A324540 -id:A324540 - cp's OEIS frontend

A355944 a(n) = smallest positive k such that n divides k*A276086(k), where A276086 is primorial base exp-function.

1, 1, 2, 4, 5, 2, 7, 8, 3, 5, 11, 4, 13, 7, 5, 16, 17, 3, 19, 8, 14, 11, 23, 8, 10, 13, 9, 28, 29, 5, 31, 32, 11, 17, 7, 4, 37, 19, 26, 8, 41, 14, 43, 44, 5, 23, 47, 16, 35, 10, 17, 52, 53, 9, 11, 32, 38, 29, 59, 8, 61, 31, 21, 64, 13, 11, 67, 68, 23, 7, 71, 16, 73, 37, 10, 76, 33, 26, 79, 16, 27, 41, 83, 28, 17, 43

Offset: 1

Antti Karttunen, Jul 27 2022

a(n) is the smallest positive k such that A324580(k) is a multiple of n.

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

Cf. A276086, A324539, A324540, A324541, A324580, A355945, A356151, A356152, A356153, A356160 (fixed points, where a(n)=n), A356161.

Cf. also A344005, A356164.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355944(n) = for(k=1, oo, if((k*A276086(k))%n==0, return(k)));

a(n) = n - A355945(n).

A324539 Number of divisors d of n such that A276086(d) = (n/d).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 10 2019

nonn

a(n) tells how many times n occurs in A324580. See also comments in A324541.

Question: Where is the next term larger than one in this sequence after a(2250) = 2 and a(5402250) = 2 ? Are there terms larger than 2 ?

Cf. A276086, A324540 (positions of zeros), A324541 (nonzeros), A324580.

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; };
A324539(n) = sumdiv(n,d,(d==A276086(n/d)));

a(n) = Sum_{d|n} [d == A276086(n/d)], where [ ] is the Iverson bracket.

A324541 Numbers that occur in range of A324580.

Original entry on oeis.org

0, 2, 6, 18, 30, 36, 70, 90, 120, 210, 270, 300, 434, 450, 650, 672, 990, 1050, 1260, 1386, 2142, 2250, 2310, 2590, 2940, 3600, 3990, 4410, 4642, 4750, 5978, 6996, 7350, 7500, 7650, 8190, 9114, 11880, 12600, 14058, 15000, 15050, 15750, 16170, 18480, 18522, 21186, 23100, 23870, 24750, 25830, 28224, 30030, 30870, 31250, 32830, 35970, 37114, 42000

Offset: 0

Antti Karttunen, Mar 10 2019

nonn

Indexing begins from 0 because the term a(0) = 0 is a special case.
Sequence A324580 sorted into ascending order, with duplicate occurrences removed. The first such duplicate is 2250 = A324580(15) = 150*15 = A324580(18) = 125*18. The next is 5402250 = A324580(105) = A276086(105)*105 = A324580(125) = A276086(125)*125.
Terms after zero are the positions of nonzero terms in A324539.

Cf. A324540 (complement).

Cf. A002110 (a subsequence), A276086, A324539, A324579, A324580.

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; };
A324539(n) = sumdiv(n,d,(d==A276086(n/d)));
for(n=1,oo,if(A324539(n)>0, print1(n, ", "))); \\ Print terms after zero.

\\ This program is better for computing many terms:
search_limit = 9699690;
A324580(n) = n*A276086(n);
A324541list(lim) = { my(s=Set([]),k); for(n=1,lim, k=A324580(n); if(k<=lim, s = setunion([k], s))); Vec(s); };
v324541 = A324541list(search_limit);
A324541(n) = if(!n,n,v324541[n]);

cp's OEIS Frontend

A355944 a(n) = smallest positive k such that n divides k*A276086(k), where A276086 is primorial base exp-function.

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A324539 Number of divisors d of n such that A276086(d) = (n/d).

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A324541 Numbers that occur in range of A324580.

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