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 1-6 of 6 results.

A355944 a(n) = smallest positive k such that n divides k*A276086(k), where A276086 is primorial base exp-function.

Original entry on oeis.org

1, 1, 2, 4, 5, 2, 7, 8, 3, 5, 11, 4, 13, 7, 5, 16, 17, 3, 19, 8, 14, 11, 23, 8, 10, 13, 9, 28, 29, 5, 31, 32, 11, 17, 7, 4, 37, 19, 26, 8, 41, 14, 43, 44, 5, 23, 47, 16, 35, 10, 17, 52, 53, 9, 11, 32, 38, 29, 59, 8, 61, 31, 21, 64, 13, 11, 67, 68, 23, 7, 71, 16, 73, 37, 10, 76, 33, 26, 79, 16, 27, 41, 83, 28, 17, 43
Offset: 1

Views

Author

Antti Karttunen, Jul 27 2022

Keywords

Comments

a(n) is the smallest positive k such that A324580(k) is a multiple of n.

Links

Crossrefs

Cf. A276086, A324539, A324540, A324541, A324580, A355945, A356151, A356152, A356153, A356160 (fixed points, where a(n)=n), A356161.
Cf. also A344005, A356164.

Programs

  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355944(n) = for(k=1, oo, if((k*A276086(k))%n==0, return(k)));

Formula

a(n) = n - A355945(n).

A355945 a(n) = n minus the smallest positive k such that n divides k*A276086(k), where A276086 is primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 0, 0, 4, 0, 0, 6, 5, 0, 8, 0, 7, 10, 0, 0, 15, 0, 12, 7, 11, 0, 16, 15, 13, 18, 0, 0, 25, 0, 0, 22, 17, 28, 32, 0, 19, 13, 32, 0, 28, 0, 0, 40, 23, 0, 32, 14, 40, 34, 0, 0, 45, 44, 24, 19, 29, 0, 52, 0, 31, 42, 0, 52, 55, 0, 0, 46, 63, 0, 56, 0, 37, 65, 0, 44, 52, 0, 64, 54, 41, 0, 56, 68, 43, 58, 0, 0, 85, 52
Offset: 1

Views

Author

Antti Karttunen, Jul 27 2022

Keywords

Links

Crossrefs

Cf. A276086, A324580, A355944, A356151, A356160 (positions of zeros), A356161.

Programs

  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355945(n) = for(k=1, oo, if((k*A276086(k))%n==0, return(n-k)));

Formula

a(n) = n - A355944(n).

A356166 Greatest common divisor of n and the smallest positive k such that n divides k*A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 3, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 6, 1, 32, 1, 2, 5, 4, 1, 2, 1, 8, 1, 2, 1, 4, 3, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 12, 1, 2, 1, 64, 1, 2, 1, 4, 1, 10, 1, 8, 1, 2, 3, 4, 7, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, 6, 1, 4, 1, 2, 1, 32, 1, 2, 1, 4, 1, 2, 1, 8, 5
Offset: 1

Views

Author

Antti Karttunen, Jul 28 2022

Keywords

Links

Crossrefs

Cf. A003961, A191002, A356164, A356165, A356167, A356168, A356171 (positions of 1's), A356172.
Cf. also A345992, A356151.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A356166(n) = for(k=1, oo, if((k*A003961(k))%n==0, return(gcd(n,k))));

Formula

a(n) = gcd(n, A356164(n)) = gcd(n, A356165(n)) = gcd(A356164(n), A356165(n)).

A356153 Denominator of n / the smallest positive k such that n divides k*A276086(k), where A276086 is primorial base exp-function.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 2
Offset: 1

Views

Author

Antti Karttunen, Jul 28 2022

Keywords

Links

Crossrefs

Cf. A276086, A324580, A355944, A355945, A356151, A356152 (numerators), A356154 (Dirichlet inverse).

Programs

  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A356153(n) = for(k=1, oo, if((k*A276086(k))%n==0, return(k/gcd(n,k))));

Formula

a(n) = A355944(n) / A356151(n) = A355944(n) / gcd(n, A355944(n)).

A356152 Numerator of n / the smallest positive k such that n divides k*A276086(k), where A276086 is primorial base exp-function.

Original entry on oeis.org

1, 2, 3, 1, 1, 3, 1, 1, 3, 2, 1, 3, 1, 2, 3, 1, 1, 6, 1, 5, 3, 2, 1, 3, 5, 2, 3, 1, 1, 6, 1, 1, 3, 2, 5, 9, 1, 2, 3, 5, 1, 3, 1, 1, 9, 2, 1, 3, 7, 5, 3, 1, 1, 6, 5, 7, 3, 2, 1, 15, 1, 2, 3, 1, 5, 6, 1, 1, 3, 10, 1, 9, 1, 2, 15, 1, 7, 3, 1, 5, 3, 2, 1, 3, 5, 2, 3, 1, 1, 18, 7, 1, 3, 2, 5, 3, 1, 14, 9, 25, 1, 6, 1, 1, 15
Offset: 1

Views

Author

Antti Karttunen, Jul 28 2022

Keywords

Links

Crossrefs

Cf. A276086, A324580, A355944, A355945, A356151, A356153 (denominators).

Programs

  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A356152(n) = for(k=1, oo, if((k*A276086(k))%n==0, return(n/gcd(n,k))));

Formula

a(n) = n / A356151(n) = n / gcd(n, A355944(n)).

A356154 Dirichlet inverse of A356153.

Original entry on oeis.org

1, -1, -2, 0, -1, 3, -1, 0, 3, 1, -1, -1, -1, 1, 3, 0, -1, -7, -1, -1, 2, 1, -1, 0, -1, 1, -5, 0, -1, -5, -1, 0, 3, 1, 1, 5, -1, 1, 2, 2, -1, -3, -1, 0, -7, 1, -1, 0, -4, 2, 3, 0, -1, 15, 1, -3, 2, 1, -1, 5, -1, 1, -3, 0, 1, -5, -1, 0, 3, -1, -1, -2, -1, 1, 2, 0, -1, -3, -1, -1, 8, 1, -1, 1, 1, 1, 3, 0, -1, 17
Offset: 1

Views

Author

Antti Karttunen, Jul 28 2022

Keywords

Links

Crossrefs

Cf. A276086, A355944, A356151, A356153.

Programs

  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A356153(n) = for(k=1, oo, if((k*A276086(k))%n==0, return(k/gcd(n,k))));
memoA356154 = Map();
A356154(n) = if(1==n,1,my(v); if(mapisdefined(memoA356154,n,&v), v, v = -sumdiv(n,d,if(dA356153(n/d)*A356154(d),0)); mapput(memoA356154,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA356153(n/d) * a(d).
Showing 1-6 of 6 results.

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