Keith Backman - cp's OEIS frontend

User: Keith Backman

Keith Backman has authored 7 sequences.

A346676 Numbers expressible as 2^x + 3^y where both x and y are positive integers.

5, 7, 11, 13, 17, 19, 25, 29, 31, 35, 41, 43, 59, 67, 73, 83, 85, 89, 91, 97, 113, 131, 137, 145, 155, 209, 245, 247, 251, 259, 265, 275, 283, 307, 337, 371, 499, 515, 521, 539, 593, 731, 733, 737, 745, 755, 761, 793, 857, 985, 1027, 1033, 1051, 1105, 1241

Offset: 1

Keith Backman, Jul 28 2021

nonn

All terms have the form 6k +- 1.

Cf. A000079, A000244, A004050.

f(x,y) = 2^x + 3^y;
lista(nn) = select(x->(x<=nn), setbinop(f, [1..logint(nn, 2)], [1..logint(nn, 3)])); \\ Michel Marcus, Jul 29 2021

def aupto(lim):
    s, pow3 = set(), 3
    while pow3 < lim:
        for j in range(1, (lim-pow3).bit_length()):
            s.add(2**j + pow3)
        pow3 *= 3
    return sorted(set(s))
print(aupto(1242)) # Michael S. Branicky, Jul 29 2021

{ A004050 } minus { A000079, A000244 }.

A332777 a(n) = k^2 mod p where p is the n-th prime and of the form 6k-1 or 6k+1.

Original entry on oeis.org

1, 1, 4, 4, 9, 9, 16, 25, 25, 36, 8, 6, 17, 28, 41, 39, 54, 2, 71, 11, 30, 47, 62, 87, 83, 3, 106, 22, 60, 91, 118, 112, 29, 21, 48, 77, 116, 149, 5, 176, 69, 59, 104, 94, 170, 31, 82, 70, 123, 166, 154, 7, 50, 95, 142, 128, 177, 242, 228, 57, 145, 216, 200, 273

Offset: 3

Keith Backman, Jun 08 2020

nonn

Offset is 3 because 5=prime(3) is the first prime of the given form. It is provable that if 6m-1 and 6m+1 are a pair of twin primes, then for all k, 0

Cf. A002476, A007528, A024699.

a(n) = {my(p=prime(n), k); if (((p-1) % 6) == 0, k = (p-1)/6, k = (p+1)/6); k^2 % p;} \\ Michel Marcus, Jun 09 2020

More terms from Michel Marcus, Jun 09 2020

A283555 Even numbers that cannot be expressed as p+3, p+5, or p+7, with p prime.

Original entry on oeis.org

98, 122, 124, 126, 128, 148, 150, 190, 192, 208, 210, 212, 220, 222, 224, 250, 252, 292, 294, 302, 304, 306, 308, 326, 328, 330, 332, 346, 348, 368, 398, 410, 418, 420, 430, 432, 458, 476, 478, 480, 488, 500, 518, 520, 522, 532, 534, 536, 538, 540, 542, 556

Offset: 1

Keith Backman, Mar 10 2017

nonn

Any even number 2n which fails the Goldbach condition (i.e., is not expressible as the sum of two primes) cannot be a prime plus 3 (by definition), but it must also be the case that the two even numbers immediately smaller than 2n (i.e., 2n-2 and 2n-4) also cannot be a prime plus 3, because if they were, 2n would be a prime plus 5 or a prime plus 7 and would satisfy Goldbach. Thus any even number which fails the Goldbach condition must fall in this sequence. Note: none of the given members of the sequence fails Goldbach.

Cf. A279040.

Select[2 Range[400], ! Or @@ PrimeQ[# - {3, 5, 7}] &] (* Giovanni Resta, Mar 10 2017 *)

A205728 Number of odd, nonsquare semiprimes <= n^2.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 5, 8, 11, 16, 19, 24, 28, 32, 41, 46, 50, 60, 66, 73, 81, 89, 100, 110, 118, 126, 140, 151, 163, 174, 187, 197, 210, 224, 239, 253, 269, 286, 298, 312, 326, 344, 363, 380, 399, 414, 435, 451, 468, 491, 513, 530, 546, 567, 591, 619, 643, 664

Offset: 1

Keith Backman, Jan 30 2012

nonn

Like A205727 (see comments thereto), this looks at odd semiprimes, but excludes squares. This then relates to the Goldbach conjecture 2j=p+q with the additional restriction that j, p, and q are not equal.

N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A205726, A205727.

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; nn = 100;  t = Select[Range[1, nn^2, 2], SemiPrimeQ[#] && ! IntegerQ[Sqrt[#]] &]; Table[Length[Select[t, # <= n^2 &]], {n, nn}] (* T. D. Noe, Jan 30 2012 *)

A205726 Number of semiprimes <= n^2.

