A113935 -id:A113935 - cp's OEIS frontend

A098090 Numbers k such that 2k-3 is prime.

3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 20, 22, 23, 25, 28, 31, 32, 35, 37, 38, 41, 43, 46, 50, 52, 53, 55, 56, 58, 65, 67, 70, 71, 76, 77, 80, 83, 85, 88, 91, 92, 97, 98, 100, 101, 107, 113, 115, 116, 118, 121, 122, 127, 130, 133, 136, 137, 140, 142, 143, 148, 155, 157, 158

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 14 2004

Supersequence of A063908.

Left edge of the triangle in A065305. - Reinhard Zumkeller, Jan 30 2012

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A085090, A086801.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), this sequence (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([1..200], k-> IsPrime(2*k-3) ); # G. C. Greubel, May 21 2019

[ n: n in [1..200] | IsPrime(2*n-3) ]; // Vincenzo Librandi, Dec 26 2010

(Prime[Range[2,100]]+3)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[200],PrimeQ[2#-3]&] (* Harvey P. Dale, Mar 05 2022 *)

is(n)=isprime(2*n-3) \\ Charles R Greathouse IV, Feb 17 2017

[n for n in (1..200) if is_prime(2*n-3) ] # G. C. Greubel, May 21 2019

Half of p + 3, where p is a prime greater than 2.

A122845(a(n), 3) = 3; a(n) = A113935(n+1)/2. - Reinhard Zumkeller, Sep 14 2006

A216624 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} gcd(c,d) for n>=1, k>=1.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 3, 4, 4, 3, 2, 8, 6, 8, 2, 4, 4, 6, 6, 4, 4, 2, 10, 4, 15, 4, 10, 2, 4, 4, 12, 6, 6, 12, 4, 4, 3, 11, 4, 16, 8, 16, 4, 11, 3, 4, 6, 8, 6, 8, 8, 6, 8, 6, 4, 2, 10, 10, 22, 4, 30, 4, 22, 10, 10, 2, 6, 4, 8, 9, 8, 8, 8, 8, 9, 8, 4, 6

Offset: 1

Peter Luschny, Sep 12 2012

T(n,k) = number of subgroups of C_n X C_k. [Hampjes et al.] - N. J. A. Sloane, Feb 02 2013

			[----1---2---3---4---5---6---7---8---9--10--11--12]
[ 1] 1,  2,  2,  3,  2,  4,  2,  4,  3,  4,  2,  6
[ 2] 2,  5,  4,  8,  4, 10,  4, 11,  6, 10,  4, 16
[ 3] 2,  4,  6,  6,  4, 12,  4,  8, 10,  8,  4, 18
[ 4] 3,  8,  6, 15,  6, 16,  6, 22,  9, 16,  6, 30
[ 5] 2,  4,  4,  6,  8,  8,  4,  8,  6, 16,  4, 12
[ 6] 4, 10, 12, 16,  8, 30,  8, 22, 20, 20,  8, 48
[ 7] 2,  4,  4,  6,  4,  8, 10,  8,  6,  8,  4, 12
[ 8] 4, 11,  8, 22,  8, 22,  8, 37, 12, 22,  8, 44
[ 9] 3,  6, 10,  9,  6, 20,  6, 12, 23, 12,  6, 30
[10] 4, 10,  8, 16, 16, 20,  8, 22, 12, 40,  8, 32
[11] 2,  4,  4,  6,  4,  8,  4,  8,  6,  8, 14, 12
[12] 6, 16, 18, 30, 12, 48, 12, 44, 30, 32, 12, 90
.
Displayed as a triangular array:
1,
2,  2,
2,  5,  2,
3,  4,  4,  3,
2,  8,  6,  8, 2,
4,  4,  6,  6, 4,  4,
2, 10,  4, 15, 4, 10, 2,
4,  4, 12,  6, 6, 12, 4,  4,
3, 11,  4, 16, 8, 16, 4, 11, 3,

Alois P. Heinz, Antidiagonals n = 1..141, flattened
M. Hampejs, N. Holighaus, L. Toth and C. Wiesmeyr, On the subgroups of the group Z_m X Z_n, 2012. - From _N. J. A. Sloane_, Feb 02 2013

Cf. A216620, A216621, A216622, A216623, A216625, A216626, A216627.
Main diagonal is A060724.
Rows give A000005, A060710, A054584, A221951, A221852.

with(numtheory):
T:= (n, k)-> add(add(igcd(c,d), c=divisors(n)), d=divisors(k)):
seq(seq(T(n, 1+d-n), n=1..d), d=1..14); # Alois P. Heinz, Sep 12 2012
T:=proc(m,n) local d; add( d*tau(m*n/d^2), d in divisors(gcd(m,n))); end; # N. J. A. Sloane, Feb 02 2013

t[n_, k_] := Sum[Sum[GCD[c, d], {c, Divisors[n]}], {d, Divisors[k]}]; Table[t[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 21 2013 *)

def A216624(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(gcd, cp))
for n in (1..12): [A216624(n,k) for k in (1..12)]

T(n,n) = A060724(n) = sum_{d|n} d*tau((n/d)^2).
T(n,1) = T(1,n) = A000005(n) = tau(n).
T(n,2) = T(2,n) = A060710(n) = sum_{d|n} (3-[d is odd]) (Iverson bracket).
T(n+1,n) = A092517(n) = tau(n+1)*tau(n).
T(prime(n),1) = A007395(n) = 2.
T(prime(n),prime(n)) = A113935(n) = prime(n)+3.

