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 18 results. Next

A171618 Number of ways of writing n=k1+k2 with k1 and k2 in A167707.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 3, 3, 3, 5, 4, 6, 5, 6, 5, 7, 6, 8, 6, 8, 8, 8, 9, 9, 10, 10, 9, 11, 10, 12, 12, 13, 11, 12, 13, 13, 15, 14, 14, 15, 14, 16, 14, 17, 17, 16, 17, 17, 18, 18, 19, 18, 19, 19, 21, 21, 19, 21, 20, 22, 24, 23, 22, 22, 23, 24, 25, 25, 24, 25, 24, 27, 26, 28, 27
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Dec 13 2009

Keywords

Examples

			a(31)=9 because 31 = 0 + 31 = 3 + 28 = 5 + 26 = 7 + 24 = 9 + 22 = 10 + 21 = 11 + 20 = 14 + 17 = 15 + 16.

Crossrefs

Cf. A062602, A062610, A167707.

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:
A167707 := proc(n) option remember; if n = 1 then 0; else for a from procname(n-1)+1 do if isA167707(a) then return a; end if; end do; end if; end proc:
A171618 := proc(n) a := 0 ; for i from 1 do p := A167707(i) ; q := n-p ; if q < p then return a ; end if; if isA167707(q) then a := a+1 ; end if; if q <= p then return a ; end if; end do: end proc:
seq(A171618(n),n=1..120) ; # R. J. Mathar, May 22 2010
  • Mathematica
    isA001097[n_] := PrimeQ[n] && (PrimeQ[n+2] || PrimeQ[n-2]);
isA164276[n_] := !PrimeQ[n] && (!PrimeQ[n+1] ||!PrimeQ[n-1]);
isA167707[n_] := isA001097[n] || isA164276[n];
A167707[n_] := A167707[n] = If[n == 1, 0, For[a = A167707[n-1]+1, True, a++, If[isA167707[a], Return@a]]];
A171618[n_] := Module[{a}, a = 0; For[i = 1, True, i++, p = A167707[i]; q = n-p; If[q < p, Return@a]; If[isA167707[q], a++]; If[q <= p, Return@a]]];
Table[A171618[n], {n, 1, 120}] (* Jean-François Alcover, Feb 23 2024, after R. J. Mathar *)

Extensions

a(29) and a(34) corrected and sequence extended by R. J. Mathar, May 22 2010

A167911 Differences between consecutive non-single (or nonisolated) numbers A167707.

Original entry on oeis.org

1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 15 2009

Keywords

Examples

			a(1)=1-0=1, a(2)=3-1=2, a(3)=5-3=2, a(4)=7-5=2, a(5)=8-7=1.

Crossrefs

Cf. A167707.

Formula

a(n) = A167707(n+1) - A167707(n). - Jason Yuen, Oct 08 2024

Extensions

Entries checked by R. J. Mathar, May 30 2010

A168045 a(n) = A167707(n) + n.

Original entry on oeis.org

1, 3, 6, 9, 12, 14, 16, 18, 20, 23, 25, 27, 29, 31, 34, 36, 38, 40, 43, 45, 47, 49, 51, 53, 56, 58, 60, 62, 64, 66, 69, 71, 73, 75, 78, 80, 82, 84, 87, 89, 91, 93, 95, 98, 100, 102, 104, 106, 108, 111, 113, 115, 117, 119, 121, 124, 126, 128, 130, 133, 135, 137, 139, 141
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 17 2009

Keywords

Crossrefs

Cf. A167707.

Formula

a(n) = 2n + n / log n + O(n / (log n)^2). - Charles R Greathouse IV, May 25 2011

A171384 a(n) = A140475(n) - A167707(n).

Original entry on oeis.org

1, 4, 4, 6, 6, 9, 10, 13, 18, 18, 23, 26, 27, 30, 34, 39, 40, 45, 47, 48, 53, 56, 61, 68, 70, 71, 74, 75, 78, 91, 93, 98, 99, 108, 108, 113, 118, 121, 125, 130, 131, 140, 141, 143, 144, 155, 166, 169, 170, 172, 177, 178, 187, 192, 197, 201, 202, 207, 210, 210, 219
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Dec 07 2009

Keywords

Examples

			a(1)=1-0=1, a(2)=5-1=4, a(3)=7-3=4, a(4)=11-5=6, a(5)=13-7=6, a(6)=17-8=9.

Crossrefs

Cf. A140475, A167707.

Extensions

Edited by Omar E. Pol, Apr 14 2010

A168501 Numbers without the decimal digits 2, 4 and 6.

