A344589 -id:A344589 - 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).

A345936 Number of divisors d of n for which A002034(d) < A002034(n), where A002034(n) is the smallest positive integer k such that n divides k!.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 4, 1, 5, 2, 2, 2, 6, 1, 2, 2, 4, 1, 4, 1, 3, 4, 2, 1, 8, 2, 4, 2, 3, 1, 6, 2, 4, 2, 2, 1, 6, 1, 2, 3, 5, 2, 4, 1, 3, 2, 4, 1, 8, 1, 2, 4, 3, 2, 4, 1, 8, 3, 2, 1, 6, 2, 2, 2, 4, 1, 8, 2, 3, 2, 2, 2, 10, 1, 4, 3, 6, 1, 4, 1, 4, 4

Offset: 1

Antti Karttunen, Jul 02 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 six divisors that do not obtain the maximal value 6 obtained at 36 itself, therefore a(36) = 6.

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

Cf. A000005, A002034, A345935.

Cf. also A344589.

a[n_]:=(m=1;While[Mod[m!,n]!=0,m++];m);Table[Length@Select[Divisors@k,a@#Giorgos Kalogeropoulos, Jul 03 2021 *)

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.
A345936(n) = { my(x=A002034(n)); sumdiv(n,d,A002034(d)

a(n) = Sum_{d|n} [A002034(d) < A002034(n)], where [ ] is the Iverson bracket.

a(n) = A000005(n) - A345935(n).

A354992 Number of divisors 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

0, 0, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 3, 4, 1, 4, 1, 3, 2, 2, 1, 7, 2, 2, 3, 5, 1, 6, 1, 5, 3, 2, 3, 6, 1, 2, 2, 7, 1, 4, 1, 5, 5, 2, 1, 8, 2, 4, 3, 3, 1, 6, 2, 5, 2, 2, 1, 11, 1, 2, 5, 6, 3, 6, 1, 3, 3, 6, 1, 7, 1, 2, 4, 5, 3, 4, 1, 7, 4, 2, 1, 11, 3, 2, 3, 7, 1, 10, 3, 5, 2, 2, 3, 11, 1, 4, 5, 6, 1, 6, 1, 7, 6

Offset: 1

Antti Karttunen, Jun 17 2022

nonn

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

Cf. A000005, A344005, A354991.

Cf. also A344589, A345936.

g:= proc(n) option remember; local t,x;
  min(map(t -> rhs(op(t)), {msolve(x*(x+1),n)}) minus {0})
end proc:
g(1):= 1: g(2):= 1:
f:= proc(n) local d,v;
  v:= g(n);
  nops(select(t -> g(t) < v, numtheory:-divisors(n)))
end proc:
map(f, [$1..100]); # Robert Israel, Jun 17 2022

s[n_] := Module[{m = 1}, While[! Divisible[m*(m + 1), n], m++]; m]; a[n_] := Module[{sn = s[n]}, DivisorSum[n, 1 &, # < n && s[#] < sn &]]; Array[a, 100] (* Amiram Eldar, Jun 17 2022 *)

A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005
A354992(n) = { my(x=A344005(n)); sumdiv(n, d, A344005(d)

a(n) = Sum_{d|n} [A344005(d) < A344005(n)], where [ ] is the Iverson bracket.

a(n) = A000005(n) - A354991(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.

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A345936 Number of divisors d of n for which A002034(d) < A002034(n), where A002034(n) is the smallest positive integer k such that n divides k!.

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A354992 Number of divisors 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).

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