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.

A342651 a(n) = A329697(A156552(n)).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 3, 2, 3, 0, 3, 1, 3, 2, 3, 0, 3, 0, 3, 1, 5, 1, 3, 0, 1, 3, 3, 0, 3, 0, 4, 2, 6, 0, 4, 1, 2, 3, 4, 0, 3, 2, 4, 5, 4, 0, 4, 0, 7, 3, 4, 1, 4, 0, 5, 1, 2, 0, 3, 0, 4, 2, 4, 1, 5, 0, 4, 2, 8, 0, 3, 3, 7, 6, 4, 0, 3, 2, 6, 4, 9, 3, 4, 0, 4, 3, 2, 0, 5, 0, 5, 3
Offset: 2

Views

Author

Antti Karttunen, Mar 18 2021

Keywords

Links

Crossrefs

Cf. A156552, A329697, A322993, A342652, A350067.
Cf. A000040 (positions of 0's), A350069 (of 1's).

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};
A329697(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A329697(f[k,1]-1)))); };
A342651(n) = A329697(A156552(n));
  • PARI
    \\ Version using the factorization file available at https://oeis.org/A156552/a156552.txt
v156552sigs = readvec("a156552.txt");
A329697(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A329697(f[k,1]-1)))); };
A342651(n) = if(isprime(n),0,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); sum(i=2-(ps[1]%2),#ps,es[i]*(1+A329697(ps[i]-1)))); \\ Antti Karttunen, Jan 29 2022

Formula

a(n) = A329697(A156552(n)) = A329697(A322993(n)).
a(n) = A329697(A342666(n)) + A342656(n).
a(p) = 0 for all primes p.
a(A003961(n)) = a(n).

A342655 Number of prime factors (counted with multiplicity) in A156552(n).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Mar 18 2021

Keywords

Links

Crossrefs

Cf. A001222, A003961, A055396, A156552, A342656, A342657, A350064.
Cf. also A323243, A324104, A324105, A324119, A342653 (sigma, phi, tau, omega and mu similarly permuted).

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};
A342655(n) = bigomega(A156552(n));

Formula

a(n) = A001222(A156552(n)).
a(n) = A342656(n) + A055396(n) - 1.
a(A003961(n)) = 1 + a(n).
a(A000040(n)) = n-1 for all n >= 1.

A350065 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350063(i) = A350063(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 5, 2, 3, 2, 3, 4, 5, 2, 3, 3, 5, 3, 5, 2, 5, 2, 3, 3, 5, 3, 6, 2, 3, 5, 5, 2, 3, 2, 3, 3, 7, 2, 3, 3, 4, 5, 3, 2, 3, 4, 3, 5, 8, 2, 3, 2, 5, 3, 8, 3, 5, 2, 5, 3, 3, 2, 5, 2, 5, 3, 5, 3, 5, 2, 3, 5, 5, 2, 8, 5, 9, 7, 7, 2, 8, 4, 5, 8, 10, 5, 5, 2, 4, 5, 5, 2, 8, 2, 3, 5
Offset: 1

Views

Author

Antti Karttunen, Jan 29 2022

Keywords

Comments

Restricted growth sequence transform of A350063.
For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j) => A324117(i) = A324117(j).
For all i, j >= 2: a(i) = a(j) => A342656(i) = A342656(j).

Links

Crossrefs

Cf. A000265, A046523, A156552, A322993, A350063, A350064, A350067, A350068.
Cf. also A305897, A324117, A342656.

Programs

  • PARI
    up_to = 3000;
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));
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 };
A350063(n) = if(1==n,0,A046523(A000265(A156552(n))));
v350065 = rgs_transform(vector(up_to, n, A350063(n)));
A350065(n) = v350065[n];
  • PARI
    \\ Version using the factorization file available at https://oeis.org/A156552/a156552.txt
v156552sigs = readvec("a156552.txt");
up_to = #v156552sigs;
A350063(n) = if(n<=2,n-1,my(es=v156552sigs[n][2]); if(n%2, es = vector(#es-1,i,es[1+i])); my(f=vecsort(es, , 4), p=0); prod(i=1, #f, (p=nextprime(p+1))^f[i]));
v350065 = rgs_transform(vector(up_to, n, A350063(n)));
A350065(n) = v350065[n]; \\ Antti Karttunen, Jan 29 2022

A342657 The difference between floor(log_2(.)) of and the number of prime factors in A156552(n) (when counted with multiplicity).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 1, 1, 0, 2, 0, 3, 1, 3, 0, 3, 0, 4, 1, 3, 0, 2, 0, 3, 3, 5, 1, 1, 0, 7, 3, 3, 0, 4, 0, 5, 2, 5, 0, 4, 0, 2, 4, 6, 0, 3, 1, 5, 5, 7, 0, 4, 0, 9, 3, 2, 3, 4, 0, 6, 7, 4, 0, 3, 0, 10, 2, 7, 1, 5, 0, 5, 1, 11, 0, 3, 3, 11, 5, 3, 0, 2, 1, 8, 7, 11, 4, 4, 0, 3, 3, 3, 0, 5, 0, 7, 2
Offset: 2

Views

Author

Antti Karttunen, Mar 18 2021

Keywords

Links

Crossrefs

Cf. A000430 (positions of 0's), A000523, A001222, A003961, A061395, A070939, A156552, A246277, A322993, A325134, A339893, A342655, A342656.
Cf. also A339895.

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};
A342657(n) = { my(u=A156552(n)); (#binary(u)-bigomega(u))-1; };

Formula

a(n) = A339893(A156552(n)) = A339893(A322993(n)) = A000523(A156552(n)) - A001222(A156552(n)).
a(n) = (A252464(n)-A342655(n))-1 = (A325134(n)-A342655(n)) - 2.
a(p) = a(p^2) = 0 for all primes p. (Second part added Jul 27 2023)
a(A003961(n)) = a(2*A246277(n)) = a(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).
Showing 1-5 of 5 results.

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