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.

Previous Showing 11-18 of 18 results.

A253556 a(1) = 0; after which, a(2n) = a(n), a(2n+1) = 1 + a(A250470(n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 12 2015

Keywords

Comments

Consider the binary tree illustrated in A252753 and A252755: If we start from any n, computing successive iterations of A253554 until 1 is reached (i.e., we are traversing level by level towards the root of the tree, starting from that vertex of the tree where n is located at), a(n) gives the number of odd numbers > 1 encountered on the path (i.e., excluding the final 1 from the count but including the starting n if it was odd).

Links

Crossrefs

One less than A253558.
Powers of two, A000079, gives the positions of zeros.
Cf. A000120, A080791, A252753, A252754, A252755, A252756, A253554, A253555, A253557, A253559.
Differs from A252735 for the first time at n=21, where a(21) = 1, while A252735(21) = 3.

Formula

a(1) = 0; after which, a(2n) = a(n), a(2n+1) = 1 + a(A250470(n)).
a(n) = A253555(n) - A253557(n).
a(n) = A253558(n) - 1.
a(n) = A080791(A252754(n)). [Number of nonleading 0-bits in A252754(n).]
Other identities. For all n >= 2:
a(n) = A000120(A252756(n)) - 1. [One less than the binary weight of A252756(n).]

A253559 a(1) = 0; for n>1: a(n) = A253557(n) - 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 2, 1, 0, 3, 1, 1, 1, 2, 0, 2, 0, 4, 2, 1, 1, 3, 0, 1, 1, 3, 0, 3, 0, 2, 3, 1, 0, 4, 1, 2, 2, 2, 0, 2, 2, 3, 2, 1, 0, 3, 0, 1, 1, 5, 1, 3, 0, 2, 3, 2, 0, 4, 0, 1, 1, 2, 1, 2, 0, 4, 2, 1, 0, 4, 2, 1, 2, 3, 0, 4, 2, 2, 4, 1, 1, 5, 0, 2, 1, 3, 0, 3, 0, 3, 3, 1, 0, 3, 0, 3, 1, 4, 0, 3, 3, 2, 3, 1, 1, 4, 1, 1, 3, 2, 2, 2, 0, 6
Offset: 1

Views

Author

Antti Karttunen, Jan 12 2015

Keywords

Comments

Consider the binary trees illustrated in A252753 and A252755: If we start from any n, computing successive iterations of A253554 until 1 is reached (i.e., we are traversing level by level towards the root of the tree, starting from that vertex of the tree where n is located), a(n) gives the number of even numbers > 2 encountered on the path (i.e., excluding the 2 from the count but including the starting n if it was even).

Links

Crossrefs

Essentially, one less than A253557.
A008578 gives the positions of zeros.
Cf. A000120, A080791, A252753, A252754, A252755, A252756, A253554, A253555, A253556, A253558.
Differs from A252736 for the first time at n=21, where a(21) = 2, while A252736(21) = 1.

Programs

Formula

a(n) = A080791(A252756(n)). [Number of nonleading 0-bits in A252756(n).]
a(1) = 0; for n>1: a(n) = A253557(n) - 1.
Other identities. For all n >= 2:
a(n) = A000120(A252754(n)) - 1. [One less than the binary weight of A252754(n).]
a(n) = A253555(n) - A253558(n).

A253558 a(n) = A253556(n) + 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 12 2015

Keywords

Comments

Consider the binary trees illustrated in A252753 and A252755: If we start from any n, computing successive iterations of A253554 until 1 is reached (i.e., we are traversing level by level towards the root of the tree, starting from that vertex of the tree where n is located at), a(n) gives the number of odd numbers encountered on the path (i.e., including both the final 1 and the starting n if it was odd).

Links

Crossrefs

One more than A253556.
Powers of two, A000079, gives the positions of ones.
Cf. A000040, A253555, A253557, A253559.
After n=1, differs from A061395 for the first time at n=21, where a(21) = 2, while A061395(21) = 4.

Programs

Formula

a(n) = A253556(n) + 1.
a(n) = A080791(A252754(n)) + 1. [One more than the number of nonleading 0-bits in A252754(n).]
Other identities.
For all n >= 1:
a(A000040(n)) = n.
For all n >= 2:
a(n) = A000120(A252756(n)). [Binary weight of A252756(n).]
a(n) = A253555(n) - A253559(n).

A324545 An analog of sigma (A000203) for nonstandard factorization based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 40, 36, 24, 60, 31, 42, 32, 56, 30, 72, 32, 63, 78, 54, 48, 91, 38, 60, 48, 90, 42, 120, 44, 84, 121, 72, 48, 124, 57, 93, 124, 98, 54, 96, 156, 120, 104, 90, 60, 168, 62, 96, 56, 127, 72, 234, 68, 126, 240, 144, 72, 195, 74, 114, 72, 140, 96, 144, 80
Offset: 1

