==0, return (0), return (#f~ - sum(i=1, #f~-1, if (primepi(f[i,1])+1 == primepi(f[i+1,1]), 1, 0))))

Python

from sympy import factorint, primepi
def a087207(n):
    f=factorint(n)
    return sum([2**primepi(i - 1) for i in f])
def a069010(n): return sum(1 for d in bin(n)[2:].split('0') if len(d)) # this function from Chai Wah Wu
def a(n): return a069010(a087207(n)) # Indranil Ghosh, Jun 06 2017

Formula

a(n) = A069010(A087207(n))