A356304 -id:A356304 - 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.

A356311 Numbers k for which A003415(k) and A276086(k) are relatively prime, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 2, 3, 4, 5, 7, 10, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 28, 29, 30, 31, 32, 34, 37, 40, 41, 42, 43, 47, 53, 54, 56, 58, 59, 60, 61, 66, 67, 70, 71, 72, 73, 78, 79, 80, 82, 83, 84, 89, 90, 96, 97, 101, 103, 104, 105, 107, 108, 109, 113, 114, 118, 120, 124, 127, 130, 131, 132, 136, 137, 138, 139, 140, 142, 144

Offset: 1

Antti Karttunen, Nov 03 2022

nonn

Index entries for sequences related to primorial base

Positions of 1's in A327858. Positions of 0's in A356304 (for n >= 2) and in A356305.
Cf. A003415, A276086, A356310 (characteristic function), A356312 (complement).
Cf. also A324583.

isA356311(n) = A356310(n); \\ See code in A356310.

A356305 The least k >= 0 such that A003415(n) and A276086(n-k) are relatively prime, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 1, 0, 2, 3, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 17, 0, 0, 0, 21, 25, 2, 0, 0, 0, 0, 0, 5, 0, 5, 6, 0, 13, 1, 0, 0, 0, 0, 2, 2, 11, 0, 20, 21, 19, 17, 24, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 2, 4, 5, 0, 0, 2, 1, 0, 0, 0, 0, 1, 10, 12, 5, 0, 0, 0, 3, 0, 0, 0, 1, 25, 1, 84, 0, 0, 1, 2, 1, 65, 5, 0, 0, 69, 8, 96

Offset: 0

Antti Karttunen, Nov 03 2022

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. A003415, A276086, A356311 (positions of 0's).

Cf. also A356303, A356304.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A356305(n) = { my(k=0,x=A003415(n)); while(gcd(A276086(n-k),x)!=1,k++); (k); };

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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A356311 Numbers k for which A003415(k) and A276086(k) are relatively prime, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A356305 The least k >= 0 such that A003415(n) and A276086(n-k) are relatively prime, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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