A356318 -id:A356318 - cp's OEIS frontend

A356302 The least k >= 0 such that n and A276086(n+k) are relatively prime, where A276086 is the primorial base exp-function.

0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 20, 0, 0, 0, 0, 15, 0, 0, 0, 0, 10, 3, 0, 0, 0, 5, 0, 3, 0, 0, 0, 0, 0, 3, 0, 175, 0, 0, 0, 3, 20, 0, 168, 0, 0, 15, 0, 0, 0, 161, 10, 3, 0, 0, 0, 5, 154, 3, 0, 0, 0, 0, 0, 147, 0, 0, 0, 0, 0, 3, 140, 0, 0, 0, 0, 15, 0, 2233, 0, 0, 10, 3, 0, 0, 126, 5, 0, 3, 0, 0, 0, 119, 0, 3, 0, 0, 0, 0, 112

Offset: 0

Antti Karttunen, Nov 03 2022

For all nonzero terms, adding a(n) to n in primorial base generates at least one carry. See the formula involving A329041.

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

Cf. A276086, A324198, A356309, A329041.
Cf. A324583 (positions of zeros), A324584 (of nonzeros), A356318 (positions where a(n) > 0 and a multiple of n), A356319 (where 0 < a(n) < n).
Cf. A358213, A358214 (conjectured positions of records and their values).
Cf. also A356303, A356304.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A356302(n) = { my(k=0); while(gcd(A276086(n+k),n)!=1,k++); (k); };

a(n) = A356309(n) - n.

If a(n) > 0, then A000035(a(n)) = A000035(n) and A329041(n, a(n)) > 1.

A356309 The least j >= n such that n and A276086(j) are relatively prime, where A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 2, 6, 4, 5, 6, 7, 8, 12, 30, 11, 12, 13, 14, 30, 16, 17, 18, 19, 30, 24, 22, 23, 24, 30, 26, 30, 28, 29, 30, 31, 32, 36, 34, 210, 36, 37, 38, 42, 60, 41, 210, 43, 44, 60, 46, 47, 48, 210, 60, 54, 52, 53, 54, 60, 210, 60, 58, 59, 60, 61, 62, 210, 64, 65, 66, 67, 68, 72, 210, 71, 72, 73, 74, 90, 76, 2310, 78

Offset: 0

Antti Karttunen, Nov 04 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 0..11550
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to primorial base

Cf. A276086, A356302, A356313.

Cf. A324583 (positions of the fixed points), A356314 (positions of the terms that are primorial numbers), A356316 (where a(n) is a multiple of n), A356318 (where a nontrivial multiple), A356319 (where n < a(n) < 2*n).

f[nn_] := Block[{m = 1, i = 1, n = nn, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m]; Array[Block[{k = #}, While[! CoprimeQ[#, f[k]], k++]; k] &, 79, 0] (* Michael De Vlieger, Nov 06 2022, after Jean-François Alcover at A276086 *)

A356309(n) = (n+A356302(n)); \\ See code in the latter sequence.

a(n) = n + A356302(n).

A356316 Numbers k such that k divides the least j >= k for which k and A276086(j) are coprime, where A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 41, 42, 43, 44, 46, 47, 48, 52, 53, 54, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 82, 83, 86, 88, 89, 90, 92, 94, 95, 96, 97, 101, 102, 103, 104, 105, 106, 107

Offset: 1

Antti Karttunen, Nov 04 2022

nonn

Numbers k such that k divides A356309(k).

Index entries for sequences related to primorial base

Cf. A276086, A356309, A356315 (characteristic function), A356317 (complement).

Disjoint union of A324583 and A356318.

PARI
```
isA356316(n) = A356315(n);
```

A358213 The index of the first occurrence of A002110(n) in A356309.

Original entry on oeis.org

1, 2, 3, 10, 35, 77, 286, 2431, 4199, 37145

Offset: 0

Antti Karttunen, Nov 05 2022

A subsequence of A356314, and probably also of A356316 (from a(2)=3 onward of A356318 as well).

Also, from a(2)=3 onward conjectured to be the positions of records in A356302 (after its initial zero), while A358214 gives the conjectured record values.

Index entries for sequences related to primorial numbers

Cf. A002110, A356302, A356309, A356314, A356316, A356318, A358214.

f[nn_] := Block[{m = 1, i = 1, n = nn, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m]; q = P = 1; Reap[Do[k = j; While[! CoprimeQ[j, f[k]], k++]; If[k == P, Sow[j]; P *= Prime[q]; q++], {j, 0, 2500}] ][[-1, -1]] (* Michael De Vlieger, Nov 06 2022 *)

\\ Very slow:
A002110(n) = prod(i=1,n,prime(i));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A356309(n) = { my(j=n); while(gcd(A276086(j),n)!=1,j++); (j); };
A358213(n) = { my(x=A002110(n)); for(i=0,oo,if(A356309(i)==x,return(i))); };

A356309(a(n)) = a(n) + A358214(n).

a(n) = A002110(n) - A358214(n).

cp's OEIS Frontend

A356302 The least k >= 0 such that n and A276086(n+k) are relatively prime, where A276086 is the primorial base exp-function.

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A356309 The least j >= n such that n and A276086(j) are relatively prime, where A276086 is the primorial base exp-function.

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A356316 Numbers k such that k divides the least j >= k for which k and A276086(j) are coprime, where A276086 is the primorial base exp-function.

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A358213 The index of the first occurrence of A002110(n) in A356309.

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