A368916 -id:A368916

A368914

a(n) = 1 if there is no prime p such that p^p divides A342001(n), otherwise 0.

+10
10

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1

COMMENTS

Question: What is the asymptotic mean of this sequence?

Answer: Apparently about 0.77... See A368920. - Antti Karttunen, Jan 14 2024

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for characteristic functions

FORMULA

a(1) = 0, and for n > 1, a(n) = A359550(A342001(n)).

a(n) = A368913(n) + A368915(n).

For all n >= 1, a(n) >= A368912(n).

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A003557(n) = (n/factorback(factorint(n)[, 1]));

A342001(n) = (A003415(n) / A003557(n));

A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };

A368914(n) = ((n>1)&&A359550(A342001(n)));

CROSSREFS

Characteristic function of A368904, whose complement A368996 gives the positions of 0's.

Cf. A003415, A342001, A359550, A368912, A368913, A368915, A368916 [= a(A276086(n))], A368905 [= A005940(1+n)], A368920.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 09 2024

STATUS

approved

A368915

a(n) = 1 if there is no prime p such that p^p divides the arithmetic derivative of n, and 0 otherwise.

+10
9

0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1

COMMENTS

Question: What is the asymptotic mean of this sequence (and its complement A341996)? Knowing the value for A360111 would solve this. See also related sequences like A354874 and A368916.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for characteristic functions

FORMULA

a(1) = 0; for n > 1, a(n) = A359550(A003415(n)).

For all n > 1, a(n) = 1 - A341996(n) = A359550(n) - A360111(n).

For all n > 1, A359550(n) >= a(n) >= A328308(n).

For all n >= 1, a(n) >= A354874(n).

a(n) = A368914(n) - A368913(n).

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };

A368915(n) = ((n>1)&&A359550(A003415(n)));

CROSSREFS

Characteristic function of A358215.

Cf. A003415, A328308, A341996 (one's complement), A354874, A359550, A360111, A368913, A368914, A368916 [= a(A276086(n))].

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 09 2024

STATUS

approved

A328306

a(n) = 1 if k-th arithmetic derivative of A276086(n) is zero for some k, otherwise 0.

+10
7

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

0

COMMENTS

a(n) = 1 if A276086(n) is in A099308, 0 otherwise.

Question: Does the sequence have asymptotic mean? See also A328307 and A351071. - Antti Karttunen, Jan 10 2024

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for characteristic functions

Index entries for sequences related to primorial base

FORMULA

a(n) = A328308(A276086(n)).

For n > 0, a(n) <= A368916(n). - Antti Karttunen, Jan 10 2024

PROG

(PARI)

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i, 2]>=f[i, 1], return(0), s += f[i, 2]/f[i, 1])); (n*s));

A276086(n) = { my(i=0, m=1, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), m*=(prime(i)^((n%nextpr)/pr)); n-=(n%nextpr)); pr=nextpr); m; };

A328308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n));

A328306(n) = A328308(A276086(n));

CROSSREFS

Characteristic function of A328116.

Cf. A003415, A099308, A276086, A328307 (partial sums), A328308, A351071, A368916.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 12 2019

STATUS

approved

A360111

a(n) = 1 if there is no prime p such that p^p divides n, but for the arithmetic derivative of n such a prime exists, otherwise a(n) = 0; a(1) = 0 by convention.

+10
6

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1

COMMENTS

Question: What is the asymptotic mean of this and sequences like A341996 and its complement A368915, and related A368916? - Antti Karttunen, Jan 11 2024

Question: Is the asymptotic mean of this sequence 1 - Product_{p prime} (1 - 1/p^(1+p)) = 0.13585792767780221591...? (I.e. complementary to that of A368916). But see also A368911. - Antti Karttunen, Jan 29 2024

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for characteristic functions

FORMULA

a(n) = A359550(n) * A341996(n).

a(n) = [-1 == A256750(n)], where [ ] is the Iverson bracket.

For n > 1, a(n) = A359550(n) - A368915(n). - Antti Karttunen, Jan 11 2024

EXAMPLE

a(12) = 0, because for both 12 and 12' = A003415(12) = 16 there is a prime p (in both cases p=2) such that p^p divides them.

a(15) = 1, because 15 = 3*5 has no such prime divisor p that p^p would divide it, while 15' = 8 is divisible by 2^2.

MATHEMATICA

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); q[n_] := AllTrue[FactorInteger[n], Last[#] < First[#] &]; q[1] = True; a[1] = 0; a[n_] := If[q[n] && ! q[d[n]], 1, 0]; Array[a, 100] (* Amiram Eldar, Jan 31 2023 *)

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };

A360111(n) = ((n>1)&&A359550(n)&&!A359550(A003415(n)));

CROSSREFS

Cf. A003415, A256750, A341996, A359550, A368911, A368915, A368916.

