A076274 -id:A076274 - cp's OEIS frontend

A023588 a(n) = sum of exponents in prime-power factorization of 2*prime(n)-1.

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 1, 1, 4, 2, 2, 3, 3, 2, 2, 2, 2, 1, 3, 2, 1, 2, 2, 2, 2, 4, 2, 3, 3, 1, 4, 2, 1, 3, 3, 3, 3, 2, 2, 3, 2, 1, 1, 2, 2, 1, 3, 3, 2, 2, 4, 4, 2, 1, 2, 3, 2, 4, 1, 4, 4, 2, 1, 1, 4, 2, 3, 2, 1, 2, 1, 4, 3, 2, 3, 2, 4, 2, 3, 2, 1, 3, 3, 2, 2, 3, 2, 3, 2, 3, 1, 3, 3, 2, 3

Offset: 1

Clark Kimberling

nonn

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

Array[Plus@@Last/@FactorInteger[2*Prime[ # ]-1]&,6! ] (* Vladimir Joseph Stephan Orlovsky, Feb 28 2010 *)
Table[Total[FactorInteger[2p-1][[;;,2]]],{p,Prime[Range[100]]}] (* Harvey P. Dale, May 30 2024 *)

a(n) = A001222(A076274(n)). - R. J. Mathar, Mar 10 2011

A166257 Odd numbers not of the form prime(k) + phi(prime(k)).

Original entry on oeis.org

1, 7, 11, 15, 17, 19, 23, 27, 29, 31, 35, 39, 41, 43, 47, 49, 51, 53, 55, 59, 63, 65, 67, 69, 71, 75, 77, 79, 83, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 135, 137, 139, 143, 147, 149, 151, 153, 155, 159, 161, 163, 167

Offset: 1

Juri-Stepan Gerasimov, Oct 10 2009

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

Cf. A000010, A000040, A005408.

L := 100; S := {}:
for i from 2 to L do
  for j from 2 to L do
    if i*j <= L then S := `union`(S, {2*i*j-1}) end if;
  end do;
end do:
{1} union S; # Peter Bala, Jan 30 2025

Module[{upto=200},Complement[Range[1,upto,2],Table[n+EulerPhi[n],{n,Prime[ Range[PrimePi[upto]]]}]]] (* Harvey P. Dale, Jun 21 2019 *)

{1} U {A005408 \ A076274 }. - R. J. Mathar, May 21 2010

Entries checked by R. J. Mathar, May 21 2010.

A195896 Numbers of the form 2p-1 or 3p-1 where p is 1 or a prime.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 13, 14, 20, 21, 25, 32, 33, 37, 38, 45, 50, 56, 57, 61, 68, 73, 81, 85, 86, 92, 93, 105, 110, 117, 121, 122, 128, 133, 140, 141, 145, 157, 158, 165, 176, 177, 182, 193, 200, 201, 205, 212, 213, 217, 218, 225, 236, 248, 253, 261, 266, 273, 277, 290, 297, 301, 302, 308

Offset: 1

Juri-Stepan Gerasimov, Sep 24 2011

nonn

			a(1)=1 because p=1 and 2*1 - 1 = 1;
a(2)=2 because p=1 and 3*1 - 1 = 2;
a(3)=3 because p=2 and 2*2 - 1 = 3;
a(4)=5 because p=2 and 3*3 - 1 = 5 or p=3 and p=2 and 3*2 - 1 = 5;
a(5)=8 because p=3 and 3*3 - 1 = 8.

Cf. A196127, A008578.

isA195896 := proc(n)
        for p in {(n+1)/2,(n+1)/3} do
        if type(p,'integer') then
                if isprime(p) or p = 1 then
                        return true;
                end if;
        end if;
        end do;
        false ;
end proc:
for n from 1 to 400 do
        if isA195896(n) then
                printf("%d,",n) ;
        end if;
end do: # R. J. Mathar, Oct 15 2011

Union[Flatten[Join[{1,2},{2#-1,3#-1}&/@Prime[Range[50]]]]] (* Harvey P. Dale, Mar 27 2015 *)

Union of A076274 and A112773.

