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.

A052331 Inverse of A052330; A binary encoding of Fermi-Dirac factorization of n, shown in decimal.

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 16, 5, 32, 9, 64, 6, 128, 17, 10, 256, 512, 33, 1024, 12, 18, 65, 2048, 7, 4096, 129, 34, 20, 8192, 11, 16384, 257, 66, 513, 24, 36, 32768, 1025, 130, 13, 65536, 19, 131072, 68, 40, 2049, 262144, 258, 524288, 4097, 514, 132, 1048576, 35
Offset: 1

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Author

Christian G. Bower, Dec 15 1999

Keywords

Comments

Every number can be represented uniquely as a product of numbers of the form p^(2^k), sequence A050376. This sequence is a binary representation of this factorization, with a(p^(2^k)) = 2^(i-1), where i is the index (A302778) of p^(2^k) in A050376. Additive with a(p^e) = sum a(p^(2^e_k)) where e = sum(2^e_k) is the binary representation of e and a(p^(2^k)) is as described above. - Franklin T. Adams-Watters, Oct 25 2005 - Index offset corrected by Antti Karttunen, Apr 17 2018

Examples

			n = 84 has Fermi-Dirac factorization A050376(5) * A050376(3) * A050376(2) = 7*4*3. Thus a(84) = 2^(5-1) + 2^(3-1) + 2^(2-1) = 16 + 4 + 2 = 22 ("10110" in binary = A182979(84)). - _Antti Karttunen_, Apr 17 2018
		

Crossrefs

Cf. A182979 (same sequence shown in binary).
One less than A064358.
Cf. also A156552.

Programs

  • PARI
    A052331=a(n)={for(i=1,#n=factor(n)~,n[2,i]>1||next; m=binary(n[2,i]); n=concat(n,Mat(vector(#m-1,j,[n[1,i]^2^(#m-j),m[j]]~)));n[2,i]%=2); n||return(0); m=vecsort(n[1,]); forprime(p=1,m[#m],my(j=0);while(p^2^j>1} \\ M. F. Hasler, Apr 08 2015
    
  • PARI
    up_to_e = 8192;
    v050376 = vector(up_to_e);
    ispow2(n) = (n && !bitand(n,n-1));
    i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
    A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); }; \\ Antti Karttunen, Apr 12 2018

Formula

a(1)=0; a(n*A050376(k)) = a(n) + 2^k for a(n) < 2^k, k=0, 1, ... - Thomas Ordowski, Mar 23 2005
From Antti Karttunen, Apr 13 2018: (Start)
a(1) = 0; for n > 1, a(n) = A000079(A302785(n)-1) + a(A302776(n)).
For n > 1, a(n) = A000079(A302786(n)-1) * A302787(n).
a(n) = A064358(n)-1.
A000120(a(n)) = A064547(n).
A069010(a(n)) = A302790(n).
(End)

A207901 Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k, also let R_k equal S_k read in reverse order; then the numbers b_k*R_k are the next 2^k terms.

Original entry on oeis.org

1, 2, 6, 3, 12, 24, 8, 4, 20, 40, 120, 60, 15, 30, 10, 5, 35, 70, 210, 105, 420, 840, 280, 140, 28, 56, 168, 84, 21, 42, 14, 7, 63, 126, 378, 189, 756, 1512, 504, 252, 1260, 2520, 7560, 3780, 945, 1890, 630, 315, 45, 90, 270, 135, 540, 1080, 360, 180, 36, 72, 216
Offset: 0

Views

Author

Paul D. Hanna, Feb 21 2012

Keywords

Comments

A permutation of the positive integers (but please note the starting offset: 0-indexed).
This sequence is a variant of A052330.
Shares with A064736, A302350, etc. the property that a(n) is either a divisor or a multiple of a(n+1). - Peter Munn, Apr 11 2018 on SeqFan-list. Note: A302781 is another such "divisor-or-multiple permutation" satisfying the same property. - Antti Karttunen, Apr 14 2018
The offset is 0 since S_0 = {1} denotes the first 2^0 = 1 terms. - Daniel Forgues, Apr 13 2018
This is "Fermi-Dirac piano played with Gray code", as indicated by Peter Munn's Apr 11 2018 formula. Compare also to A303771 and A302783. - Antti Karttunen, May 16 2018

Examples

			Start with [1]; appending 2*[1] results in [1,2];
appending 3*[2,1] results in [1,2, 6,3];
appending 4*[3,6,2,1] results in [1,2,6,3, 12,24,8,4];
appending 5*[4,8,24,12,3,6,2,1]
results in [1,2,6,3,12,24,8,4, 20,40,120,60,15,30,10,5];
next append 7*[5,10,30,15,60,120,40,20,4,8,24,12,3,6,2,1],
multiplying by 7 since 6 is already found in the previous terms.
Each new factor is in A050376: [2,3,4,5,7,9,11,13,16,17,19,23,25,29,...].
Continue in this way to generate all the terms of this sequence.
		