Views

Author

Antti Karttunen, Mar 06 2019

Keywords

Links

Crossrefs

Cf. A000203, A020639, A078898, A250246, A252754, A302044, A302045, A324054, A324535, A324544, A324546.
Cf. also A302051, A302055, A323243.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
A324545(n) = sigma(A250246(n));
  • PARI
    \\ Or alternatively, using also A078898 defined above:
A000265(n) = (n/2^valuation(n, 2));
A001511(n) = 1+valuation(n,2);
A302045(n) = A001511(A078898(n));
A302044(n) = { my(c = A000265(A078898(n))); if(1==c,1,my(p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p)); };
A324545(n) = if(1==n,n,my(p=A020639(n)); (((p^(A302045(n)+1))-1)/(p-1))*A324545(A302044(n)));

Formula

a(n) = A000203(A250246(n)) = A324535(n) + A250246(n).
a(1) = 1; for n > 1, let p = A020639(n) [the smallest prime factor of n], then a(n) = (((p^(1+A302045(n)))-1) / (p-1)) * a(A302044(n)).
a(n) = A324054(A252754(n)).

A253789 Fixed points of f(n) = A252753(n-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 128, 160, 192, 256, 320, 384, 481, 512, 640, 768, 962, 1024, 1280, 1536, 1924, 2048, 2560, 3072, 3848, 4096, 5120, 6144, 7696, 8192, 10240, 12288, 15392, 16384, 20480, 24576
Offset: 1

Views

Author

Antti Karttunen, Jan 13 2015

Keywords

Crossrefs

Cf. A252753.
Also fixed points of f(n) = A252754(n)+1.
Differs from a subsequence A029747 for the first time at n=25, where a(25) = 481, while A029747 contains no odd terms after 1, 3 and 5.
No other odd numbers can occur than those listed at A253790.

A269378 Permutation of natural numbers: a(1) = 0, after which a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A269370(n)).

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 4, 7, 8, 13, 10, 11, 16, 9, 32, 15, 14, 17, 12, 27, 64, 21, 26, 23, 128, 33, 24, 19, 22, 65, 256, 31, 512, 29, 18, 35, 1024, 25, 20, 55, 30, 129, 2048, 43, 48, 53, 34, 47, 4096, 257, 8192, 67, 54, 49, 96, 39, 40, 45, 42, 131, 28, 513, 16384, 63, 46, 1025, 32768, 59, 65536, 37, 66, 71, 131072, 2049, 262144, 51, 38, 41
Offset: 1

Views

Author

Antti Karttunen, Mar 01 2016

Keywords

Comments

Note the indexing: Domain starts from 1, range from 0.

Links

Crossrefs

Inverse: A269377.
Cf. A269370.
Related permutation: A269376.
Cf. also A252754, A269388.

Formula

a(1) = 0, after which, a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A269370(n)).
As a composition of related permutations:
a(n) = A269388(A260741(n)).

A279349 a(1) = 1, for n > 1, if n is even, a(n) = A055938(a(n/2)), otherwise a(n) = A005187(a(A268674(n))).

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 7, 12, 8, 9, 11, 13, 19, 14, 10, 27, 35, 17, 67, 20, 22, 24, 131, 28, 15, 40, 16, 29, 259, 21, 515, 58, 23, 72, 18, 36, 1027, 136, 25, 43, 2051, 45, 4099, 51, 50, 264, 8195, 59, 26, 30, 32, 83, 16387, 33, 41, 60, 38, 520, 32771, 44, 65539, 1032, 46, 121, 31, 48, 131075, 147, 53, 37, 262147, 75, 524291, 2056, 78, 275, 34, 52
Offset: 1

Views

Author

Antti Karttunen, Dec 12 2016

Keywords

Links

Crossrefs

Inverse: A279348.
Cf. A005187, A055938, A268674.
Related or similar permutations: A250246, A252754, A252756, A279339, A279342, A279344.

Programs

Formula

a(1) = 1, for n > 1, if n is even, a(n) = A055938(a(n/2)), otherwise a(n) = A005187(a(A268674(n))).
As a composition of other permutations:
a(n) = A279339(A250246(n)).
a(n) = A279344(A252754(n)).
a(n) = A279342(A252756(n)).

A324544 a(n) = A009194(A250246(n)) = gcd(A250246(n), A324545(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 3, 10, 1, 6, 1, 4, 1, 2, 1, 4, 1, 1, 1, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 18, 1, 2, 15, 2, 1, 3, 1, 2, 3, 4, 1, 6, 1, 2, 3, 2, 1, 4, 1, 2, 3, 4, 1, 3, 1, 4, 1, 2, 1, 12, 1, 1, 1, 1, 1, 6, 1, 2, 1
Offset: 1

Views

Author

Antti Karttunen, Mar 06 2019

Keywords

Comments

Fixed points are: 1, 6, 28, 120, 496, 8128, etc,
Positions where a(n) == A250246(n) are: 1, 6, 28, 120, 496, 864, 8128, 11424, 15240, ..., which is sequence A250245(A007691(n)) sorted into ascending order.

Links

Crossrefs

Cf. A007691, A009194, A250246, A252754, A324394, A324545, A324546.
Differs from A009194 for the first time at n=39. Here a(39) = 3.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
A009194(n) = gcd(n, sigma(n));
A324544(n) = A009194(A250246(n));

Formula

a(n) = A009194(A250246(n)) = gcd(A250246(n), A324545(n)).
a(n) = A324394(A252754(n)).
Previous Showing 11-18 of 18 results.

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