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 1-5 of 5 results.

A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

Original entry on oeis.org

3, 2, 5, 13, 11, 19, 17, 31, 29, 43, 41, 61, 59, 73, 71, 103, 101, 109, 107, 139, 137, 151, 149, 181, 179, 193, 191, 199, 197, 229, 227, 241, 239, 271, 269, 283, 281, 313, 311, 349, 347, 421, 419, 433, 431, 463, 461, 523, 521, 571, 569, 601, 599, 619, 617
Offset: 1

Views

Author

Clark Kimberling, Dec 15 2015

Keywords

Comments

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...).
See A265772 and A265774 for definitions of lower POBA and upper POBA. In the following guide, for example, A001359/A006512 represents (conjecturally in some cases) the Lower POBAs p(n)/q(n) to 1, where p = A001359 and q = A006512 except for first terms in some cases. Every POBA is either a lower POBA or an upper POBA.
x Lower POBA Upper POBA POBA
1 A001359/A006512 A006512/A001359 A265759/A265760
3/2 A104163/A158708 A162336/A158709 A265761/A222565
2 A005383/A005382 A005385/A005384 A079149/A120628
3 A091180/A088878 A094525/A023208 A265763/A265764
4 A162857/A062737 A090866/A023212 A265765/A120639
5 A265766/A158318 A265767/A023217 A265768/A265769
6 A227756/A158015 A051644/A007693 A265770/A265771
sqrt(2) A265772/A265773 A265774/A265775 A265776/A265777
sqrt(3) A265778/A265779 A265780/A265781 A265782/A265783
sqrt(5) A265784/A265785 A265786/A265787 A265788/A265789
sqrt(8) A265790/A265791 A265792/A265793 A265794/A265795
tau A265796/A265797 A265798/A265799 A265800/A265801
1/tau A265799/A265798 A265797/A265796 A265806/A265807
pi A265808/A265809 A265810/A265811 A265812/A265813
e A265814/A265815 A265816/A265817 A265818/A265819

Examples

			The POBAs for 1 start with 3/2, 2/3, 5/7, 13/11, 11/13, 19/17, 17/19, 31/29, 29/31, 43/41, 41/43, 61/59, 59/61. For example, if p and q are primes and q > 13, then 11/13 is closer to 1 than p/q is.

Crossrefs

Cf. A000040, A001359, A006512, A265759, A265760.

Programs

  • Mathematica
    x = 1; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265759/A265760 *)
Numerator[tL]   (* A001359 *)
Denominator[tL] (* A006512 *)
Numerator[tU]   (* A006512 *)
Denominator[tU] (* A001359 *)
Numerator[y]    (* A265759 *)
Denominator[y]  (* A265760 *)

A227758 a(n) = sigma(sigma(n)) - sigma(n) - n, where sigma(n) = A000203(n) = sum of the divisors of n.

Original entry on oeis.org

-1, -1, 0, -3, 1, 10, 0, 1, -8, 11, 5, 16, -3, 22, 21, -15, 4, -1, 3, 34, 10, 33, 13, 84, -24, 28, 23, 36, 13, 93, 0, 9, 43, 32, 41, -15, -15, 70, 25, 104, 13, 114, -3, 96, 45, 77, 29, 52, -26, -15, 72, 21, 13, 186, 68, 184, 49, 86, 49, 252, -27, 94, 43, -63
Offset: 1

Views

Author

Jaroslav Krizek, Jul 26 2013

Keywords

Comments

Conjecture: a(n) = 0 for Mersenne primes (A000668). [This is easily proved: For Mersenne primes n=2^p-1, sigma(n)=n+1=2^p, sigma(2^p)=2^(p+1)-1, thus a(n)=0. - M. F. Hasler, Jul 30 2013]
a(n) < 0 for numbers n from A227759, a(n) > 0 for numbers n from A227760.
Sequence contains anomalous increased frequency of values 13 (see A227756).

Examples

			For n = 6; a(n) = sigma(sigma(6)) - sigma(6) - 6 = 28 - 12 - 6 = 10.

Links

Crossrefs

Cf. A051027, A000203, A000668, A227759, A227760, A227756.

Formula

a(n) = A051027(n) - A000203(n) - n.

