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.

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A112797 Pierpont 3-almost primes. 3-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

28, 244, 325, 385, 730, 1025, 1729, 2188, 5185, 6562, 7777, 16385, 26245, 36865, 46657, 49153, 55297, 82945, 93313, 221185, 354295, 419905, 531442, 559873, 589825, 663553, 708589, 884737, 1119745, 1572865, 1594324, 1889569, 2985985
Offset: 1

Views

Author

Jonathan Vos Post, Nov 08 2005

Keywords

Examples

			a(1) = 28 = (2^0)*(3^3)+1 = 2 * 2 * 7.
a(2) = 244 = (2^0)*(3^5)+1 = 2 * 2 * 61.
a(3) = 325 = (2^2)*(3^4)+1 = 5 * 5 * 13.
a(4) = 385 = (2^7)*(3^1)+1 = 5 * 7 * 11.
a(11) = 7777 = (2^5)*(3^5)+1 = 7 * 11 * 101.
a(115) = 94143178828 = (2^0)*(3^23)+1 = 2 * 2 * 23535794707.
a(119) = 137438953473 = (2^37)*(3^0)+1 = 3 * 1777 * 25781083.
a(196) = 281474976710657 = (2^48)*(3^0)+1 = 193 * 65537 * 22253377.

Links

Crossrefs

Intersection of A014612 and A055600.
A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.
A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.
A111344 gives the Pierpont 4-almost primes, of the form (2^K)*(3^L)+1.
A111345 gives the Pierpont 5-almost primes, of the form (2^K)*(3^L)+1.
A111346 gives the Pierpont 6-almost primes, of the form (2^K)*(3^L)+1.
A113739 gives the Pierpont 7-almost primes, of the form (2^K)*(3^L)+1.
A113740 gives the Pierpont 8-almost primes, of the form (2^K)*(3^L)+1.
A113741 gives the Pierpont 9-almost primes, of the form (2^K)*(3^L)+1.

Programs

  • Mathematica
    Take[Select[2^#[[1]] 3^#[[2]] + 1 & /@ Tuples[Range[0, 20], 2],
PrimeOmega[ #]  ==  3 &] // Union, 40] (* Harvey P. Dale, Jan 02 2021 *)
  • PARI
    list(lim)=my(v=List(), L=lim\1-1); for(e=0, logint(L, 3), my(t=3^e); while(t<=L, if(bigomega(t+1)==3, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Formula

a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 3.

Extensions

Extended by Ray Chandler, Nov 08 2005

A069357 Numbers of form 2^i*3^j + (i+j) with i, j >= 0.

Original entry on oeis.org

1, 3, 4, 6, 8, 11, 15, 20, 21, 28, 30, 37, 40, 53, 58, 70, 77, 85, 102, 113, 135, 150, 167, 199, 222, 248, 264, 295, 330, 392, 439, 492, 521, 584, 655, 735, 777, 872, 979, 1034, 1161, 1304, 1465, 1546, 1737, 1952, 2059, 2194, 2314, 2601, 2924, 3083, 3466, 3897
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 18 2002

Keywords

Crossrefs

Cf. A055600, A069358, A064800, A022328, A022329.

Formula

Distinct values of A003586(k) + A069352(k). [Corrected by Georg Fischer, Dec 11 2022, further clarification by Sean A. Irvine, Apr 28 2024]

Extensions

Missing a(1)=1 inserted and duplicate values removed by Sean A. Irvine, Apr 28 2024

A113433 Semi-Pierpont semiprimes: products of exactly two Pierpont primes A005109.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 25, 26, 34, 35, 38, 39, 49, 51, 57, 65, 74, 85, 91, 95, 111, 119, 133, 146, 169, 185, 194, 218, 219, 221, 247, 259, 289, 291, 323, 326, 327, 361, 365, 386, 481, 485, 489, 511, 514, 545, 579, 629, 679, 703, 763, 771, 815, 866, 949, 965
Offset: 1

Views

Author

Jonathan Vos Post, Nov 01 2005

Keywords

Comments

Semiprime both of whose prime factors are Pierpont primes (A005109), which are primes of the form (2^K)*(3^L)+1. Not to be confused with A113432: Pierpont semiprimes [Semiprimes of the form (2^K)*(3^L)+1]. This terminology itself is by analogy to what Tomaszewski used for the Sophie Germain counterparts A111153 and A111206.

