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.

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A351441 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003557(i) = A003557(j) and A351450(i) = A351450(j) for all i, j >= 1.

Original entry on oeis.org

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

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Author

Antti Karttunen, Feb 12 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A003557(n), A351450(n)].
For all i, j >= 1:
a(i) = a(j) => A326042(i) = A326042(j),
a(i) = a(j) => A351455(i) = A351455(j).

Links

Crossrefs

Cf. A003557, A003961, A048250, A064989, A326042, A351450, A351455.
Cf. also A291751, A351461.

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; };
A003557(n) = (n/factorback(factorint(n)[, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A064989(n) = { my(f = factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A351450(n) = A064989(A048250(A003961(n)));
Aux351441(n) = [A003557(n), A351450(n)];
v351441 = rgs_transform(vector(up_to, n, Aux351441(n)));
A351441(n) = v351441[n];

A351449 a(n) = A064989(A295294(A003961(n))).

Original entry on oeis.org

1, 1, 1, 11, 1, 1, 1, 3, 29, 1, 1, 11, 1, 1, 1, 49, 1, 29, 1, 11, 1, 1, 1, 3, 34, 1, 22, 11, 1, 1, 1, 55, 1, 1, 1, 319, 1, 1, 1, 3, 1, 1, 1, 11, 29, 1, 1, 49, 85, 34, 1, 11, 1, 22, 1, 3, 1, 1, 1, 11, 1, 1, 29, 1091, 1, 1, 1, 11, 1, 1, 1, 87, 1, 1, 34, 11, 1, 1, 1, 49, 469, 1, 1, 11, 1, 1, 1, 3, 1
Offset: 1

Views

Author

Antti Karttunen, Feb 11 2022

Keywords

Links

Crossrefs

Cf. A003961, A048250, A057521, A064989, A151800, A295294, A326042, A351450, A351451.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
A064989(n) = { my(f = factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A295294(n) = sigma(A057521(n));
A351449(n) = A064989(A295294(A003961(n)));

Formula

Multiplicative with a(p) = 1 and for e > 1, a(p^e) = A064989((q^(e+1)-1)/(q-1)), where q = nextPrime(p) = A151800(p).
a(n) = A064989(A295294(A003961(n))) = A326042(A057521(n)).
a(n) = A326042(n) / A351451(n).

A351451 a(n) = A064989(A092261(A003961(n))).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 5, 2, 4, 2, 2, 1, 3, 1, 2, 1, 4, 5, 6, 2, 1, 4, 1, 2, 1, 2, 17, 1, 10, 3, 2, 1, 10, 2, 8, 1, 7, 4, 2, 5, 1, 6, 8, 2, 1, 1, 6, 4, 6, 1, 5, 2, 4, 1, 29, 2, 13, 17, 2, 1, 4, 10, 4, 3, 12, 2, 31, 1, 3, 10, 2, 2, 10, 8, 10, 1, 1, 7, 12, 4, 3, 2, 2, 5, 25, 1, 8, 6, 34, 8, 2
Offset: 1

Views

Author

Antti Karttunen, Feb 11 2022

Keywords

Links

Crossrefs

Cf. A003961, A048250, A055231, A064989, A092261, A151800, A351449, A351450.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A055231(n) = {my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > 1, f[k, 2] = 0); ); factorback(f); } \\ From A055231
A064989(n) = { my(f = factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A351450(n) = A064989(A048250(A003961(n)));
A351451(n) = A351450(A055231(n));

Formula

Multiplicative with a(p) = A064989(q+1) and a(p^e) = 1 for e > 1, where q = nextPrime(p) = A151800(p).
a(n) = A351450(A055231(n)) = A064989(A092261(A003961(n))).
a(n) = A326042(n) / A351449(n).

A351455 a(n) = A064989(A001615(A003961(n))).

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 4, 6, 1, 5, 4, 4, 2, 2, 8, 3, 6, 2, 2, 4, 5, 6, 8, 5, 4, 18, 4, 1, 2, 17, 16, 10, 3, 2, 12, 10, 2, 8, 4, 7, 4, 2, 10, 6, 6, 8, 16, 14, 5, 6, 8, 6, 18, 5, 8, 4, 1, 29, 4, 13, 17, 12, 32, 4, 10, 4, 6, 12, 2, 31, 24, 3, 10, 10, 4, 10, 8, 10, 8, 54, 7, 12, 8, 3, 2, 2, 20, 25, 6, 8
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2022

Keywords

Links

Crossrefs

Cf. A001615, A003557, A003961, A064989, A151800, A347385, A351450.
Coincides with A326042 on squarefree numbers (A005117, and apparently on no other numbers).
Cf. also A351441.

Programs

  • PARI
    A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f = factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A351455(n) = A064989(A001615(A003961(n)));

Formula

Multiplicative with a(p^e) = A064989((q+1)*q^(e-1)), where q = nextPrime(p) = A151800(p).
a(n) = A064989(A001615(A003961(n))) = A064989(A347385(A003961(n))).
a(n) = A003557(n) * A351450(n).
Showing 1-4 of 4 results.

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