A356319 -id:A356319 - 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).

A356318 Numbers k such that the least j >= k for which k and A276086(j) are coprime is a nontrivial multiple of k, where A276086 is the primorial base exp-function.

Original entry on oeis.org

3, 10, 15, 35, 42, 70, 77, 105, 154, 231, 286, 330, 385, 429, 462, 715, 770, 858, 1001, 1155, 1430, 2002, 2145, 2431, 2730, 3003, 3094, 3315, 4199, 4290, 4641, 4862, 5005, 6006, 6630, 7293, 7735, 8398, 9282, 10010, 12155, 12597, 14586, 15015, 15470, 17017, 20995, 23205, 24310, 25194, 29393, 33915, 34034, 35530, 36465

Offset: 1

Antti Karttunen, Nov 04 2022

nonn

Numbers k such that k divides A356309(k) and A356309(k) > k.

Numbers k for which A356302(k) is a nonzero multiple of k.

Index entries for sequences related to primorial base

Setwise difference A356316 \ A324583. Intersection of A324584 and A356316.

Cf. A276086, A356162, A356302, A356309, A356315, A356319.

isA356318(n) = { my(u=A356302(n)); (u && !(u%n)); }; \\ Needs also code from A356302.

A356317 Numbers k such that k does not divide 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

9, 20, 21, 25, 27, 33, 39, 40, 45, 49, 50, 51, 55, 56, 57, 63, 69, 75, 80, 81, 84, 85, 87, 91, 93, 98, 99, 100, 110, 111, 112, 115, 117, 119, 123, 126, 129, 130, 133, 135, 140, 141, 145, 147, 153, 159, 160, 161, 165, 168, 170, 171, 175, 177, 182, 183, 189, 190, 195, 196, 200, 201, 203, 205, 207, 213, 219, 220, 225

Offset: 1

Antti Karttunen, Nov 04 2022

nonn

Numbers k such that k does not divide A356309(k).

Index entries for sequences related to primorial base

Subsequence of A324584.
Positions of 0's in A356315.
Cf. A276086, A356309, A356316 (complement), A356319 (subsequence).

PARI
```
isA356317(n) = !A356315(n);
```

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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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A356318 Numbers k such that the least j >= k for which k and A276086(j) are coprime is a nontrivial multiple of k, where A276086 is the primorial base exp-function.

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A356317 Numbers k such that k does not divide the least j >= k for which k and A276086(j) are coprime, where A276086 is the primorial base exp-function.

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