cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A327859 a(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, 2, 9, 2, 18, 2, 25, 5, 10, 2, 225, 2, 30, 15, 21, 2, 750, 2, 625, 45, 50, 2, 525, 45, 150, 3750, 21, 2, 14, 2, 18375, 75, 250, 25, 49, 2, 750, 225, 735, 2, 630, 2, 875, 210, 1250, 2, 385875, 75, 1050, 375, 13125, 2, 36750, 225, 1029, 1125, 14, 2, 1029, 2, 42, 5250, 2941225, 125, 98, 2, 1225, 1875, 78750
Offset: 0

Views

Author

Antti Karttunen, Sep 30 2019

Keywords

Comments

Sequence contains only terms of A048103.
Are there fixed points other than 1, 2, 10, 15, 5005? (There are none in the range 5006 .. 402653184.) See A369650.
Records occur at n = 0, 2, 4, 6, 8, 12, 18, 27, 32, 48, 64, 80, 144, 224, 256, 336, 448, 480, 512, 1728, ... (see also A131117).
a(n) and n are never multiples of 9 at the same time, thus the fixed points certainly exclude any terms of A008591. For a proof, consider my comment in A047257 and that A003415(9*n) is always a multiple of 3. - Antti Karttunen, Feb 08 2024

Links

Crossrefs

Cf. A003415, A008591, A048103, A131117, A276086, A327858, A327860, A341517 [= mu(a(n))], A341518 (k where a(k) is squarefree), A369641 (composite k where a(k) is squarefree), A369642.
Cf. A370114 (where a(k) is a multiple of k), A370115 (where k is a multiple of a(k)), A369650.

Programs

  • PARI
    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); };
A327859(n) = A276086(A003415(n));

Formula

a(n) = A276086(A003415(n)).
a(p) = 2 for all primes p.

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.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Feb 11 2024

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • PARI
    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

Views

Author

Antti Karttunen, Feb 11 2024

Keywords

Comments

Sequence contains only terms of A048103.

Links

Crossrefs

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

Programs

  • PARI
    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)); };

Formula

a(n) = A327859(n) / A369964(n) = A327859(n) / gcd(n, A276086(A003415(n))).
a(n) = A276086(A370120(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

Views

Author

Antti Karttunen, Feb 11 2024

Keywords

Links

Crossrefs

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

Programs

  • PARI
    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)); };

Formula

a(n) = A276085(A370117(n)).
Showing 1-4 of 4 results.

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