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 9, 10, 11, 13, 15, 17, 18, 19, 30, 31, 33, 35, 37, 38, 39, 50, 51, 53, 55, 57, 58, 59, 70, 71, 73, 75, 77, 78, 79, 80, 81, 83, 85, 87, 88, 89, 90, 91, 93, 95, 97, 98, 99, 100, 101, 103, 105, 107, 108, 109, 110, 111, 113, 115, 117, 118, 119, 130, 131, 133
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 27 2009

Keywords

Comments

Numbers that contain non-single or nonisolated digits only (see the first seven values of A167707 for the digits that are not 2, 4, or 6).

Links

Crossrefs

Cf. A014263, A167707.

Programs

  • Magma
    [n: n in [0..200] | IsEmpty(Set([2,4,6]) meet Set(Intseq(n)))]; // Bruno Berselli, Jul 25 2016
  • Mathematica
    Select[Select[Select[Range[0, 4500], FreeQ[IntegerDigits@#, 2] &], FreeQ[IntegerDigits@#, 4] &], FreeQ[IntegerDigits@#, 6] &] (* G. C. Greubel, Jul 24 2016 *)

Extensions

Corrected (114 removed) by R. J. Mathar, Jun 04 2010

A171561 (n-th prime number)^(n-th non-single or nonisolated number) with duplicates removed.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Dec 11 2009, Dec 14 2009

Keywords

Comments

The definition is: numbers of the form A000040(i)^A167707(j), sorted, without repetition, any i,j>=1. [From R. J. Mathar, May 22 2010]

Examples

			a(1)=2^0=1, a(2)=2^1=2, a(3)=3^1=3, a(4)=5^1=5, a(5)=7^1=7, a(6)=2^3=8.

Crossrefs

Cf. A000040 (the prime numbers), A167707 (the non-single or nonisolated numbers).

Extensions

Entries checked by R. J. Mathar, May 22 2010

A168042 The non-single or nonisolated numbers whose digits are all square.

Original entry on oeis.org

0, 1, 9, 10, 11, 14, 19, 40, 41, 44, 49, 90, 91, 94, 99, 100, 101, 104, 109, 110, 111, 114, 119, 140, 141, 144, 149, 190, 191, 194, 199, 400, 404, 410, 411, 414, 419, 440, 441, 444, 490, 494, 900, 901, 904, 909, 910, 914, 940, 944, 949, 990, 994, 999, 1000, 1001
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 17 2009

Keywords

Comments

Square digits are 0, 1, 4 or 9. The non-single or nonisolated numbers are 0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21,..

Crossrefs

Cf. A000290, A157908, A167707.

Extensions

Entries checked by R. J. Mathar, May 30 2010

A168086 Numbers k such that d(k) is not an isolated number.

Original entry on oeis.org

1, 4, 9, 16, 24, 25, 30, 36, 40, 42, 48, 49, 54, 56, 64, 66, 70, 78, 80, 81, 88, 100, 102, 104, 105, 110, 112, 114, 120, 121, 128, 130, 135, 136, 138, 144, 152, 154, 162, 165, 168, 169, 170, 174, 176, 182, 184, 186, 189, 190, 192, 195, 196, 208, 210, 216, 222
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 18 2009

Keywords

Comments

A000005 = A167759 U A168086. Where 0,1,3,5,7,8,9,10,11,13,14,15,16,17,19,20,21,22,.. are nonisolated numbers A167707. The nonisolated numbers of divisors of n. The positions of isolated numbers in A000005.

Examples

			A000005(a(1)=1)=1, A000005(a(2)=4)=3, A000005(a(3)=9)=3.

Crossrefs

Cf. A000005, A167759, A167707.

Formula

A000005(a(n)) = nonisolated number.

Extensions

Corrected (132, 140, 148 removed, 152 inserted etc.) by R. J. Mathar, Jun 04 2010

A168252 Numbers n such that n is squarefree and n is non-single or nonisolated.

Original entry on oeis.org

1, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 26, 29, 31, 33, 34, 35, 38, 39, 41, 43, 46, 51, 55, 57, 58, 59, 61, 62, 65, 66, 69, 70, 71, 73, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 101, 103, 105, 106, 107, 109, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 21 2009

Keywords

Crossrefs

Cf. A000469 (nonprime squarefree numbers), A005117 (squarefree numbers), A120944 (composite squarefree numbers), A167707 (non-single or nonisolated numbers).

Formula

A005117 INTERSECT A167707. [From R. J. Mathar, Apr 24 2010]

Extensions

Edited by N. J. A. Sloane, Jan 23 2010
Corrected (114 inserted) by R. J. Mathar, Apr 24 2010

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
Showing 1-10 of 18 results. Next

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