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 11-19 of 19 results.

A093516 Transform of the prime sequence by the Rule137 cellular automaton.

Original entry on oeis.org

1, 3, 10, 16, 22, 26, 27, 28, 34, 35, 36, 40, 46, 50, 51, 52, 56, 57, 58, 64, 65, 66, 70, 76, 77, 78, 82, 86, 87, 88, 92, 93, 94, 95, 96, 100, 106, 112, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 130, 134, 135, 136, 142, 143, 144, 145, 146, 147, 148, 154
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Keywords

Comments

As described in A051006, a monotonic sequence can be mapped into a fractional real. Then the binary digits of that real can be treated (transformed) by an elementary cellular automaton. Taken resulted sequence of binary digits as a fractional real, it can be mapped back into a sequence, as in A092855.

Links

Crossrefs

Cf. A092855, A051006, A093510, A093511, A093512, A093513, A093514, A093515, A093517.

Programs

  • PARI
    {ca_tr(ca,v)= /* Calculates the Cellular Automaton transform of the vector v by the rule ca */
local(cav=vector(8),a,r=[],i,j,k,l,po,p=vector(3));
a=binary(min(255,ca));k=matsize(a)[2];forstep(i=k,1,- 1,cav[k-i+1]=a[i]);
j=0;l=matsize(v)[2];k=v[l];po=1;
for(i=1,k+2,j*=2;po=isin(i,v,l,po);j=(j+max(0,sign(po)))% 8;if(cav[j+1],r=concat(r,i)));
return(r) /* See the function "isin" at A092875 */}

A093517 Transform of the prime sequence by the Rule225 cellular automaton.

Original entry on oeis.org

1, 4, 5, 7, 10, 13, 16, 19, 22, 26, 27, 28, 31, 34, 35, 36, 40, 43, 46, 50, 51, 52, 56, 57, 58, 61, 64, 65, 66, 70, 73, 76, 77, 78, 82, 86, 87, 88, 92, 93, 94, 95, 96, 100, 103, 106, 109, 112, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 130, 134, 135, 136, 139
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Keywords

Comments

As described in A051006, a monotonic sequence can be mapped into a fractional real. Then the binary digits of that real can be treated (transformed) by an elementary cellular automaton. Taken resulted sequence of binary digits as a fractional real, it can be mapped back into a sequence, as in A092855.

Links

Crossrefs

Cf. A092855, A051006, A093510, A093511, A093512, A093513, A093514, A093515, A093516.

Programs

  • PARI
    {ca_tr(ca,v)= /* Calculates the Cellular Automaton transform of the vector v by the rule ca */
local(cav=vector(8),a,r=[],i,j,k,l,po,p=vector(3));
a=binary(min(255,ca));k=matsize(a)[2];forstep(i=k,1,- 1,cav[k-i+1]=a[i]);
j=0;l=matsize(v)[2];k=v[l];po=1;
for(i=1,k+2,j*=2;po=isin(i,v,l,po);j=(j+max(0,sign(po)))% 8;if(cav[j+1],r=concat(r,i)));
return(r) /* See the function "isin" at A092875 */}

A092858 "Sum" of the sequences of primes and the triangular numbers (A000217).

Original entry on oeis.org

5, 6, 7, 10, 11, 13, 15, 17, 19, 21, 23, 28, 29, 31, 36, 37, 41, 43, 45, 47, 53, 55, 59, 61, 66, 67, 71, 73, 78, 79, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 120, 127, 131, 136, 137, 139, 149, 151, 153, 157, 163, 167, 171, 173, 179, 181, 190, 191, 193, 197, 199
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Keywords

Comments

If two monotonic sequences are mapped into the real codomain of (0,1) as it is defined in A051006, then the fractional part of the sum of the two reals can be mapped back into a sequence as defined in A092855, yielding the "sum" of the two sequences.

Links

Crossrefs

Cf. A092855, A051006, A092857, A092859, A092860, A092861, A092862, A092863, A092874.

Programs

  • PARI
    {ssum(a,b)= /*Returns the "sum" monotonic sequences a and b */ return(mtinv(mt(a)+mt(b))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A092859 "Difference" of the sequences of triangular numbers (A000217) and the primes (cf. A092858).

Original entry on oeis.org

3, 4, 5, 7, 12, 13, 16, 18, 19, 22, 23, 30, 31, 38, 39, 40, 42, 43, 46, 48, 49, 50, 51, 52, 53, 56, 57, 58, 60, 61, 68, 69, 70, 72, 73, 80, 81, 82, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 98, 99, 100, 102, 103, 106, 108, 110, 111, 112, 113, 121, 122, 123, 124, 125, 126
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Keywords

Comments

Here the complement of the sequence of primes (1 and the composites) is "added" to the sequence of triangulars, according to the definition outlined in A092858.

Links

Crossrefs

Cf. A092855, A051006, A092857, A092858, A092860, A092861, A092862, A092863, A092874.

Programs

  • PARI
    {sdif(a,b)= /*Returns the "difference" of monotonic sequences a and b */ return(mtinv(mt(a)+mt(compl(b)))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ } {compl(v)=/* Returns the complement of v monotonic positive sequence */ local(n,p=0,vv=[]);n=matsize(v)[2];for(i=1,n, for(j=p+1,v[i]-1,vv=concat (vv,j));p=v[i]);return(vv)}

A092860 "3 times the prime sequence".