Characteristic function of A327934.

After n=1 differs from A360109 for the next time at n=81, where a(81) = 0, while A360109(81) = 1.

Differs from A353479 for the first time at n=158, where a(158) = 1, while A353479(158) = 1.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 31 2023

STATUS

approved

A368905

a(n) = 1 if A342001(A005940(1+n)) is not divisible by p^p for any prime p, otherwise 0.

+10
4

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for characteristic functions

FORMULA

a(0) = 0, and for n > 0, a(n) = A359550(A366801(n)) = A368914(A005940(1+n)).

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A003557(n) = (n/factorback(factorint(n)[, 1]));

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };

A342001(n) = (A003415(n) / A003557(n));

A366801(n) = A342001(A005940(1+n));

A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };

A368905(n) = if(!n, 0, A359550(A366801(n)));

CROSSREFS

Cf. A003415, A005940, A342001, A359550, A366801, A368914, A368906 (partial sums).

Cf. also A368907, A368916.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 11 2024

STATUS

approved

A368917

Numbers k for which there is no prime p such that p^p divides the arithmetic derivative of A276086(k), where A276086 is the primorial base exp-function.

+10
4

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Numbers k such that A327860(k) [or equally, A342002(k)] is in A048103.

LINKS

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

Index entries for sequences related to primorial base

FORMULA

For n >= 1, A368918(a(n)) = n.

PROG

(PARI) \\ See A368916.

CROSSREFS

Complement of A342018.

Cf. A003415, A048103, A276086, A341996, A327860, A342002, A342019, A359550, A368915, A368916 (characteristic function), A368918 (its partial sums).

Cf. A328116 (subsequence, after its initial zero).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 09 2024

STATUS

approved

A368918

a(n) is the number of integers m in the range 0..n such that the arithmetic derivative of A276086(m) has no divisors of the form p^p.

+10
4

0, 1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 44, 45, 46, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 57, 58, 59, 60, 61, 62, 63, 64, 64, 65, 66, 67, 68, 69, 70, 71, 71, 72

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for sequences related to primorial base

FORMULA

a(0) = 0, for n > 0, a(n) = a(n-1) + A368916(n) = a(n-1) + A359550(A327860(n)).

For all n >= 1, a(A368917(n)) = n.

For all n >= 0, a(n) >= A328307(n) - 1.

PROG

(PARI)

up_to = 65537;

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };

A368916(n) = { my(u=A276086(n)); ((u>1)&&A359550(A003415(u))); };

A368918list(up_to) = { my(v=vector(up_to), s=A368916(0)); for(i=1, up_to, s +=

A368916(i); v[i] = s); (v); };

v368918 = A368918list(up_to);

A368918(n) = if(!n, 0, v368918[n]);

CROSSREFS

Partial sums of A368916.

Left inverse of A368917.

Cf. A003415, A276086, A327860, A328307, A359550.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 10 2024

STATUS

approved

A368919

Number of terms of A368917 less than 10^n, where A368917 lists the numbers k for which there is no prime p such that p^p divides the arithmetic derivative of A276086(k), where A276086 is the primorial base exp-function.

+10
3

8, 88, 863, 8634, 86407, 864150, 8641439, 86414292

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Value a(n) / 10^n seems to converge to 1 - lim_{n->oo} (A368911(n) / 10^n).

LINKS

Table of n, a(n) for n=1..8.

PROG

(PARI)

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };

A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };

A368916(n) = ((n>0)&&A359550(A327860(n)));

tp=10; s=0; for(n=1, 10^10, s+=A368916(n); if(1+n==tp, print1(s, ", "), if(n==tp, tp *= 10)));

CROSSREFS

Cf. A003415, A327860, A368911, A368916, A368917, A368920.

KEYWORD

nonn,more

AUTHOR

Antti Karttunen, Jan 14 2024

STATUS

approved

A369006

a(n) = 1 if there is no prime p such that p^p divides n' / gcd(n,n'), and 0 otherwise, where n' stands for the arithmetic derivative of n, A003415(n).

+10
2

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1

COMMENTS

Question: Is there a formula for the asymptotic mean, which seems to be around 0.813...? Consider A369004 and A369007.

LINKS

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

Index entries for characteristic functions

FORMULA

a(1) = 0, for n > 1, a(n) = A359550(A083345(n)).

A368914(n) <= a(n) <= 1 - A369004(n).

PROG