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 1-3 of 3 results.

A261614 Numbers that are neither prime (A000040) nor practical (A005153).

Original entry on oeis.org

9, 10, 14, 15, 21, 22, 25, 26, 27, 33, 34, 35, 38, 39, 44, 45, 46, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 70, 74, 75, 76, 77, 81, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 102, 105, 106, 110, 111
Offset: 1

Views

Author

Frank M Jackson, Nov 18 2015

Keywords

Examples

			a(5)=21 and it is neither prime nor practical. It is the 5th such occurrence.

Crossrefs

Cf. A000040, A005153, A225223.

Programs

  • Mathematica
    PracticalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; Select[Range[1, 1000], ! PracticalQ[#] && ! PrimeQ[#] &] (* using T. D. Noe's program A005153 *)
  • PARI
    isok(n) = bittest(n, 0) && return(n==1); my(P=1); n && !for(i=2, #n=factor(n)~, n[1, i]>1+(P*=sigma(n[1, i-1]^n[2, i-1])) && return) ;
for(n=1, 200, if(!isok(n) && !isprime(n), print1(n, ", "))) \\ Altug Alkan, Nov 19 2015

A260698 Practical numbers of the form p - 1 where p is a prime.

Original entry on oeis.org

1, 2, 4, 6, 12, 16, 18, 28, 30, 36, 40, 42, 60, 66, 72, 78, 88, 96, 100, 108, 112, 126, 150, 156, 162, 180, 192, 196, 198, 210, 228, 240, 256, 270, 276, 280, 306, 312, 330, 336, 348, 352, 378, 396, 400, 408, 420, 432, 448, 456, 460, 462, 486, 520, 522, 540, 546
Offset: 1

Views

Author

Frank M Jackson, Nov 16 2015

Keywords

Comments

Intersection of A005153 and A006093. - Michel Marcus, Nov 16 2015

Examples

			a(5)=12 as 12 is a practical number and 12+1=13 is prime. It is the 5th such practical number.

Crossrefs

Cf. A005153, A006093, A225223.

Programs

  • Mathematica
    PracticalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; Select[Table[Prime[n]-1, {n, 1, 200}], PracticalQ] (* using T. D. Noe's program A005153 *)
  • PARI
    is(n) = bittest(n, 0) && return(n==1); my(P=1); n && !for(i=2, #n=factor(n)~, n[1, i]>1+(P*=sigma(n[1, i-1]^n[2, i-1])) && return);
forprime(p=2, 1000, if(is(p-1), print1(p-1", "))) \\ Altug Alkan, Nov 16 2015

A346794 Primes p such that the largest Dyck path of the symmetric representation of sigma(p) does not touch the largest Dyck path of the symmetric representation of sigma(p+1).

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 47, 53, 59, 71, 79, 83, 89, 103, 107, 127, 131, 139, 149, 167, 179, 191, 197, 199, 223, 227, 233, 239, 251, 263, 269, 271, 293, 307, 311, 359, 367, 379, 383, 389, 419, 431, 439, 449, 461, 463, 467, 479, 499, 503, 509, 521
Offset: 1

Views

Author

Omar E. Pol, Aug 04 2021

Keywords

Comments

This property of a(n) is because the symmetric representation of sigma(a(n)+1) has only one part.
First differs from both A085498 and A225223 at a(40).

Crossrefs

Primes in A343621.
Cf. A000040, A000203, A085498, A174973, A196020, A225223, A235791, A236104, A237270, A237271, A237591, A237593, A238443, A239931, A239932, A239933, A239934, A262626.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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