Original entry on oeis.org

3, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17, 18, 19, 22, 23, 28, 29, 30, 31, 36, 37, 40, 41, 42, 43, 46, 47, 52, 53, 58, 59, 60, 61, 66, 67, 70, 71, 72, 73, 78, 79, 82, 83, 88, 89, 96, 97, 100, 101, 102, 103, 106, 107, 108, 109, 112, 113, 126, 127, 130, 131, 136, 137, 138, 139
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Keywords

Comments

By iterating the addition to itself a monotonic sequence, according to the definition given in A092858, we can multiply the monotonic sequences by natural numbers.
Note, that it is easy to see that for an i natural and a v monotonic sequence, i(x)compl(v)=compl(i(x)v); where the "(x)" mark stands for the "integer multiplication of a sequence" and the function "compl" produces the complement of a positive monotonic sequence.

Links

Crossrefs

Cf. A092855, A051006, A092857, A092858, A092859, A092861, A092862, A092863, A092874.

Programs

  • PARI
    {imulv(i,v)= /*Returns "i (x) v" monotonic sequence */ return(mtinv(i*mt(v))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A092861 "Product" of the sequence of primes and the "evil" numbers (A001969).

Original entry on oeis.org

4, 7, 9, 12, 14, 15, 18, 19, 21, 25, 26, 33, 35, 36, 37, 40, 41, 42, 44, 47, 48, 50, 54, 55, 58, 59, 60, 64, 65, 66, 69, 72, 77, 78, 79, 80, 84, 86, 87, 88, 89, 90, 91, 97, 99, 100, 105, 106, 107, 108, 110, 111, 112, 114, 115, 116, 117, 118, 120, 122, 123, 125, 127, 128
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Keywords

Comments

If two monotonic sequences are mapped into the real section of (0,1), as it is defined in A051006 and the product of the two reals mapped back into the set of monotonic sequences as defined in A092855, then we have the "product" of the two sequences.

Links

Crossrefs

Cf. A092855, A051006, A092857, A092858, A092859, A092860, A092862, A092863, A092874.

Programs

  • PARI
    {prod(a,b)= /*Returns the "product" of monotonic sequences a and b */ return(mtinv(mt(a)*mt(b))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A092862 "Square" of the prime sequence.

Original entry on oeis.org

3, 5, 6, 14, 16, 17, 19, 21, 22, 25, 27, 31, 32, 34, 36, 37, 41, 42, 44, 45, 48, 49, 52, 54, 57, 59, 60, 62, 64, 65, 69, 74, 75, 78, 81, 88, 90, 91, 92, 94, 97, 98, 100, 103, 104, 108, 109, 114, 118, 119, 121, 123, 124, 125, 127, 128, 129, 130, 131, 133, 135, 136, 137
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Keywords

Comments

By following the definition outlined in A092861, one can multiply a monotonic sequence by itself, thus squaring it.

Links

Crossrefs

Cf. A092855, A051006, A092857, A092858, A092859, A092860, A092861, A092863, A092874.

Programs

  • PARI
    {pow(a,n)= /*Returns the "n-th power" of monotonic sequence a */ return(mtinv(mt(a)^n)) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A092863 Prime sequence to the power Pi.

Original entry on oeis.org

4, 7, 10, 16, 18, 20, 22, 23, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 40, 42, 46, 51, 57, 60, 65, 66, 67, 68, 69, 70, 72, 73, 74, 77, 78, 80, 81, 82, 84, 85, 89, 91, 92, 93, 94, 95, 99, 101, 103, 107, 108, 110, 111, 112, 115, 117, 122, 123, 124, 125, 127, 128, 129, 130
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Keywords

Links

Crossrefs

Cf. A092855, A051006, A092857, A092858, A092859, A092860, A092861, A092862, A092874.

Programs

  • PARI
    {prow(a,r)= /*Returns the "r-th power" of monotonic sequence a */ return(mtinv(mt(a)^r)) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A239976 Positions of ones in binary representation of log(2).

Original entry on oeis.org

1, 3, 4, 8, 10, 11, 12, 15, 20, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 36, 40, 41, 42, 45, 46, 47, 48, 50, 51, 52, 53, 56, 57, 59, 61, 63, 64, 65, 66, 69, 72, 73, 74, 75, 79, 80, 81, 83, 84, 87, 88, 89, 92, 93, 103, 104, 105, 106, 107, 108, 111, 113, 114, 115, 116, 118, 119, 121, 123
Offset: 1

Views

Author

Ralf Stephan, Mar 30 2014

Keywords

Examples

			log(2) = [0], [1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, ...

Links

Crossrefs

Cf. A002162, A092855.

Programs

  • Mathematica
    Position[RealDigits[Log[2],2,200][[1]],1]//Flatten (* Harvey P. Dale, Aug 29 2016 *)
  • PARI
    v=binary(log(2))[2]; for(i=1,#v,if(v[i],print1(i,",")));
Previous Showing 11-19 of 19 results.

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