A253242 Least k>=0 such that n^(2^k)+1 is prime (for even n), or (n^(2^k)+1)/2 is prime (for odd n); -1 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, -1, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 2, 1, 0, 1, -1, 0, 1, 0

Offset: 2

Eric Chen, Apr 19 2015

Least k such that the generalized Fermat number in base n (GFN(k,n)) is prime.
a(n) = -1 if n is in A070265 (perfect powers with an odd exponent).
a(n) is currently unknown for n = {31, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 97, 98, 99, 104, 107, 109, 122, 123, 127, 135, 137, ...}
Corresponding primes are {3, 2, 5, 3, 7, 1201, 0, 5, 11, 61, 13, 7, 197, 113, 17, 41761, 19, 181, 401, 11, 23, 139921, 577, 13, 677, 0, 29, 421, 31, ...}. (use 0 if a(n) = -1)
All 2 <= n <= 1500 and 0 <= k <= 14 are checked, the first occurrence of k (start with k = 0) in a(n) are {2, 11, 7, 43, 41, 75, 274, 234, 331, 1342, 824, ...}.

			a(7) = 2 since (7^(2^0)+1)/2 and (7^(2^1)+1)/2 are not primes, but (7^(2^2)+1)/2 = 1201 is prime.
a(14) = 1 since 14^(2^0)+1 is not prime, but 14^(2^1)+1 = 197 is prime.

Eric Chen, Table of n, a(n) for n = 2..1500 status (for the -1s, only a(n) for n in A070265 are proved, all other -1s are only conjectured)
Gary Barnes, List of generalized Fermat primes in even bases up to 1030
Eric Chen, List of generalized Fermat primes in bases up to 1000
Chris Caldwell, Generalized Fermat number
Richard Fischer, List of generalized Fermat primes in odd bases
Yves Gallot, Generalized Fermat prime search
Wilfrid Keller, Factorization of GFN(n,2)
Wilfrid Keller, Factorization of GFN(n,3)
Wilfrid Keller, Factorization of GFN(n,5)
Wilfrid Keller, Factorization of GFN(n,6)
Wilfrid Keller, Factorization of GFN(n,10)
Wilfrid Keller, Factorization of GFN(n,12)
Jeppe Stig Salling Nielsen, List of generalized Fermat primes in even bases up to 1000
MathWorld, Generalized Fermat number
OEIS wiki, Generalized Fermat number
Wikipedia, Generalized Fermat number

Cf. A079706, A228101, A084712, A123669, A058064, A057856, A130536, A080121, A077659, A122900.

Table[k=0; While[p=If[EvenQ[n], (2n)^(2^k)+1, ((2n)^(2^k)+1)/2]; k<12 && !PrimeQ[p], k=k+1]; If[k==12, -1, k], {n, 2, 1500}]

f(n) = for(k=0, 11, if(ispseudoprime(n^(2^k)+1), return(k))); -1
g(n) = for(k=0, 11, if(ispseudoprime((n^(2^k)+1)/2), return(k))); -1
a(n) = if(n%2==0, f(n), g(n))

f(n,k)=if(n%2, (n^(2^k)+1)/2, n^(2^k)+1)
a(n)=if(ispower(-n), -1, my(k); while(!ispseudoprime(f(n,k)), k++); k) \\ Charles R Greathouse IV, Apr 20 2015

a(2n) = A228101(n) = log_2(A079706(n)).

a(A006093(n)) = 0, a(A076274(n)) = 0, a(A070265(n)) = -1.

A330508 Numbers k such that k + 6^t is semiprime for t = 0 to 9.

Original entry on oeis.org

61273, 109441, 160213, 274501, 275473, 311593, 360673, 394201, 477181, 486061, 514993, 522085, 617137, 620053, 715477, 725485, 803833, 812677, 847117, 1063585, 1146913, 1182577, 1215865, 1232917, 1409425, 1508113, 1587241, 1768993, 1863073, 1895413, 2085517, 2095177

Offset: 1

K. D. Bajpai, Dec 16 2019

nonn

a(2620) = 530079693 is the first multiple of 3 in this sequence; there are no multiples of 2. - Charles R Greathouse IV, Dec 20 2019

			a(1) = 61273:
  61273 + 6^0  =    61274 =   2 *  30637;
  61273 + 6^1  =    61279 = 233 *    263;
  61273 + 6^2  =    61309 =  37 *   1657;
  61273 + 6^3  =    61489 =  17 *   3617;
  61273 + 6^4  =    62569 =  13 *   4813;
  61273 + 6^5  =    69049 =  29 *   2381;
  61273 + 6^6  =   107929 =  37 *   2917;
  61273 + 6^7  =   341209 =  11 *  31019;
  61273 + 6^8  =  1740889 = 197 *   8837;
  61273 + 6^9  = 10138969 =  89 * 113921;
all ten results are semiprime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A076274.

Cf. A001358, A082919, A096173, A104238, A105041.

f:=func; [k:k in [1..2100000]|forall{m:m in [0..9]|f(k+6^m)}]; // Marius A. Burtea, Dec 20 2019

fX[n_] = PrimeOmega[n] == 2; Select[Range[2000000], AllTrue[# + 6^{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, fX] &]

issemi(n)=bigomega(n)==2
is(n)=for(t=0,9, if(!issemi(n+6^t), return(0))); 1 \\ Charles R Greathouse IV, Dec 20 2019

A369493 Table read by rows: row n is the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = prime(n) and the short leg "a" is odd.

