A295235 -id:A295235 - cp's OEIS frontend

A335406 First position of n in the sequence of run-lengths of the sequence of prime gaps.

1, 2, 49, 633353, 6706139

Offset: 1

Gus Wiseman, Jun 10 2020

Prime gaps are differences between adjacent prime numbers.

Positions of first appearances in A333254.
The unequal version is 7, 1, 4, 15, 10, 36, 5, 6, 84, ...
The weakly decreasing version is 1, 2, 7, 23, 26, ...
The weakly increasing version is 5, 2, 3, 1, 81, 193, ...
The strictly decreasing version is 1, 4, 8, 150, 160, ...
The strictly increasing version is 6, 1, 4, 38, 221, ...
Prime gaps are A001223.
The first term of the first length-n arithmetic progression of consecutive primes is A006560(n), with index A089180(n).
Positions of adjacent equal prime gaps are A064113.
Positions of adjacent unequal prime gaps are A333214.
Cf. A000040, A031217, A054800, A059044, A084758, A090832, A106356, A124767, A238279, A295235, A006560.

qe=Length/@Split[Differences[Array[Prime,10000]],SameQ];
Table[Position[qe,i][[1,1]],{i,Union[qe]}]

a(5) from Giovanni Resta, Jun 11 2020

A333762 Fixed points of A333692.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 21, 24, 28, 30, 31, 32, 33, 34, 36, 40, 42, 48, 55, 56, 60, 62, 63, 64, 65, 66, 68, 72, 73, 80, 84, 85, 96, 103, 110, 112, 120, 124, 126, 127, 128, 129, 130, 132, 136, 144, 146, 160, 168, 170, 181

Offset: 1

Rémy Sigrist, Apr 04 2020

For any n >= 0, n belongs to this sequence iff 2*n also belongs to this sequence.

This sequence contains A000079, A000225, A018900, A023758, A295235.

Cf. A000079, A000225, A018900, A023758, A295235, A333692, A333693.

is(n, base=2)={ my (b=digits(n, base), p=[]); for (k=1, #b, p=concat(p, b[k]); if (b[k], p=Vecrev(p))); n==fromdigits(p, base) }

A327368 The positions of ones in the reversed binary expansion of n have integer mean and integer geometric mean.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 130, 256, 257, 512, 1024, 2048, 2084, 2316, 4096, 8192, 16384, 32768, 32776, 32777, 65536, 131072, 131074, 131200, 131457, 131462, 133390, 165920, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 8388640, 8388897, 8390688, 8519840

Offset: 1

Gus Wiseman, Sep 27 2019

nonn

			The sequence of terms together with their binary indices begins:
  2      {2}
  4      {3}
  8      {4}
  16     {5}
  32     {6}
  64     {7}
  128    {8}
  130    {2,8}
  256    {9}
  257    {1,9}
  512    {10}
  1024   {11}
  2048   {12}
  2084   {3,6,12}
  2316   {3,4,9,12}
  4096   {13}
  8192   {14}
  16384  {15}
  32768  {16}
  32776  {4,16}

A superset of A327777.
Numbers whose binary indices have integer mean: A326669
Numbers whose binary indices have integer geometric mean: A326673
Cf. A000120, A029931, A048793, A070939, A096111, A291166, A295235, A326029, A326643, A326699/A326700, A327777.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]

ok(n)={my(s=0,p=1,k=0); for(i=0, logint(n,2), if(bittest(n,i), s+=i+1; p*=i+1; k++)); s%k==0 && ispower(p,k)}
{ for(n=1, 10^7, if(ok(n), print1(n, ", "))) } \\ Andrew Howroyd, Sep 29 2019

a(33)-a(40) from Andrew Howroyd, Sep 29 2019

A330220 Numbers whose representation in base 2^w contains only the digit 2^k for some w and k such that 0 <= k < w.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 15, 16, 17, 18, 21, 31, 32, 33, 34, 36, 42, 63, 64, 65, 66, 68, 73, 85, 127, 128, 129, 130, 132, 136, 146, 170, 255, 256, 257, 258, 260, 264, 273, 292, 341, 511, 512, 513, 514, 516, 520, 528, 546, 585, 682, 1023, 1024, 1025, 1026

Offset: 1

Rémy Sigrist, Dec 06 2019

This is a subsequence of A295235.
For any k > 0, there are k nonzero terms with k binary digits.
Odd terms are A064896.

			The representation of 546 in base 2^4 is "222", so 546 belongs to the sequence.

Cf. A064896, A295235.

is(n) = { for (w=1, max(1, #binary(n)), my (d=if (n, digits(n,2^w), [0])); if (#Set(d)==1 && hammingweight(d[1])<=1, return (1))); return (0) }

cp's OEIS Frontend

A335406 First position of n in the sequence of run-lengths of the sequence of prime gaps.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A333762 Fixed points of A333692.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A327368 The positions of ones in the reversed binary expansion of n have integer mean and integer geometric mean.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A330220 Numbers whose representation in base 2^w contains only the digit 2^k for some w and k such that 0 <= k < w.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI