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 1-5 of 5 results.

A351236 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344025(i) = A344025(j) and A351085(i) = A351085(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 28, 40, 41, 21, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 46, 55, 2, 56, 57, 58, 2, 59, 41, 60, 61, 62, 2, 63, 64, 65
Offset: 1

Views

Author

Antti Karttunen, Feb 06 2022

Keywords

Comments

Restricted growth sequence transform of the 4-tuple [A003415(n), A003557(n), A327858(n), A345000(n)].
Question: If an image-analysis algorithm were to classify the scatter plot of this sequence, where it would cluster it? Nearer to A344025 than to A351085?

Links

Crossrefs

Cf. A003415, A003557, A327858, A345000, A351085.
Differs from A344025 for the first time at n=91, where a(91) = 64, while A344025(91) = 37.
Cf. also A305800, A351235, A351260.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327858(n) = gcd(A003415(n),A276086(n));
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
Aux351236(n) = [A003415(n), A003557(n), A327858(n), A345000(n)];
v351236 = rgs_transform(vector(up_to, n, Aux351236(n)));
A351236(n) = v351236[n];

A345000 a(n) = gcd(A003415(n), A003415(A276086(n))), where A003415(n) is the arithmetic derivative of n, and A276086(n) gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 16, 1, 3, 1, 2, 5, 1, 1, 4, 5, 5, 1, 2, 1, 1, 1, 10, 1, 1, 3, 12, 1, 1, 1, 2, 1, 1, 1, 4, 1, 5, 1, 2, 1, 5, 5, 4, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 12, 3, 1, 1, 2, 1, 1, 1, 12, 1, 1, 55, 10, 3, 1, 1, 16, 1, 1, 1, 2, 1, 5, 1, 140, 1, 3, 1, 16, 1, 49, 3, 2, 1, 7, 1, 28, 1, 7, 1, 2, 1
Offset: 0

Views

Author

Antti Karttunen, Jul 21 2021

Keywords

Links

Crossrefs

Cf. A003415, A276086, A327860, A347958 (inverse Möbius transform), A347959, A351083, A351085, A351086, A351235, A351236.
Cf. A166486 (a(n) mod 2, parity of terms, see comment in A327860).
Cf. also A324198, A327858.

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); };
A345000(n) = gcd(A003415(n), A003415(A276086(n)));

Formula

a(n) = gcd(A003415(n), A327860(n)) = gcd(A003415(n), A003415(A276086(n))).

A351080 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324198(i) = A324198(j) and A351083(i) = A351083(j) for all i, j >= 0.

Original entry on oeis.org

1, 2, 2, 3, 4, 2, 2, 5, 6, 3, 7, 2, 4, 2, 2, 8, 9, 2, 10, 2, 11, 3, 2, 2, 12, 13, 2, 3, 4, 2, 2, 2, 4, 3, 14, 15, 16, 2, 2, 17, 18, 2, 19, 2, 12, 8, 2, 2, 4, 19, 13, 3, 12, 2, 10, 20, 21, 3, 2, 2, 4, 2, 2, 22, 12, 2, 2, 2, 4, 3, 23, 2, 24, 2, 2, 25, 4, 26, 2, 2, 27, 3, 2, 2, 28, 20, 2, 3, 6, 2, 10, 29, 12, 3, 2, 2, 4
Offset: 0

Views

Author

Antti Karttunen, Feb 03 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A324198(n), A351083(n)].
For all i, j: a(i) = a(j) => A351084(i) = A351084(j).

Links

Crossrefs

Cf. A324198, A351083, A351084.
Cf. also A351085.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
A351083(n) = gcd(n, A327860(n));
Aux351080(n) = [A324198(n), A351083(n)];
v351080 = rgs_transform(vector(1+up_to,n,Aux351080(n-1)));
A351080(n) = v351080[1+n];

A351235 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j), A327858(i) = A327858(j) and A345000(i) = A345000(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 7, 2, 15, 16, 17, 18, 8, 2, 19, 2, 20, 21, 7, 22, 23, 2, 24, 10, 25, 2, 19, 2, 26, 27, 28, 2, 29, 30, 31, 14, 32, 2, 33, 10, 25, 10, 7, 2, 34, 2, 9, 27, 35, 36, 19, 2, 13, 10, 19, 2, 37, 2, 9, 38, 39, 36, 19, 2, 40, 41, 7, 2, 34, 10, 17, 10, 42, 2
Offset: 1

Views

Author

Antti Karttunen, Feb 06 2022

Keywords

Comments

Restricted growth sequence transform of the triplet [A046523(n), A327858(n), A345000(n)].
For all i, j >= 1:
A305800(i) = A305800(j) => a(i) = a(j) => A351085(i) = A351085(j).

Links

Crossrefs

Cf. A003415, A046523, A276086, A327858, A345000, A351086.
Cf. also A101296, A305800, A329620, A351085, A351236.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327858(n) = gcd(A003415(n),A276086(n));
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
Aux351235(n) = [A046523(n), A327858(n), A345000(n)];
v351235 = rgs_transform(vector(up_to,n,Aux351235(n)));
A351235(n) = v351235[n];

A351086 a(n) = gcd(A003415(n), A328572(n)), where A003415 is the arithmetic derivative and A328572 converts the primorial base expansion of n into its prime product form, but with 1 subtracted from all nonzero digits.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 35, 1, 1, 1, 1, 1, 49, 3, 1, 1, 7, 1, 7, 1, 7
Offset: 0

Views

Author

Antti Karttunen, Feb 03 2022

Keywords

Links

Crossrefs

Cf. A003415, A276086, A327858, A345000, A351084, A351085.

Programs

  • PARI
    A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };
A351086(n) = gcd(A003415(n), A328572(n));
  • PARI
    A351086(n) = { my(m=1, p=2, orgn=A003415(n)); while(n, if(n%p, m *= (p^min((n%p)-1, valuation(orgn, p)))); n = n\p; p = nextprime(1+p)); (m); };

Formula

a(n) = gcd(A003415(n), A328572(n)).
a(n) = gcd(A327858(n), A345000(n)).
Showing 1-5 of 5 results.

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