Original entry on oeis.org

0, 1, 3, 6, 9, 13, 17, 22, 26, 34, 40, 48, 56, 62, 75, 82, 90, 103, 114, 126, 135, 149, 164, 179, 190, 202, 220, 236, 253, 270, 289, 304, 320, 340, 360, 381, 404, 425, 443, 462, 484, 508, 533, 556, 581, 604, 634, 655, 678, 709, 738, 761, 783, 813, 846, 881

Offset: 1

Keith Backman, Jan 30 2012

nonn

See A205727 and A205728 for related sequences and relationship to Goldbach conjecture.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A038108, A205727, A205728.

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; nn = 100;  t = Select[Range[nn^2], SemiPrimeQ]; Table[Length[Select[t, # <= n^2 &]], {n, nn}] (* T. D. Noe, Jan 30 2012 *)
Module[{nn=60,sp},sp=Accumulate[Table[If[PrimeOmega[n]==2,1,0],{n,nn^2}]];Table[sp[[i^2]],{i,nn}]] (* Harvey P. Dale, May 29 2014 *)

from sympy import prime, primepi
def A205726(n): return int(sum(primepi(n**2//prime(k))-k+1 for k in range(1,primepi(n)+1))) # Chai Wah Wu, Jul 23 2024

a(n) = A072000(A000290(n)). - Michel Marcus, Sep 02 2013

A205727 Number of odd semiprimes <= n^2.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 8, 11, 14, 19, 23, 28, 33, 37, 46, 51, 56, 66, 73, 80, 88, 96, 108, 118, 126, 134, 148, 159, 172, 183, 197, 207, 220, 234, 249, 263, 280, 297, 309, 323, 338, 356, 376, 393, 412, 427, 449, 465, 482, 505, 527, 544, 561, 582, 606, 634, 658

Offset: 1

Keith Backman, Jan 30 2012

nonn

The Goldbach conjecture being true would imply that for every integer j, there exists at least one integer k such that (j^2)-(k^2) is an odd semiprime; i.e., for 2j=p+q, j=(p+q)/2 and k=(p-q)/2 results in (j^2)-(k^2)=pq. [Note that in many cases, 2j can be expressed as the sum of more than one set of two primes.] See A205728 for related series where p must be distinct from q.

Cf. A046315, A205726, A205728.

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])];nn = 100;  t = Select[Range[1, nn^2, 2], SemiPrimeQ]; Table[Length[Select[t, # <= n^2 &]], {n, nn}] (* T. D. Noe, Jan 30 2012 *)
With[{osp=Table[{n,PrimeOmega[n]},{n,1,10001,2}]},Table[ Count[ Select[ osp,#[[1]]<=k^2&],?(#[[2]]==2&)],{k,60}]] (* _Harvey P. Dale, Dec 29 2017 *)

a(n) = sum(k=1, n^2, (k%2) && (bigomega(k) == 2)); \\ Michel Marcus, Feb 24 2018

A174840 Least k such that the primes 3 to prime(k+1) form a complete residue system (mod prime(n)).

Original entry on oeis.org

3, 7, 9, 13, 26, 26, 42, 32, 65, 63, 84, 74, 89, 162, 110, 126, 177, 169, 144, 171, 214, 196, 237, 238, 323, 297, 363, 344, 327, 515, 441, 543, 420, 481, 612, 494, 604, 543, 646, 552, 645, 644, 519, 742, 593, 737, 644, 851, 1012, 787, 1204, 727, 899, 800, 1046

Offset: 1

Keith Backman, Mar 30 2010

nonn

If the value of the terminal prime is given rather than its index in the list of odd primes, the sequence becomes 7 19 29 43 103 103 191 137 347 311 439

In other words, the odd primes no larger than 7 form a complete residue set mod 3, the odd primes no larger than 19 form a complete residue set mod 5, and so forth

Table[p=Prime[n]; k=1; While[u=Union[Mod[Prime[Range[2,k]], p]]; u != Range[0,p-1], k++ ]; k-1, {n,2,100}] (* T. D. Noe, Apr 02 2010 *)

Name improved by T. D. Noe, Apr 05 2010

Corrected and extended by T. D. Noe, Apr 02 2010

cp's OEIS Frontend

User: Keith Backman

A346676 Numbers expressible as 2^x + 3^y where both x and y are positive integers.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Python

Formula

A332777 a(n) = k^2 mod p where p is the n-th prime and of the form 6k-1 or 6k+1.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Extensions

A283555 Even numbers that cannot be expressed as p+3, p+5, or p+7, with p prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A205728 Number of odd, nonsquare semiprimes <= n^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A205726 Number of semiprimes <= n^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A205727 Number of odd semiprimes <= n^2.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

A174840 Least k such that the primes 3 to prime(k+1) form a complete residue system (mod prime(n)).

Views

Author

Keywords

Comments

Programs

Mathematica

Extensions