Examples

			a(1) = 4 = 2^2 = [(2^0)*(3^0)+1]*[(2^1)*(3^0)+1] = A005109(1)*A005109(1).
a(2) = 6 = 2*3 = [(2^0)*(3^0)+1]*[(2^1)*(3^0)+1] = A005109(1)*A005109(2).
a(3) = 9 = 3^2 = [(2^1)*(3^0)+1]*[(2^1)*(3^0)+1] = A005109(2)*A005109(2).
a(4) = 10 = 2*5 = [(2^0)*(3^0)+1]*[(2^2)*(3^0)+1] = A005109(1)*A005109(3).
a(5) = 14 = 2*7 = [(2^0)*(3^0)+1]*[(2^1)*(3^1)+1] = A005109(1)*A005109(4).
a(6) = 15 = 3*5 = [(2^1)*(3^0)+1]*[(2^2)*(3^0)+1] = A005109(2)*A005109(3).

Links

Crossrefs

Cf. A001358, A003586, A005109, A055600, A111153, A111206, A113432, A113434.

Programs

  • Mathematica
    Select[Range[10^3], Plus @@ Last /@ FactorInteger[ # ] == 2 && And @@ (Max @@ First /@ FactorInteger[ # - 1] < 5 &) /@ First /@ FactorInteger[ # ] &] (* Ray Chandler, Jan 24 2006 *)

Formula

{a(n)} = Semiprimes A001358 both of whose factors are of the form (2^K)*(3^L)+1. {a(n)} = {A005109(i)*A005109(j) for integers i and j not necessarily distinct}.

A113434 Semi-Pierpont semiprimes which are also Pierpont semiprimes.

Original entry on oeis.org

4, 9, 10, 25, 49, 65, 289
Offset: 1

Views

Author

Jonathan Vos Post, Nov 01 2005

Keywords

Comments

Semiprimes both of whose prime factors are Pierpont primes (A005109), which are primes of the form (2^K)*(3^L)+1 and where the semiprime is itself of the form (2^K)*(3^L)+1.
No more under 10^50; what is the next element of this sequence?
No more terms <= 10^100. - Robert Israel, Mar 10 2017
This sequence is complete, see Links. - Charlie Neder, Feb 04 2019

Examples

			a(1) = 4 = 2^2 = [(2^0)*(3^0)+1]*[(2^0)*(3^0)+1] = (2^0)*(3^1)+1.
a(2) = 9 = 3^2 = [(2^1)*(3^0)+1]*[(2^1)*(3^0)+1] = (2^3)*(3^0)+1.
a(3) = 10 = 2*5 = [(2^0)*(3^0)+1]*[(2^2)*(3^0)+1] = (2^0)*(3^2)+1.
a(4) = 25 = 5^2 = [(2^2)*(3^0)+1]*[(2^2)*(3^0)+1] = (2^3)*(3^1)+1.
a(5) = 49 = 7^2 = [(2^1)*(3^1)+1]*[(2^1)*(3^1)+1] = (2^4)*(3^1)+1.
a(6) = 65 = 5*13 = [(2^2)*(3^0)+1]*[(2^2)*(3^1)+1] = (2^6)*(3^0)+1.
a(7) = 289 = 17^2 = [(2^4)*(3^0)+1]*[(2^4)*(3^0)+1] = (2^5)*(3^2)+1.

Links

Crossrefs

Cf. A001358, A003586, A005109, A055600, A111153, A111206, A113432, A113433.

Programs

  • Maple
    N:= 10^100: # to get all terms <= N
PP:= select(isprime, {seq(seq(1+2^i*3^j, i=0..ilog2((N-1)/3^j)),j=0..floor(log[3](N-1)))}):
SP:= select(t -> t <= N and t = 1+2^padic:-ordp(t-1,2)*3^padic:-ordp(t-1,3), [seq(seq(PP[i]*PP[j], j=1..i),i=1..nops(PP))]):
sort(convert(SP,list)); # Robert Israel, Mar 10 2017

Formula

{a(n)} = intersection of A113432 and A113433. {a(n)} = Semiprimes A001358 of the form (2^K)*(3^L)+1 both of whose factors are of the form (2^K)*(3^L)+1. {a(n)} = {integers P such that, for nonnegative integers I, J, K, L, m, n there is a solution to (2^I)*(3^J)+1 = [(2^K)*(3^L)+1]*[(2^m)*(3^n)+1] where both [(2^K)*(3^L)+1] and [(2^m)*(3^n)+1] are prime}.
Previous Showing 11-14 of 14 results.

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