Crossrefs

Cf. A064736, A281978, A282291, A302350, A302781, A302783, A303751, A303771, A304085, A304531, A304755 for other divisor-or-multiple permutations or conjectured permutations.
Cf. A302033 (a squarefree analog), A304745.

Programs

  • Mathematica
    a = {1}; Do[a = Join[a, Reverse[a]*Min[Complement[Range[Max[a] + 1], a]]], {n, 1, 6}]; a (* Ivan Neretin, May 09 2015 *)
  • PARI
    {A050376(n)= local(m, c, k, p); n--; if(n<=0, 2*(n==0), c=0; m=2; while( cA050376(n-1)*Vec(Polrev(A))));A[n]}
    for(n=0,63,print1(a(n),",")) \\ edited for offsets by Michel Marcus, Apr 04 2019
    
  • PARI
    up_to_e = 13;
    v050376 = vector(up_to_e);
    A050376(n) = v050376[n];
    ispow2(n) = (n && !bitand(n,n-1));
    i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
    A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
    A003188(n) = bitxor(n, n>>1);
    A207901(n) = A052330(A003188(n)); \\ Antti Karttunen, Apr 13 2018

Formula

a(n) = A052330(A003188(n)). - Peter Munn, Apr 11 2018
a(n) = A302781(A302843(n)) = A302783(A064706(n)). - Antti Karttunen, Apr 16 2018
a(n+1) = A059897(a(n), A050376(A001511(n+1))). - Peter Munn, Apr 01 2019

Extensions

Offset changed from 1 to 0 by Antti Karttunen, Apr 13 2018

A302784 Inverse permutation to A302783: a(n) = A003188(A052331(n)).

Original entry on oeis.org

0, 1, 3, 6, 12, 2, 24, 7, 48, 13, 96, 5, 192, 25, 15, 384, 768, 49, 1536, 10, 27, 97, 3072, 4, 6144, 193, 51, 30, 12288, 14, 24576, 385, 99, 769, 20, 54, 49152, 1537, 195, 11, 98304, 26, 196608, 102, 60, 3073, 393216, 387, 786432, 6145, 771, 198, 1572864, 50, 108, 31, 1539, 12289, 3145728, 9, 6291456, 24577, 40, 390, 204, 98, 12582912, 774, 3075, 21
Offset: 1

Views

Author

Antti Karttunen, Apr 16 2018

Keywords

Crossrefs

Cf. A302783 (inverse).
Cf. also A302029.

Programs

  • PARI
    up_to = 4096;
    v050376 = vector(up_to);
    ispow2(n) = (n && !bitand(n,n-1));
    i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
    A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
    A003188(n) = bitxor(n, n>>1);
    A302784(n) = A003188(A052331(n));

Formula

a(n) = A003188(A052331(n)).

A302030 a(n) = 1+A006068(A052331(n)).

Original entry on oeis.org

1, 2, 4, 8, 16, 3, 32, 7, 64, 15, 128, 5, 256, 31, 13, 512, 1024, 63, 2048, 9, 29, 127, 4096, 6, 8192, 255, 61, 25, 16384, 14, 32768, 511, 125, 1023, 17, 57, 65536, 2047, 253, 10, 131072, 30, 262144, 121, 49, 4095, 524288, 509, 1048576, 8191, 1021, 249, 2097152, 62, 113, 26, 2045, 16383, 4194304, 12, 8388608, 32767, 33, 505, 241, 126
Offset: 1

Views

Author

Antti Karttunen, Apr 13 2018

Keywords

Comments

This is the inverse of A207901 if it is considered with a starting offset 1.

Crossrefs

One more than A302029.

Programs

  • PARI
    up_to_e = 8192;
    v050376 = vector(up_to_e);
    ispow2(n) = (n && !bitand(n,n-1));
    i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
    A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
    A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ After code in A006068
    A302030(n) = (1+A006068(A052331(n)));

Formula

a(n) = 1+A302029(n) = 1+A006068(A052331(n)).
Showing 1-4 of 4 results.