A175222 a(n) = prime(n) + 5.

Original entry on oeis.org

7, 8, 10, 12, 16, 18, 22, 24, 28, 34, 36, 42, 46, 48, 52, 58, 64, 66, 72, 76, 78, 84, 88, 94, 102, 106, 108, 112, 114, 118, 132, 136, 142, 144, 154, 156, 162, 168, 172, 178, 184, 186, 196, 198, 202, 204, 216, 228, 232, 234, 238, 244, 246, 256, 262, 268, 274, 276

Offset: 1

Jaroslav Krizek, Mar 06 2010

a(n) = A000040(n) + 5 = A008864(n) + 4 = A052147(n) + 3 = A113395(n) + 2 = A175221 (n) + 1 = A139049(n) - 1 = A175223(n) - 2 = A175224(n) - 3 = A140353(n) - 4 = A175225(n) - 5.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A008864, A052147, A086801, A113395, A113935, A139049, A140353, A175221, A175223, A175224, A175225.

Filtered([1..300], k-> IsPrime(k) ) + 5; # G. C. Greubel, May 20 2019

[p+5: p in PrimesUpTo(500)]; // Vincenzo Librandi, Dec 04 2010

Prime[Range[120]]+5 (* Vladimir Joseph Stephan Orlovsky, Nov 21 2010 *)

{a(n) = prime(n) + 5}; \\ G. C. Greubel, May 20 2019

[nth_prime(n) + 5 for n in (1..100)] # G. C. Greubel, May 20 2019

More terms from Vincenzo Librandi, Mar 14 2010

A116366 Triangle read by rows: even numbers as sums of two odd primes.

Original entry on oeis.org

6, 8, 10, 10, 12, 14, 14, 16, 18, 22, 16, 18, 20, 24, 26, 20, 22, 24, 28, 30, 34, 22, 24, 26, 30, 32, 36, 38, 26, 28, 30, 34, 36, 40, 42, 46, 32, 34, 36, 40, 42, 46, 48, 52, 58, 34, 36, 38, 42, 44, 48, 50, 54, 60, 62, 40, 42, 44, 48, 50, 54, 56, 60, 66, 68, 74, 44, 46, 48, 52, 54, 58, 60, 64, 70, 72, 78, 82

Offset: 1

Reinhard Zumkeller, Feb 06 2006

T(n,k) = 2*A065305(n,k) = A065342(n+1,k+1);
Row sums give A116367; central terms give A116368;
T(n,1) = A113935(n+1);
T(n,n-2) = A048448(n) for n>2;
T(n,n-1) = A001043(n) for n>1;
T(n,n) = A001747(n+2) = A100484(n+1).

			Triangle begins:
  6;
  8,  10;
  10, 12, 14;
  14, 16, 18, 22;
  16, 18, 20, 24, 26;
  20, 22, 24, 28, 30, 34;
  22, 24, 26, 30, 32, 36, 38;
  26, 28, 30, 34, 36, 40, 42, 46;
  32, 34, 36, 40, 42, 46, 48, 52, 58;
  34, 36, 38, 42, 44, 48, 50, 54, 60, 62;
  40, 42, 44, 48, 50, 54, 56, 60, 66, 68, 74;
  44, 46, 48, 52, 54, 58, 60, 64, 70, 72, 78, 82; etc. - _Bruno Berselli_, Aug 16 2013

G. C. Greubel, Rows n = 1..100 of triangle, flattened
Index entries for sequences related to Goldbach conjecture

Cf. A001043, A001747, A048448, A065305, A065342, A100484, A113935, A116367, A116368.

[NthPrime(n+1)+NthPrime(k+1): k in [1..n], n in [1..15]]; // Bruno Berselli, Aug 16 2013

Table[Prime[n+1] + Prime[k+1], {n,1,12}, {k,1,n}]//Flatten (* G. C. Greubel, May 12 2019 *)

{T(n,k) = prime(n+1) + prime(k+1)}; \\ G. C. Greubel, May 12 2019

[[nth_prime(n+1) + nth_prime(k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 12 2019

T(n,k) = prime(n+1) + prime(k+1), 1 <= k <= n.

A298252 Even integers n such that n-3 is prime.