Original entry on oeis.org

3, 4, 5, 5, 12, 13, 9, 40, 41, 13, 84, 85, 21, 220, 221, 25, 312, 313, 33, 544, 545, 37, 684, 685, 45, 1012, 1013, 57, 1624, 1625, 61, 1860, 1861, 73, 2664, 2665, 81, 3280, 3281, 85, 3612, 3613, 93, 4324, 4325, 105, 5512, 5513, 117, 6844, 6845, 121, 7320, 7321, 133, 8844, 8845, 141, 9940, 9941

Offset: 1

Miguel-Ángel Pérez García-Ortega, Jan 24 2024

See Exercise 3.5. of the reference.

			Table begins:
  n=1:   3,   4,   5;
  n=2:   5,  12,  13;
  n=3:   9,  40,  41;
  n=4:  13,  84,  85;
  n=5:  21, 220, 221;

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.

Miguel-Ángel Pérez García-Ortega, Ejercicio 3.5.

Cf. A000040, A076274 (short leg), A006093 (inradius).

Row n = (a, b, c) = (2*p - 1, 2*p^2 - 2*p, 2*p^2 - 2*p + 1), where p = prime(n) = A000040(n).

A178623 Triangle T(n,m) read by rows: T(n,0)= prime(n); T(n,m)=1 if m>=1.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29

Offset: 0

Paul Curtz, May 31 2010

The sequence reflects a conjecture on the denominator of inverse Bernoulli polynomials in A178340: if the row index is one less than one of the primes in A008578, the row of denominators starts with that prime and contains 1's in the remaining entries.

[Row sums in A178252 are A159069(n+1), unless there is a common factor in numerator and denominator. The row sum over columns with index of the same parity as the row index in the table of fractions of the [x^m] B^{-1}(n,x) in A178252 are: 1, 1, 1/3+1=4/3, 1+1=2, 1/5+2+1=16/5, 1+10/3+1=16/3, 1/7+3+5+1=64/7, 16, 256/9, 256/5, 1024/11, 512/3, 496/13, ... =A084623(n+1)/A000265(n+1).]

			1;
2,1;
3,1,1;
5,1,1,1,1;
7,1,1,1,1,1,1;
11,1,1,1,1,1,1,1,1,1,1;
13,1,1,1,1,1,1,1,1,1,1,1,1;
17,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
19,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
23,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
29,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;

Cf. A076274 (row sums).

T(n,0) = A008578(n+1). T(n,m) =1, 1<=m<=A008578(n+1)-1.

A211979 Numbers n formed by p 1's followed by p - 1 0's, where p is prime(n).

Original entry on oeis.org

110, 11100, 111110000, 1111111000000, 111111111110000000000, 1111111111111000000000000, 111111111111111110000000000000000, 1111111111111111111000000000000000000, 111111111111111111111110000000000000000000000

Offset: 1

Omar E. Pol, Dec 12 2012

			For n = 3, the third prime is 5, so a(3) = 111110000 (five 1's followed by four 0's).

Binary representation of A060286.
a(n) has A076274(n) digits.
Cf. A007088, A076274, A109241, A135650.

(* Technically this is in base 10 *) Table[10^(Prime[n] - 1)((10^Prime[n] - 1)/9), {n, 20}] (* Alonso del Arte, Dec 12 2012 *)
FromDigits[Join[PadRight[{},#,1],PadRight[{},#-1,0]]]&/@ Prime[ Range[ 10]] (* Harvey P. Dale, Aug 30 2015 *)

cp's OEIS Frontend

A023588 a(n) = sum of exponents in prime-power factorization of 2*prime(n)-1.

Views

Author

Keywords

Links

Programs

Mathematica

Formula

A166257 Odd numbers not of the form prime(k) + phi(prime(k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A195896 Numbers of the form 2*p-1 or 3*p-1 where p is 1 or a prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A253242 Least k>=0 such that n^(2^k)+1 is prime (for even n), or (n^(2^k)+1)/2 is prime (for odd n); -1 if no such k exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A330508 Numbers k such that k + 6^t is semiprime for t = 0 to 9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A369493 Table read by rows: row n is the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = prime(n) and the short leg "a" is odd.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A178623 Triangle T(n,m) read by rows: T(n,0)= prime(n); T(n,m)=1 if m>=1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A211979 Numbers n formed by p 1's followed by p - 1 0's, where p is prime(n).

Views

Author

Keywords

Examples

Crossrefs

Programs

A195896 Numbers of the form 2p-1 or 3p-1 where p is 1 or a prime.