A345935 -id:A345935 - 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).

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).

A345944 Numbers k such that A002034(d) < A002034(k) for all the proper divisors d of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, 256, 257, 263, 269, 271, 277, 281

Offset: 1

Antti Karttunen, Jul 04 2021

nonn

Index entries for sequences related to factorial numbers

Subsequence of A000961.
Cf. A000040 (subsequence), A002034, A345945 (complement).
Positions of 1's in A345935 and in A345950 (characteristic function).
Cf. also A344881.

PARI
```
isA345944(n) = (1==A345950(n));
```

A345945 Numbers k that have such a proper divisor d for which A002034(d) = A002034(k).

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117

Offset: 1

Antti Karttunen, Jul 04 2021

nonn

			21 = 3*7 is present because A002034(21) = 7 = A002034(7).

Index entries for sequences related to factorial numbers

Cf. A002034, A345944 (complement).
Positions of terms > 1 in A345935.
Union of A024619 and A345946.
Cf. also A344882.

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.
A345950(n) = { my(x=A002034(n)); fordiv(n,d,if(A002034(d)==x,return(d==n))); };
isA345945(n) = !A345950(n);

A345934 Ordinal transform of Kempner numbers, A002034.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 02 2021

Number of values of k, 1 <= k <= n, with A002034(k) = A002034(n).

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

Cf. A002034, A345935.

Cf. also A344770.

Table[Length@Select[Table[m=1;While[Mod[m!,k]!=0,m++];m,{k,n}],#==(m=1;While[Mod[m!,n]!=0,m++];m)&],{n,100}] (* Giorgos Kalogeropoulos, Jul 03 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
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.
v345934 = ordinal_transform(vector(up_to,n,A002034(n)));
A345934(n) = v345934[n];

a(n) >= A345935(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).

A345946 Prime powers k that have such a proper divisor d for which A002034(d) = A002034(k).

Original entry on oeis.org

8, 64, 81, 128, 1024, 6561, 8192, 15625, 16384, 32768, 262144, 531441, 1594323, 2097152, 4194304, 5764801, 33554432, 129140163, 244140625, 268435456, 536870912, 1073741824, 2147483648, 10460353203, 17179869184, 137438953472, 274877906944, 847288609443, 2199023255552

Offset: 1

Antti Karttunen, Jul 04 2021

nonn

Numbers k such that A345950(k) = 0 (or equivalently, A345935(k) > 1), and A010055(k) = 1.

			8 = 2^3 is present because A002034(8) = 4 = A002034(4).

Index entries for sequences related to factorial numbers

Cf. A002034, A010055, A345935, A345950.

Intersection of A000961 and A345945.

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.
A345950(n) = { my(x=A002034(n)); fordiv(n,d,if(A002034(d)==x,return(d==n))); };
isA345946(n) = (!isprime(n) && isprimepower(n) && !A345950(n));

More terms from Jinyuan Wang, Jul 07 2021

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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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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A345944 Numbers k such that A002034(d) < A002034(k) for all the proper divisors d of k.

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A345945 Numbers k that have such a proper divisor d for which A002034(d) = A002034(k).

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A345934 Ordinal transform of Kempner numbers, A002034.

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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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A345946 Prime powers k that have such a proper divisor d for which A002034(d) = A002034(k).

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