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 1-10 of 20 results. Next

A167759 Numbers k such that d(k) is an isolated number (A167706).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 11 2009

Keywords

Comments

Isolated numbers (A167706) are 2, 4, 6, 12, 18, 23, 30, 37, .... Sequence lists numbers k such that the number of divisors of k is isolated number. Also, the positions of isolated numbers in A000005.

Examples

			A000005(a(1)=2)=2; A000005(a(2)=3)=2; A000005(a(3)=5)=2; A000005(a(4)=6)=4.

Crossrefs

Cf. A000005, A002035, A167706.

Programs

  • Maple
    isA007510 := proc(n) if isprime(n) then not isprime(n+2) and not isprime(n-2) ; else false; end if; end proc: isA014574 := proc(n) isprime(n+1) and isprime(n-1) ; end proc: isA167706 := proc(n) isA007510(n) or isA014574(n) ; end proc: isA167759 := proc(n) isA167706(numtheory[tau](n)) ; end proc: for n from 1 to 100 do if isA167759(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Nov 16 2009

Formula

A000005(a(n)) is in A167706.

Extensions

Edited by Jon E. Schoenfield, May 10 2019

A167909 Differences between consecutive single (or isolated) numbers A167706.

Original entry on oeis.org

2, 2, 6, 6, 5, 7, 7, 5, 5, 6, 7, 7, 5, 7, 4, 6, 8, 5, 6, 5, 14, 4, 7, 12, 7, 6, 4, 6, 7, 12, 6, 13, 12, 5, 5, 7, 11, 6, 6, 7, 7, 5, 11, 14, 5, 5, 14, 6, 11, 5, 6, 8, 6, 6, 4, 6, 8, 4, 8, 11, 12, 7, 4, 6, 8, 5, 5, 12, 8, 4, 8, 4, 6, 13, 19, 6, 10, 6, 7, 7, 10, 6, 7, 7, 6, 5, 13, 11, 5, 6, 7, 13, 4, 6, 8, 10
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 15 2009

Keywords

Examples

			a(1)=4-2=2, a(2)=6-4=2, a(3)=12-6=6.

Crossrefs

Cf. A167706.

Extensions

Corrected at 3 or more placed by R. J. Mathar, May 30 2010

A167707 The non-single or nonisolated numbers. The union of non-single (or nonisolated or twin) primes and non-single (or nonisolated) nonprimes.

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 10 2009

Keywords

Comments

A167707 = A001097 U A164276. A167706 U A167707 = A001477.

Links

Crossrefs

Cf. A001097, A001477, A164276, A167706.

Programs

  • Maple
    isA001097 := proc(n) isprime(n) and (isprime(n+2) or isprime(n-2)) ; end proc: isA164276 := proc(n) not isprime(n) and ( not isprime(n+1) or not isprime(n-1) ) ; end proc: isA167707 := proc(n) isA001097(n) or isA164276(n) ; end proc: for n from 0 to 100 do if isA167707(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Mar 18 2010
  • Mathematica
    Union[Select[Range[0, 300], !PrimeQ[#] && (! PrimeQ[# - 1] || ! PrimeQ[# + 1]) & ], Select[Prime[Range[300]], PrimeQ[# - 2] || PrimeQ[# + 2] &]] (* G. C. Greubel, Jul 07 2016 *)

Formula

a(n) = n + n / log n + O(n / (log n)^2) by Brun's theorem. [Charles R Greathouse IV, Mar 15 2011]

A167777 Even single (or even isolated) numbers.

Original entry on oeis.org

2, 4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, 642, 660, 810, 822, 828, 858, 882, 1020, 1032, 1050, 1062, 1092, 1152, 1230, 1278, 1290, 1302, 1320, 1428, 1452, 1482, 1488
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 11 2009

Keywords

Comments

Two together with average of twin prime pairs.

Crossrefs

Cf. A014574 (average of twin prime pairs), A167706 (the single or isolated numbers).

Programs

  • Maple
    Contribution from R. J. Mathar, Apr 14 2010: (Start)
isA007510 := proc(n) if isprime(n) then not isprime(n+2) and not isprime(n-2) ; else false; end if; end proc:
isA014574 := proc(n) if not isprime(n) then isprime(n+1) and isprime(n-1) ; else false; end if; end proc:
A167777 := proc(n) if n = 1 then 2; else for a from procname(n-1)+2 by 2 do if isA007510(a) or isA014574(a) then return(a) ; end if; end do ; end if; end proc:
seq(A167777(n),n=1..60) ; (End)

A168543 pi(n-th single or isolated number).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 35, 37, 38, 39, 40, 41, 43, 45, 47, 48, 49, 51, 52, 54, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 29 2009

Keywords

Formula

a(n)=A000720(A167706(n)).

