A354992 -id:A354992 - cp's OEIS frontend

A354991 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).

1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 2, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2

Offset: 1

Antti Karttunen, Jun 17 2022

nonn

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

Cf. A000005, A344005, A354990, A354992, A354994 (positions of 1's).

Cf. also A344590, A345935.

s[n_] := Module[{m = 1}, While[! Divisible[m*(m + 1), n], m++]; m]; a[n_] := Module[{sn = s[n]}, 1 + 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
A354991(n) = { my(x=A344005(n)); sumdiv(n, d, A344005(d)==x); };

a(n) = Sum_{d|n} [A344005(d) = A344005(n)], where [ ] is the Iverson bracket.
a(n) = A000005(n) - A354992(n).
a(n) <= A354990(n).

A354999 a(n) = n divided by the 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, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 6, 1, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 5, 7, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 3, 1, 1, 6, 1, 5, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 3

Offset: 1

Antti Karttunen, Jun 17 2022

nonn

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

Cf. A344005, A354991, A354992, A354994 (positions of 1's), A354998.

Cf. also A344759, A346089.

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[n/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
A354999(n) = { my(x=A344005(n)); fordiv(n, d, if(A344005(d)==x, return(n/d))); };

a(n) = n / A354998(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

A354991 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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A354999 a(n) = n divided by the 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).

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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).

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