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.

A350067 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A342666(n), A350063(n)] for n > 1, with f(1) = 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 29 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A342666(n), A350063(n)], when assuming that A342666(1) = 0.
Restricted growth sequence transform of the function f(1) = 0, f(n) = A336470(A156552(n)) for n > 1.
For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j) => A350065(i) = A350065(j).
For all i, j >= 2: a(i) = a(j) => A342651(i) = A342651(j).

Links

Crossrefs

Cf. A156552, A322993, A336466, A342666, A350063, A336470.
Cf. also A305897, A350065, A342651.

Programs

  • PARI
    up_to = 3003;
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; };
A000265(n) = (n>>valuation(n,2));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A336466(n) = { my(f=factor(n)); prod(k=1, #f~, A000265(f[k, 1]-1)^f[k, 2]); };
Aux350067(n) = if(1==n,1,my(u=A000265(A156552(n))); [A046523(u),A336466(u)]);
v350067 = rgs_transform(vector(up_to, n, Aux350067(n)));
A350067(n) = v350067[n];

A342666 a(n) = A336466(A156552(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 9, 1, 5, 1, 11, 1, 3, 3, 3, 1, 3, 1, 15, 1, 21, 1, 1, 1, 1, 5, 3, 1, 9, 1, 33, 5, 9, 1, 23, 1, 1, 3, 65, 1, 7, 1, 35, 21, 5, 1, 21, 1, 341, 9, 3, 1, 11, 1, 27, 1, 5, 1, 5, 1, 15, 3, 51, 1, 27, 1, 39, 1, 1365, 1, 1, 5, 49, 9, 1, 1, 1, 1, 117, 5, 825, 3, 9, 1, 9, 3, 1, 1, 7, 1
Offset: 2

Views

Author

Antti Karttunen, Mar 18 2021

Keywords

Links

Crossrefs

Cf. A000265, A003961, A156552, A322993, A336466.
Cf. also A324104, A342651, A342656, A350067.

Programs

  • PARI
    A000265(n) = (n>>valuation(n,2));
A336466(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1))^f[k,2])); };
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
A342666(n) = A336466(A156552(n));
  • PARI
    \\ Version using the factorization file available at https://oeis.org/A156552/a156552.txt
v156552sigs = readvec("a156552.txt");
A000265(n) = (n>>valuation(n,2));
A342666(n) = if(isprime(n),1,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,A000265(ps[i]-1)^es[i])); \\ Antti Karttunen, Jan 29 2022

Formula

a(n) = A336466(A156552(n)) = A336466(A322993(n)).
a(p) = 1 for all primes p.
a(A003961(n)) = a(n).

A342652 a(n) = A331410(A156552(n)).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 3, 2, 3, 0, 2, 0, 3, 2, 3, 0, 2, 1, 4, 1, 3, 0, 2, 0, 1, 3, 4, 2, 3, 0, 5, 3, 3, 0, 4, 0, 4, 2, 6, 0, 2, 1, 4, 4, 4, 0, 4, 2, 3, 4, 7, 0, 3, 0, 7, 3, 3, 3, 3, 0, 5, 5, 3, 0, 4, 0, 8, 2, 5, 2, 4, 0, 3, 3, 8, 0, 5, 3, 11, 6, 5, 0, 4, 2, 5, 7, 8, 4, 5, 0, 2, 3, 4, 0, 6, 0, 4, 2
Offset: 2

Views

Author

Antti Karttunen, Mar 18 2021

Keywords

Comments

Positions of ones is given by a subsequence of A053810, i.e., prime powers whose exponent is one of the primes in A000043. See also A324201, A335431.

Links

Crossrefs

Cf. A000043, A003961, A053810, A156552, A324201, A331410, A335431, A342651.

Programs

  • PARI
    A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); };
A342652(n) = A331410(A156552(n));

Formula

a(n) = A331410(A156552(n)).
a(p) = 0 for all primes p.
a(A003961(n)) = a(n).

A350069 Semiprimes k such that 1+(2^(1+A243055(k))) is a Fermat prime, where A243055(k) gives the difference between the indices of the smallest and the largest prime divisor of k.

Original entry on oeis.org

4, 6, 9, 14, 15, 25, 33, 35, 38, 49, 65, 69, 77, 106, 119, 121, 143, 145, 169, 177, 209, 217, 221, 289, 299, 305, 323, 361, 407, 437, 469, 493, 529, 533, 589, 667, 731, 781, 841, 851, 893, 899, 949, 961, 1147, 1189, 1219, 1333, 1343, 1369, 1517, 1577, 1681, 1711, 1739, 1763, 1849, 1891, 2021, 2047, 2173, 2209, 2479
Offset: 1

Views

Author

Antti Karttunen, Jan 29 2022

Keywords

Examples

			9 is a semiprime (9 = 3*3), and as the difference between the indices of the smallest (3) and the largest prime (3) dividing 9 is 0, we have 1+(2^(1+A243055(k))) = 3, which is in A019434, and therefore 9 is included in this sequence, like all squares of primes (A001248).
177 = 3 * 59 = prime(2) * prime(17), therefore A243055(177) = 17-2 = 15, and as 1+(2^16) = 65537 is also in A019434, 177 is included in this sequence.

Links

Crossrefs

Positions of ones in A342651.
Cf. A019434, A209229, A243055, A334101.
Subsequence of A001358. A001248 is a subsequence.

Programs

  • PARI
    A209229(n) = (n && !bitand(n,n-1));
A243055(n) = if(1==n,0,my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); (primepi(gpf)-primepi(lpf)));
isA350069(n) = if(2!=bigomega(n),0,my(d=1+A243055(n)); (A209229(d) && isprime(1+(2^d))));
Showing 1-4 of 4 results.

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