A345936 -id:A345936 - cp's OEIS frontend

A345935 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!.

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 4, 1, 4, 1, 3, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 2, 4, 2, 2, 1, 6, 1, 2, 3, 2, 2, 4, 1, 3, 2, 4, 1, 4, 1, 2, 2, 3, 2, 4, 1, 2, 2, 2, 1, 6, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 2, 1, 2, 3, 3, 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 three divisors that obtain the maximal value 6 obtained at 36 itself, therefore a(36) = 3.

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

Cf. A000005, A002034, A345934, A345936, A345944 (positions of 1's), A345945 (of terms > 1), A345950.

Cf. also A344590.

a[n_]:=(m=1;While[Mod[m!,n]!=0,m++];m);Table[Length@Select[Divisors@k,a@#==a@k&],{k,100}] (* 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.
A345935(n) = { my(x=A002034(n)); sumdiv(n,d,A002034(d)==x); };

a(n) = Sum_{d|n} [A002034(d) = A002034(n)], where [ ] is the Iverson bracket.
a(n) = A000005(n) - A345936(n).
a(n) <= A345934(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).

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).

A346089 a(n) = n divided by the 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, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 6, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 5, 4, 1, 2, 3, 8, 1, 6, 1, 4, 5, 2, 1, 3, 1, 2, 3, 4, 1, 2, 5, 8, 3, 2, 1, 12, 1, 2, 9, 2, 5, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 6, 1, 5, 3, 2, 1, 12, 5, 2, 3, 8, 1, 10, 7, 4, 3, 2, 5, 3, 1, 2, 9, 4, 1, 6, 1, 8, 15

Offset: 1

Antti Karttunen, Jul 05 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A002034, A345935, A345936, A345944 (positions of 1's), A346088.
Cf. also A344759.
Differs from A302776 for the first time at n=27, where a(27) = 1, while A302776(27) = 3.

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.
A346089(n) = { my(x=A002034(n)); fordiv(n,d,if(A002034(d)==x, return(n/d))); };

a(n) = n / A346088(n).

cp's OEIS Frontend

A345935 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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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.

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A346089 a(n) = n divided by the 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.

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