A328323 -id:A328323 - 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)

A328316 Iterates of A276086 starting from 0.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 18, 125, 43218, 258413198822535882125

Offset: 0

Antti Karttunen, Oct 14 2019

nonn

The unique infinite sequence such that a(0) = 0, a(n) = A276085(a(n+1)) for n >= 0, and A129251(a(n)) = 0 for n >= 1, i.e., all nonzero terms must be in A048103.

a(10) is 240 decimal digits long (can be found in b-file), and a(11) is too big to fit even into a b-file as it is 32700 decimal digits long, but it can be found in the given a-file.

Antti Karttunen, Table of n, a(n) for n = 0..10
Antti Karttunen, Terms a(0) .. a(11), without running index
Index entries for sequences related to primorial base

Cf. A002110, A048103, A129251, A276085, A276086, A328317 (the smallest prime not dividing a(n)), A328318, A328319 (digit sum in primorial base), A328322 (max. digit), A328323.
Cf. A153013, and also A109162, A179016, A219666, A259934 for more or less analogous sequences.
Cf. also A328313.

A276086[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
NestList[A276086, 0, 10] (* Jean-François Alcover, Dec 01 2021, after Antti Karttunen in A276086 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328316(n) = if(!n,0,A276086(A328316(n-1)));

a(0) = 0; and for n > 0, a(n) = A276086(a(n-1)).

A328317 Smallest prime not dividing A328316(n), with a(0) = 1 by convention; Equally, for n > 0, smallest prime dividing A328316(1+n).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 14 2019

a(2n+1) = 2 for all n >= 0. Does the pattern of 5's in the even bisection also continue?

Cf. A020639, A053669, A276086, A326810, A328316, A328318, A328319, A328322, A328323, A328585, A328586, A328633.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328316(n) = if(!n,0,A276086(A328316(n-1)));
A053669(n) = forprime(p=2, , if(n%p, return(p))); \\ From A053669
A328317(n) = if(0==n,1,A053669(A328316(n)));
\\ Or alternatively as:
A020639(n)=if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1)
A328317(n) = A020639(A328316(1+n));

a(0) = 1; and for n > 0, a(n) = A053669(A328316(n)).
a(n) = A020639(A328316(1+n)).
For n >= 1, a(n) = A326810(A328316(n-1)). - Antti Karttunen, Nov 15 2019

a(12)-a(13) from Jinyuan Wang, Jul 20 2020

A328322 Maximal digit value used when A328316(n) is written in primorial base; maximal prime exponent in A328316(1+n).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 3, 4, 7, 49, 430, 74814

Offset: 0

Antti Karttunen, Oct 14 2019

nonn

Index entries for sequences related to primorial base

Cf. A051903, A276086, A328114, A328316, A328317, A328318, A328319, A328323.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328316(n) = if(!n,0,A276086(A328316(n-1)));
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A328322(n) = A328114(A328316(n));
\\ Or alternatively as, more slowly:
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328322(n) = A051903(A328316(1+n));

a(n) = A328114(A328316(n)).

a(n) = A051903(A328316(1+n)).

cp's OEIS Frontend

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

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A328316 Iterates of A276086 starting from 0.

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A328317 Smallest prime not dividing A328316(n), with a(0) = 1 by convention; Equally, for n > 0, smallest prime dividing A328316(1+n).

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A328322 Maximal digit value used when A328316(n) is written in primorial base; maximal prime exponent in A328316(1+n).

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