A370117 -id:A370117 - cp's OEIS frontend

A370120 a(n) = A276085(A370117(n)), where A370117(n) is the denominator of n/A276086(A003415(n)), A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and A276085 is its inverse, the primorial base log-function.

0, 0, 0, 1, 4, 1, 2, 1, 12, 6, 0, 1, 14, 1, 8, 0, 32, 1, 18, 1, 18, 8, 12, 1, 42, 4, 14, 25, 2, 1, 30, 1, 80, 12, 18, 6, 60, 1, 20, 14, 62, 1, 8, 1, 48, 31, 24, 1, 110, 14, 32, 18, 56, 1, 78, 10, 62, 20, 30, 1, 90, 1, 32, 19, 192, 12, 60, 1, 72, 24, 22, 1, 156, 1, 38, 43, 80, 18, 68, 1, 170, 108, 42, 1, 92, 16, 44

Offset: 0

Antti Karttunen, Feb 11 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A003415, A276085, A276086, A327859, A369964, A370115 (positions of 0's), A370117.

A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
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); };
A370120(n) = { my(u=A276086(A003415(n))); A276085(u/gcd(n, u)); };

a(n) = A276085(A370117(n)).

A369964 a(n) = gcd(n, A276086(A003415(n))), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 10, 1, 3, 1, 2, 15, 1, 1, 6, 1, 5, 3, 2, 1, 3, 5, 2, 3, 7, 1, 2, 1, 1, 3, 2, 5, 1, 1, 2, 3, 5, 1, 42, 1, 1, 15, 2, 1, 3, 1, 50, 3, 1, 1, 6, 5, 7, 3, 2, 1, 3, 1, 2, 21, 1, 5, 2, 1, 1, 3, 70, 1, 1, 1, 2, 25, 1, 1, 6, 1, 5, 1, 2, 1, 21, 5, 2, 3, 1, 1, 6, 1, 1, 3, 2, 5, 3, 1, 98, 3, 25

Offset: 0

Antti Karttunen, Feb 11 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A003415, A276086, A327859, A370114 (fixed points, see also A369650), A370116, A370117.

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); };
A369964(n) = gcd(n, A276086(A003415(n)));

a(n) = gcd(n, A327859(n)) = gcd(n, A276086(A003415(n))).
For n >= 1, a(n) = n / A370116(n).
For n >= 0, a(n) = A327859(n) / A370117(n).

A370115 Numbers k for which k is a multiple of A276086(A003415(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 2, 10, 15, 161, 2189, 5005, 27030, 29861, 510221, 223092341

Offset: 1

Antti Karttunen, Feb 11 2024

nonn

Question: Is the squarefreeness a necessary condition for the nonzero terms of this sequence?
Many of the terms occur also in A368703, because the arithmetic derivative of those terms is one of the primorial numbers, A002110.
If it exists, a(13) > 1241513984.

Positions of 1's in A370117, positions of 0's in A370120.
Intersection of A048103 and A369650 is a subsequence of this sequence. See the comments in latter.
Cf. A002110, A003415, A276086, A327859, A368703.
Cf. also A369970, A370114.

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); };
isA370115(n) = !(n%A276086(A003415(n)));

A370116 Numerator of n/A276086(A003415(n)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 1, 7, 8, 9, 1, 11, 4, 13, 7, 1, 16, 17, 3, 19, 4, 7, 11, 23, 8, 5, 13, 9, 4, 29, 15, 31, 32, 11, 17, 7, 36, 37, 19, 13, 8, 41, 1, 43, 44, 3, 23, 47, 16, 49, 1, 17, 52, 53, 9, 11, 8, 19, 29, 59, 20, 61, 31, 3, 64, 13, 33, 67, 68, 23, 1, 71, 72, 73, 37, 3, 76, 77, 13, 79, 16, 81, 41, 83, 4, 17, 43

Offset: 0

Antti Karttunen, Feb 11 2024

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A003415, A276086, A327859, A369964, A370114 (positions of 1's), A370117 (denominators).

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); };
A370116(n) = (n/gcd(n, A276086(A003415(n))));

a(n) = n / A369964(n) = n / gcd(n, A276086(A003415(n))).

cp's OEIS Frontend

A370120 a(n) = A276085(A370117(n)), where A370117(n) is the denominator of n/A276086(A003415(n)), A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and A276085 is its inverse, the primorial base log-function.

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A369964 a(n) = gcd(n, A276086(A003415(n))), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A370115 Numbers k for which k is a multiple of A276086(A003415(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A370116 Numerator of n/A276086(A003415(n)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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