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 1-3 of 3 results.

A328461 a(n) = A276156(n) / A002110(A007814(n)).

Original entry on oeis.org

1, 1, 3, 1, 7, 4, 9, 1, 31, 16, 33, 6, 37, 19, 39, 1, 211, 106, 213, 36, 217, 109, 219, 8, 241, 121, 243, 41, 247, 124, 249, 1, 2311, 1156, 2313, 386, 2317, 1159, 2319, 78, 2341, 1171, 2343, 391, 2347, 1174, 2349, 12, 2521, 1261, 2523, 421, 2527, 1264, 2529, 85, 2551, 1276, 2553, 426, 2557, 1279, 2559, 1, 30031, 15016, 30033, 5006
Offset: 1

Views

Author

Antti Karttunen, Oct 16 2019

Keywords

Comments

A276156(n) converts the binary expansion of n to a number whose primorial base representation has the same digits of 0's and 1's, thus each one of its terms is a unique sum of distinct primorial numbers. In this sequence that sum is then divided by the largest primorial that divides it, which only depends on the position of the least significant 1-bit in the binary expansion of the original n, that is, the 2-adic valuation of n.

Links

Crossrefs

Cf. A000265, A002110, A007814, A111701, A276154, A276156.
Cf. A328462 (bisection, also row 1 of array A328464 which shows the same information in tabular form).
Cf. A328471, A328472, A328473, A328474.

Programs

  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328461(n) = (A276156(n)/A002110(valuation(n,2)));

Formula

a(n) = A276156(n) / A002110(A007814(n)).
a(n) = A111701(A276156(n)).

A328474 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A276156(i)) = A046523(A276156(j)) for all i, j.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 18 2019

Keywords

Links

Crossrefs

Cf. A002110, A276156, A328461, A328471, A328473.

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; };
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
v328474 = rgs_transform(vector(up_to, n, A046523(A276156(n))));
A328474(n) = v328474[n];

A328472 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A328461(i)) = A278226(A328461(j)) for all i, j.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 18 2019

Keywords

Comments

Restricted growth sequence transform of function f(n) = A046523(A276086(A276156(n)/A002110(A007814(n)))).

Links

Crossrefs

Cf. A002110, A007814, A046523, A276156, A276086, A278226, A328461, A328471.

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; };
A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328461(n) = (A276156(n)/A002110(valuation(n,2)));
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); };
A278226(n) = A046523(A276086(n));
v328472 = rgs_transform(vector(up_to, n, A278226(A328461(n))));
A328472(n) = v328472[n];
Showing 1-3 of 3 results.

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