Extensions

Corrected (25 inserted) by R. J. Mathar, Nov 30 2009

A168343 n-th single or isolated number minus n.

Original entry on oeis.org

1, 2, 3, 8, 13, 17, 23, 29, 33, 37, 42, 48, 54, 58, 64, 67, 72, 79, 83, 88, 92, 105, 108, 114, 125, 131, 136, 139, 144, 150, 161, 166, 178, 189, 193, 197, 203, 213, 218, 223, 229, 235, 239, 249, 262, 266, 270, 283, 288, 298, 302, 307, 314, 319, 324, 327, 332, 339
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 23 2009

Keywords

Programs

  • Maple
    isA007510 := proc(n) if isprime(n) then not isprime(n+2) and not isprime(n-2) ; else false; end if; end proc: isA014574 := proc(n) if not isprime(n) then isprime(n+1) and isprime(n-1) ; else false; end if; end proc: A167706 := proc(n) if n = 1 then 2; else for a from procname(n-1)+1 do if isA007510(a) or isA014574(a) then return(a) ; end if; end do ; end if; end proc: A168343 := proc(n) A167706(n)-n ; end proc: seq(A168343(n),n=1..80) ; # R. J. Mathar, Nov 24 2009

Formula

a(n) = A167706(n) - A000027(n) = A167706(n) - n.

Extensions

Corrected by R. J. Mathar, Nov 24 2009

A171421 Euler totient function of the n-th single or isolated number.

Original entry on oeis.org

1, 2, 2, 4, 6, 22, 8, 36, 12, 46, 52, 16, 66, 24, 78, 82, 88, 96, 32, 36, 112, 126, 130, 44, 40, 156, 162, 166, 172, 48, 64, 60, 210, 222, 72, 232, 64, 250, 256, 262, 72, 276, 92, 292, 306, 96, 316, 330, 336, 112, 352, 358, 366, 372, 378, 382, 388, 396, 400, 408
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Dec 08 2009

Keywords

Links

Crossrefs

Cf. A000010, A167706, A168543.

Formula

a(n) = A000010(A167706(n)).

Extensions

More terms from Amiram Eldar, Feb 29 2020

A167885 n-th single or isolated number*n-th non-single or nonisolated number.

Original entry on oeis.org

0, 4, 18, 60, 126, 184, 270, 370, 462, 611, 742, 900, 1072, 1224, 1501, 1660, 1869, 2134, 2448, 2700, 2938, 3429, 3668, 4002, 4650, 5024, 5379, 5678, 6055, 6480, 7296, 7722, 8440, 9143, 9804, 10252, 10800, 11546, 12336, 12887, 13500, 14127, 14664
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 14 2009, Nov 15 2009

Keywords

Examples

			a(1)=2*0=0, a(2)=4*1=4, a(3)=6*3=18, a(4)=12*5=60.

Crossrefs

Cf. A167706, A167707.

Formula

a(n)=A167706(n)*A167707(n).

Extensions

All terms > 1900 corrected by R. J. Mathar, May 30 2010

A167886 n-th single or isolated number minus n-th non-single or nonisolated number.

Original entry on oeis.org

2, 3, 3, 7, 11, 15, 21, 27, 31, 34, 39, 45, 51, 55, 60, 63, 68, 75, 78, 83, 87, 100, 103, 109, 119, 125, 130, 133, 138, 144, 154, 159, 171, 182, 185, 189, 195, 205, 209, 214, 220, 226, 230, 239, 252, 256, 260, 273, 278, 287, 291, 296, 303, 308, 313, 315, 320, 327
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 14 2009

Keywords

Examples

			a(1)=2-0=2, a(2)=4-1=3, a(3)=6-3=3, a(4)=12-5=7.

Crossrefs

Cf. A167706, A167707.

Formula

a(n)=A167706(n)-A167707(n).

Extensions

All numbers > 68 corrected by R. J. Mathar, May 30 2010

A167887 n-th single or isolated number plus n-th non-single or nonisolated number.

Original entry on oeis.org

2, 5, 9, 17, 25, 31, 39, 47, 53, 60, 67, 75, 83, 89, 98, 103, 110, 119, 126, 133, 139, 154, 159, 167, 181, 189, 196, 201, 208, 216, 230, 237, 251, 264, 271, 277, 285, 297, 305, 312, 320, 328, 334, 347, 362, 368, 374, 389, 396, 409, 415, 422, 431, 438, 445, 451
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 14 2009

Keywords

Examples

			a(1)=2+0=2, a(2)=4+1=5, a(3)=6+3=9, a(4)=12+5=17.

Crossrefs

Cf. A167706, A167707.

Formula

a(n)=A167706(n)+A167707(n).

Extensions

All terms > 110 corrected by R. J. Mathar, May 30 2010
Showing 1-10 of 20 results. Next

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