A323878 -id:A323878 - cp's OEIS frontend

A324198 a(n) = gcd(n, A276086(n)), where A276086 is the primorial base exp-function.

1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 15, 1, 1, 1, 1, 5, 3, 1, 1, 1, 25, 1, 3, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 3, 5, 1, 7, 1, 1, 15, 1, 1, 1, 7, 25, 3, 1, 1, 1, 5, 7, 3, 1, 1, 1, 1, 1, 21, 1, 1, 1, 1, 1, 3, 35, 1, 1, 1, 1, 75, 1, 7, 1, 1, 5, 3, 1, 1, 7, 5, 1, 3, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 49, 3, 5, 1, 1, 1, 1, 105

Offset: 0

Antti Karttunen, Feb 25 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003989, A276086, A323878, A324284, A324350, A324351, A324384, A324577, A324580, A324646, A327858, A328323, A327168, A328316, A328323, A328386, A328584, A346242 (Dirichlet inverse), A351254.
Cf. A324583 (positions of ones), A324584 (and terms larger than one).
Cf. A371098 (odd bisection), A371099 [= a(36n+9)].
Cf. also A328231.

Array[Block[{i, m, n = #, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; GCD[#, m]] &, 106, 0] (* Michael De Vlieger, Feb 04 2022 *)

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; };
A324198(n) = gcd(n,A276086(n));

A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p,valuation(orgn,p))); n = n\p; p = nextprime(1+p)); (m); }; \\ Antti Karttunen, Oct 21 2019

a(n) = gcd(n, A276086(n)).
From Antti Karttunen, Oct 21 2019: (Start)
A000005(a(n)) = A327168(n).
a(A328316(n)) = A328323(n).
a(n) = A324580(n) / A328584(n).
(End)

A323879 Number of divisors d of n such that A276154(d) divides n.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 5, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 5, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 1, 0

Offset: 1

Antti Karttunen, Feb 07 2019

Antti Karttunen, Table of n, a(n) for n = 1..30030
Index entries for sequences related to primorial base

Cf. A002110, A276154.

Cf. also A323878.

A276151(n) = { my(s=1); forprime(p=2, , if(n%p, return(n-s), s *= p)); };
A276152(n) = { my(s=1); forprime(p=2, , if(n%p, return(s*p), s *= p)); };
A276154(n) = if(!n,n,(A276152(n) + A276154(A276151(n))));
A323879(n) = sumdiv(n,d,!(n%A276154(d)));

a(n) = Sum_{d|n} [A276154(d)|n], where [ ] is the Iverson bracket.

A327168 Number of common divisors of n and A276086(n), with a(0) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 4, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 6, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 8

Offset: 0

Antti Karttunen, Sep 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to primorial base

Cf. A000005, A001222, A276086, A324198, A327167.

Cf. also A323878.

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; };
A327168(n) = numdiv(gcd(n,A276086(n)));

a(n) = A000005(A324198(n)).

a(n) = 1+A001222(A327167(n)) for n >= 1.

A323880 Number of divisors d > 1 of n such that A003415(d) divides n, where A003415 gives the arithmetic derivative of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 07 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415.

Cf. also A173441, A323878, A323879.

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A323880(n) = sumdiv(n,d,(d>1)&&!(n%A003415(d)));

a(n) = Sum_{d|n, d>1} [A003415(d)|n], where [ ] is the Iverson bracket, and A003415 gives the arithmetic derivative of n.

A324283 Multiplicative with a(p^e) = A276086(p^e).

Original entry on oeis.org

1, 3, 6, 9, 18, 18, 10, 15, 30, 54, 90, 54, 50, 30, 108, 225, 450, 90, 250, 162, 60, 270, 2250, 90, 1250, 150, 3750, 90, 11250, 324, 14, 21, 540, 1350, 180, 270, 70, 750, 300, 270, 630, 180, 350, 810, 540, 6750, 3150, 1350, 1750, 3750, 2700, 450, 15750, 11250, 1620, 150, 1500, 33750, 78750, 972, 98, 42, 300, 441, 900

Offset: 1

Antti Karttunen, Feb 20 2019

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..30030
Index entries for sequences related to primorial base

Cf. A276086, A319708, A323878, A324284.

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; };
A324283(n) = { my(f=factor(n)); prod(i=1, #f~, A276086(f[i,1]^f[i,2])); };

cp's OEIS Frontend

A324198 a(n) = gcd(n, A276086(n)), where A276086 is the primorial base exp-function.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A323879 Number of divisors d of n such that A276154(d) divides n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A327168 Number of common divisors of n and A276086(n), with a(0) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A323880 Number of divisors d > 1 of n such that A003415(d) divides n, where A003415 gives the arithmetic derivative of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A324283 Multiplicative with a(p^e) = A276086(p^e).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI