A369964 -id:A369964 - cp's OEIS frontend

A370114 Numbers k for which A276086(A003415(k)) is a multiple of k, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

1, 2, 6, 10, 15, 42, 50, 70, 98, 105, 245, 294, 330, 343, 375, 726, 770, 825, 1029, 1078, 1155, 1210, 1375, 1617, 1694, 1815, 1925, 2310, 2450, 2541, 2662, 2695, 3025, 3822, 3850, 4290, 4550, 4719, 5005, 5145, 5390, 6125, 6655, 7098, 7150, 7546, 7865, 8450, 8470, 8575, 8918, 9317, 9438, 9555, 10010, 11319, 11375

Offset: 1

Antti Karttunen, Feb 11 2024

nonn

Numbers k that divide A327859(k). Sequence contains only terms of A048103 and does not contain any multiples of 9.

Antti Karttunen, Table of n, a(n) for n = 1..7501

Fixed points of A369964. Positions of 1's in A370116.
Intersection of A048103 and A369650 is a subsequence of this sequence. See the comments in latter.
Cf. A003415, A276086, A327859, A370115.

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); };
isA370114(n) = ((n>0) && !(A276086(A003415(n))%n));

A370117 Denominator of 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, 1, 2, 9, 2, 3, 2, 25, 5, 1, 2, 75, 2, 15, 1, 21, 2, 125, 2, 125, 15, 25, 2, 175, 9, 75, 1250, 3, 2, 7, 2, 18375, 25, 125, 5, 49, 2, 375, 75, 147, 2, 15, 2, 875, 14, 625, 2, 128625, 75, 21, 125, 13125, 2, 6125, 45, 147, 375, 7, 2, 343, 2, 21, 250, 2941225, 25, 49, 2, 1225, 625, 1125, 2, 84035, 2, 105, 350

Offset: 0

Antti Karttunen, Feb 11 2024

Sequence contains only terms of A048103.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A003415, A276086, A327859, A369964, A370115 (positions of 1's), A370116 (numerators), A370120.

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); };
A370117(n) = { my(u=A276086(A003415(n))); (u/gcd(n, u)); };

a(n) = A327859(n) / A369964(n) = A327859(n) / gcd(n, A276086(A003415(n))).

a(n) = A276086(A370120(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))).

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.

Original entry on oeis.org

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)).

cp's OEIS Frontend

A370114 Numbers k for which A276086(A003415(k)) is a multiple of k, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A370117 Denominator of n/A276086(A003415(n)), 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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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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