A265770 Numerators of primes-only best approximates (POBAs) to 6; see Comments.

Original entry on oeis.org

13, 11, 19, 17, 31, 29, 43, 41, 67, 79, 103, 101, 113, 139, 137, 173, 223, 257, 283, 281, 317, 353, 367, 401, 439, 499, 607, 619, 617, 643, 641, 653, 677, 761, 787, 823, 821, 907, 941, 977, 1039, 1087, 1181, 1193, 1361, 1373, 1399, 1433, 1447, 1543, 1579
Offset: 1

Views

Author

Clark Kimberling, Dec 20 2015

Keywords

Comments

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences.

Examples

			The POBAs to 6 start with 13/2, 11/2, 19/3, 17/3, 31/5, 29/5, 43/7, 41/7, 67/11, 79/13, 103/17, 101/17. For example, if p and q are primes and q > 17, then 103/17 (and 101/17) is closer to 6 than p/q is.

Crossrefs

Cf. A000040, A265759, A227756, A158015, A051644, A007693, A265771.

Programs

  • Mathematica
    x = 6; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265770/A265771 *)
Numerator[tL]   (* A227756 *)
Denominator[tL] (* A158015 *)
Numerator[tU]   (* A051644 *)
Denominator[tU] (* A007693 *)
Numerator[y]    (* A222570 *)
Denominator[y]  (* A265771 *)

A265771 Denominators of primes-only best approximates (POBAs) to 6; see Comments.

Original entry on oeis.org

2, 2, 3, 3, 5, 5, 7, 7, 11, 13, 17, 17, 19, 23, 23, 29, 37, 43, 47, 47, 53, 59, 61, 67, 73, 83, 101, 103, 103, 107, 107, 109, 113, 127, 131, 137, 137, 151, 157, 163, 173, 181, 197, 199, 227, 229, 233, 239, 241, 257, 263, 269, 271, 277, 283, 283, 293, 311
Offset: 1

Views

Author

Clark Kimberling, Dec 20 2015

Keywords

Comments

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences.

Examples

			The POBAs to 6 start with 13/2, 11/2, 19/3, 17/3, 31/5, 29/5, 43/7, 41/7, 67/11, 79/13, 103/17, 101/17. For example, if p and q are primes and q > 17, then 103/17 (and 101/17) is closer to 6 than p/q is.

Crossrefs

Cf. A000040, A265759, A227756, A158015, A051644, A007693, A265770.

Programs

  • Mathematica
    x = 6; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265770/A265771 *)
Numerator[tL]   (* A227756 *)
Denominator[tL] (* A158015 *)
Numerator[tU]   (* A051644 *)
Denominator[tU] (* A007693 *)
Numerator[y]    (* A222570 *)
Denominator[y]  (* A265771 *)

A227757 Numbers k such that antisigma(k) = antisigma(k+1) + 12; where antisigma(k) = A024816(k).

Original entry on oeis.org

23, 29, 41, 53, 101, 113, 137, 173, 257, 281, 317, 353, 401, 617, 641, 653, 677, 761, 821, 941, 977, 1181, 1193, 1359, 1361, 1373, 1433, 1613, 1697, 1877, 1901, 2081, 2153, 2237, 2273, 2297, 2333, 2381, 2633, 2657, 2693, 2741, 2777, 2801, 3137, 3413, 3461, 3557
Offset: 1

Views

Author

Jaroslav Krizek, Jul 26 2013

Keywords

Comments

Supersequence of A227756 (primes p such that sigma(p+1) = 2*p+14).
The composites in this sequence are rare: 1359, 453967, ... (no more up to 10^6).

Examples

			1359 is in sequence because antisigma(1359) = 922144 = antisigma(1360) + 12 = 922132 + 12.

Links

Crossrefs

Cf. A024816, A227756.

Programs

  • Mathematica
    Select[Range[3600],DivisorSigma[1,#+1]-DivisorSigma[1,#]==#+13 &] (* Stefano Spezia, Apr 18 2025 *)
  • PARI
    is(n)=sigma(n+1)-sigma(n)==n+13 \\ Charles R Greathouse IV, Nov 25 2014
Showing 1-5 of 5 results.

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