A344758 -id:A344758 - cp's OEIS frontend

A344590 Number of divisors d of n for which A011772(d) = A011772(n), where A011772(n) is the smallest number m such that m(m+1)/2 is divisible by n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 2

Offset: 1

Antti Karttunen, May 31 2021

nonn

			36 has 9 divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36. When A011772 is applied to them, one obtains values [1, 3, 2, 7, 3, 8, 8, 8, 8], thus there are four divisors that obtain the maximal value 8 obtained at 36 itself, therefore a(36) = 4.

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

Cf. A000005, A011772, A344588 (positions of records), A344589, A344758, A344770, A344881 (positions of ones), A344882 (of terms > 1).

A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];
a[n_] := With[{m = A011772[n]}, Count[Divisors[n], d_ /; A011772[d] == m]];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)

A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344590(n) = { my(x=A011772(n)); sumdiv(n,d,A011772(d)==x); };

from itertools import combinations
from functools import reduce
from operator import mul
from sympy import factorint, divisors
from sympy.ntheory.modular import crt
def A011772(n):
    plist = [p**q for p, q in factorint(2*n).items()]
    if len(plist) == 1:
        return n-1 if plist[0] % 2 else 2*n-1
    return min(min(crt([m,2*n//m],[0,-1])[0],crt([2*n//m,m],[0,-1])[0]) for m in (reduce(mul, d) for l in range(1,len(plist)//2+1) for d in combinations(plist,l)))
def A344590(n):
    m = A011772(n)
    return sum(1 for d in divisors(n) if A011772(d) == m) # Chai Wah Wu, Jun 02 2021

a(n) = Sum_{d|n} [A011772(d) = A011772(n)], where [ ] is the Iverson bracket.
a(n) = A000005(n) - A344589(n).
a(n) <= A344770(n).

A344759 a(n) = n divided by the smallest divisor d of n for which A011772(d) = A011772(n), where A011772(n) is the smallest number m such that m(m+1)/2 is divisible by n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 01 2021

nonn

It seems that A006516 gives the positions of records after its initial zero.

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

Cf. A006516, A011772, A344758, A344881 (positions of ones), A344882 (of terms > 1).

A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344759(n) = { my(x=A011772(n)); fordiv(n,d,if(A011772(d)==x, return(n/d))); };
(Python 3.8+)
from itertools import combinations
from math import prod
from sympy import factorint, divisors
from sympy.ntheory.modular import crt
def A011772(n):
    plist = [p**q for p, q in factorint(2*n).items()]
    if len(plist) == 1:
        return n-1 if plist[0] % 2 else 2*n-1
    return min(min(crt([m,2*n//m],[0,-1])[0],crt([2*n//m,m],[0,-1])[0]) for m in (prod(d) for l in range(1,len(plist)//2+1) for d in combinations(plist,l)))
def A344759(n):
    m = A011772(n)
    for d in divisors(n):
        if A011772(d) == m:
            return n//d # Chai Wah Wu, Jun 03 2021

a(n) = n / A344758(n).

A346088 Smallest divisor d of n for which A002034(d) = A002034(n), where A002034(n) is the smallest positive integer k such that k! is a multiple of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 4, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 5, 7, 11, 23, 4, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 5, 41, 7, 43, 11, 9, 23, 47, 16, 49, 25, 17, 13, 53, 27, 11, 7, 19, 29, 59, 5, 61, 31, 7, 32, 13, 11, 67, 17, 23, 7, 71, 9, 73, 37, 25, 19, 11, 13, 79, 16, 27, 41, 83, 7, 17, 43, 29, 11, 89

Offset: 1

Antti Karttunen, Jul 05 2021

nonn

			36 has 9 divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36. When A002034 is applied to them, one obtains values [1, 2, 3, 4, 3, 6, 4, 6, 6], thus there are three divisors that obtain the maximal value 6 obtained at 36 itself, of which divisor 9 is the smallest, and therefore a(36) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A002034, A345935, A345936, A346089.
Cf. also A344758.
Differs from A223491 for the first time at n=27, where a(27) = 27, while A223491(27) = 9.

A002034(n) = if(1==n,n,my(s=factor(n)[, 1], k=s[#s], f=Mod(k!, n)); while(f, f*=k++); (k)); \\ After code in A002034.
A346088(n) = { my(x=A002034(n)); fordiv(n,d,if(A002034(d)==x, return(d))); };

a(n) = n / A346089(n).

A354998 Smallest divisor d of n for which A344005(d) = A344005(n), where A344005(n) is the smallest positive integer m such that n divides m*(m+1).

Original entry on oeis.org

1, 1, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 15, 16, 17, 9, 19, 5, 7, 11, 23, 24, 25, 13, 27, 28, 29, 15, 31, 32, 33, 17, 35, 9, 37, 19, 13, 40, 41, 7, 43, 44, 45, 23, 47, 16, 49, 25, 51, 13, 53, 27, 11, 8, 19, 29, 59, 60, 61, 31, 63, 64, 65, 33, 67, 17, 69, 35, 71, 9, 73, 37, 25, 76, 77, 13, 79, 16, 81, 41, 83, 84

Offset: 1

Antti Karttunen, Jun 17 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A344005, A354991, A354992, A354999.

Cf. also A344758, A346088.

s[n_] := Module[{m = 1}, While[! Divisible[m*(m + 1), n], m++]; m]; a[n_] := Module[{sn = s[n], ds = Divisors[n]}, Do[If[s[d] == sn, Return[d]], {d, ds}]]; Array[a, 100] (* Amiram Eldar, Jun 17 2022 *)

A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005
A354998(n) = { my(x=A344005(n)); fordiv(n, d, if(A344005(d)==x, return(d))); };

a(n) = n / A354999(n).

cp's OEIS Frontend

A344590 Number of divisors d of n for which A011772(d) = A011772(n), where A011772(n) is the smallest number m such that m(m+1)/2 is divisible by n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A344759 a(n) = n divided by the smallest divisor d of n for which A011772(d) = A011772(n), where A011772(n) is the smallest number m such that m(m+1)/2 is divisible by n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A346088 Smallest divisor d of n for which A002034(d) = A002034(n), where A002034(n) is the smallest positive integer k such that k! is a multiple of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A354998 Smallest divisor d of n for which A344005(d) = A344005(n), where A344005(n) is the smallest positive integer m such that n divides m*(m+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula