A095162 -id:A095162 - cp's OEIS frontend

A095163 a(n) = smallest divisor d of n that occurs earlier in the sequence fewer than d times.

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 4, 13, 7, 5, 4, 17, 6, 19, 4, 7, 11, 23, 6, 5, 13, 9, 7, 29, 5, 31, 8, 11, 17, 7, 6, 37, 19, 13, 8, 41, 6, 43, 11, 9, 23, 47, 6, 7, 10, 17, 13, 53, 6, 11, 7, 19, 29, 59, 10, 61, 31, 9, 8, 13, 11, 67, 17, 23, 10, 71, 8, 73, 37, 15, 19, 11, 13, 79, 8, 9, 41, 83

Offset: 1

Amarnath Murthy, Jun 01 2004

nonn

Agrees with A033677 for the first 19 and many further terms; A095787 gives those n for which A033677 and the present sequence disagree.

			For n = 12 we have divisors 1, 2, 3, 4, 6, 12; 1 occurs earlier once, 2 occurs earlier twice, 3 occurs earlier 3 times, but 4 occurs earlier only once, hence a(12) = 4.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A033677, A095787, A095161, A095162, A095164, A095165.

nn = 120; c[] = 0; Do[k = SelectFirst[Divisors[n], c[#] < # &]; a[n] = k; c[k]++, {n, nn}]; Array[a, nn] (* _Michael De Vlieger, Oct 14 2022 *)

{m=83;v=vector(m);for(n=1,m,d=divisors(n);j=1;while(v[d[j]]>=d[j],j++);print1(d[j],",");v[d[j]]=v[d[j]]+1)}

from sympy import divisors
from collections import Counter
from itertools import count, islice
def agen():
    c = Counter()
    for n in count(1):
        an = next(d for d in divisors(n) if c[d] < d)
        c[an] += 1
        yield an
print(list(islice(agen(), 83))) # Michael S. Branicky, Oct 14 2022

a(n) >= n^(1/3). - Charles R Greathouse IV, Oct 14 2022

Edited and extended by Klaus Brockhaus Jun 05 2004 and Jun 09 2004

A095161 n occurs n times, with distinct successive terms.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 3, 5, 4, 5, 4, 5, 4, 5, 6, 5, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 10, 9, 10, 9, 10, 9, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 12, 11, 12, 11, 12, 11

Offset: 0

Amarnath Murthy, Jun 01 2004

nonn

A different sequence with the same definition is 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 10, .. This would follow a more regular construction principle, that is, placing low numbers as early as possible: here a(7)=4 instead of a(7)=5 etc. - R. J. Mathar, Sep 25 2006

Cf. A095162.

A095164 Index of the first occurrence of n in A095163.

Original entry on oeis.org

1, 2, 3, 8, 5, 18, 7, 32, 27, 50, 11, 84, 13, 98, 75, 128, 17, 162, 19, 200, 147, 242, 23, 312, 125, 338, 243, 392, 29, 510, 31, 512, 363, 578, 245, 684, 37, 722, 507, 920, 41, 882, 43, 968, 765, 1058, 47, 1392, 343, 1250, 867, 1352, 53, 1458, 605, 1624, 1083

Offset: 1

Amarnath Murthy, Jun 01 2004

nonn

Is this the same as A075384? - R. J. Mathar, Oct 28 2008

Cf. A095161, A095162, A095163, A095165.

A095163 := proc(nmax) local a,dvs,d,n; a := [1,2,3] ; for n from 4 to nmax do dvs := sort(convert(numtheory[divisors](n),list)) ; for d in dvs do if ListTools[Occurrences](d,a) < d then a := [op(a),d] ; break; fi; od: od: a ; end: A095164 := proc(n,a095163) local i ; for i from 1 to nops(a095163) do if op(i,a095163) = n then RETURN(i) ; fi; od: RETURN(-1) ; end: a095163 := A095163(3700) ; for n from 1 do a095 := A095164(n,a095163) ; if a095 < 0 then break; else printf("%d,",a095) ; fi; od: # R. J. Mathar, Oct 28 2008

More terms from Nadia Heninger, Jul 07 2005

More terms from R. J. Mathar, Oct 28 2008

A095165 n divided by A095163(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 4, 1, 3, 1, 5, 3, 2, 1, 4, 5, 2, 3, 4, 1, 6, 1, 4, 3, 2, 5, 6, 1, 2, 3, 5, 1, 7, 1, 4, 5, 2, 1, 8, 7, 5, 3, 4, 1, 9, 5, 8, 3, 2, 1, 6, 1, 2, 7, 8, 5, 6, 1, 4, 3, 7, 1, 9, 1, 2, 5, 4, 7, 6, 1, 10, 9, 2, 1, 7, 5, 2, 3, 11, 1, 10, 7, 4, 3, 2, 5, 12, 1, 7, 11, 10, 1, 6

Offset: 1

Amarnath Murthy, Jun 01 2004

nonn

Differs from A059981 and A033676 at n = 20. - Franklin T. Adams-Watters, Dec 07 2006

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A095161, A095162, A095163, A095164.

Cf. A116548.

nn = 102; c[] = 0; Do[k = SelectFirst[Divisors[n], c[#] < # &]; a[n] = k; c[k]++, {n, nn}]; Array[#/a[#] &, nn] (* _Michael De Vlieger, Oct 14 2022 *)

{ for (n=1, #nb=vector(102), fordiv (n, d, if (nb[d]Rémy Sigrist, Oct 14 2022

More terms from Franklin T. Adams-Watters, Dec 07 2006

A094173 a(n) = m if m has already occurred at least once and n=k+i*(m+1) where k is the index of the first occurrence of n and i=1,...,max(n-1,1), otherwise a(n) = least positive integer that has not yet occurred.

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 5, 3, 6, 7, 4, 3, 5, 8, 9

Offset: 1

Amarnath Murthy, Jun 01 2004

Is the sequence possible for all n?

a(16) does not exist: it would have to be 4 (since a(6) = a(11) = 4 and so a(6 + 2*(4 + 1)) = a(16) = 4) and also 6 (since a(9) = 6 and so a(9 + 1*(6 + 1)) = a(16) = 6) simultaneously. - Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

Cf. A095162, A095163, A095164.

Edited by Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

cp's OEIS Frontend

A095163 a(n) = smallest divisor d of n that occurs earlier in the sequence fewer than d times.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A095161 n occurs n times, with distinct successive terms.

Views

Author

Keywords

Comments

Crossrefs

A095164 Index of the first occurrence of n in A095163.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Extensions

A095165 n divided by A095163(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A094173 a(n) = m if m has already occurred at least once and n=k+i*(m+1) where k is the index of the first occurrence of n and i=1,...,max(n-1,1), otherwise a(n) = least positive integer that has not yet occurred.

Views

Author

Keywords

Comments

Crossrefs

Extensions