Original entry on oeis.org

6, 8, 10, 14, 16, 20, 22, 26, 32, 34, 40, 44, 46, 50, 56, 62, 64, 70, 74, 76, 82, 86, 92, 100, 104, 106, 110, 112, 116, 130, 134, 140, 142, 152, 154, 160, 166, 170, 176, 182, 184, 194, 196, 200, 202, 214, 226, 230, 232, 236, 242, 244, 254, 260, 266, 272, 274, 280

Offset: 1

David James Sycamore, Jan 15 2018

Subsequence of A005843, same as A113935 with first term (5) excluded, since it is odd, not even. Index in A056240 of terms in A288313 (except for first two terms 2,4 of latter).
The terms in this sequence, combined with those in A297925 and A298366 form a partition of A005843(n); n>=3 (nonnegative numbers>=6). This is because any even integer n>=6 satisfies either(i) n-3 is prime, (ii) n-5 prime but n-3 composite, or (iii) n-5 and n-3 both composite.
a(n) is the smallest even number e > prime(n+1) such that e has a Goldbach partition containing prime(n+1). - Felix Fröhlich, Aug 18 2019

			a(1)=6 because 6-3=3; prime, and no smaller even number has this property; also a(1)=A113935(2)=6.  a(2)=8 because 8-3=5 is prime; also A113935(3)=8.
12 is not in the sequence because 12-3 = 9, composite.

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Cf. A005843, A056240, A113935, A288313.

Filtered([1..300],n->IsEvenInt(n) and IsPrime(n-3)); # Muniru A Asiru, Mar 23 2018

[NthPrime(n+1) +3: n in [1..70]]; // G. C. Greubel, May 21 2019

N:=200
  for n from 6 to N by 2 do
if isprime(n-3) then print(n);
end if
end do

Select[2 Range@125, PrimeQ[# - 3] &] (* Robert G. Wilson v, Jan 15 2018 *)
Select[Prime[Range[100]]+3,EvenQ] (* Harvey P. Dale, Mar 07 2022 *)

a(n) = prime(n + 1) + 3 \\ David A. Corneth, Mar 23 2018

[nth_prime(n+1) +3 for n in (1..70)] # G. C. Greubel, May 21 2019

a(n) = A113935(n+1), n>=1.
A056240(a(n)) = A288313(n+2).
a(n) = prime(n + 1) + 3 = A113935(n + 1). - David A. Corneth, Mar 23 2018

A023576 Greatest prime divisor of prime(n)+3.

Original entry on oeis.org

5, 3, 2, 5, 7, 2, 5, 11, 13, 2, 17, 5, 11, 23, 5, 7, 31, 2, 7, 37, 19, 41, 43, 23, 5, 13, 53, 11, 7, 29, 13, 67, 7, 71, 19, 11, 5, 83, 17, 11, 13, 23, 97, 7, 5, 101, 107, 113, 23, 29, 59, 11, 61, 127, 13, 19, 17, 137, 7, 71, 13, 37, 31, 157, 79, 5, 167, 17, 7, 11, 89

Offset: 1

Clark Kimberling

nonn

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

Cf. A006530, A113935.

FactorInteger[#][[-1, 1]] & /@ (Prime[Range[100]] + 3) (* Harvey P. Dale, Sep 05 2014 *)

from sympy import primefactors, prime
def a(n): return primefactors(prime(n) + 3)[-1]
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, May 03 2021

a(n) = A006530(A113935(n)). - Michel Marcus, May 04 2021

A114557 a(2n-1) = 2*(p-1) and a(2n) = p + 3, where p=prime(n).

Original entry on oeis.org

2, 5, 4, 6, 8, 8, 12, 10, 20, 14, 24, 16, 32, 20, 36, 22, 44, 26, 56, 32, 60, 34, 72, 40, 80, 44, 84, 46, 92, 50, 104, 56, 116, 62, 120, 64, 132, 70, 140, 74, 144, 76, 156, 82, 164, 86, 176, 92, 192, 100, 200, 104, 204, 106, 212, 110, 216, 112, 224, 116, 252, 130, 260, 134

Offset: 1

Roger L. Bagula, Feb 15 2006

G. C. Greubel, Table of n, a(n) for n = 1..10000

[((3-(-1)^n)*NthPrime(Floor((n+1)/2)) + (1+5*(-1)^n))/2: n in [1..70]]; // G. C. Greubel, May 20 2019

Flatten[Table[Abs[Coefficient[Expand[(x+2)(x -(1 +Sqrt[Prime[n]]))*(x - (1 - Sqrt[Prime[n]]))], x, m]], {n, 1, 50}, {m, 0, 1}]]
With[{p = Prime[Floor[(n+1)/2]]}, Table[If[OddQ[n], 2*(p-1), p+3], {n, 1, 70}]] (* G. C. Greubel, May 20 2019 *)

{a(n) = ((3-(-1)^n)*prime(floor((n+1)/2)) + (1+5*(-1)^n))/2}; \\ G. C. Greubel, May 20 2019

[( (3-(-1)^n)*nth_prime(floor((n+1)/2))+ (1+5*(-1)^n))/2 for n in (1..70)] # G. C. Greubel, May 20 2019

a(2n-1) = A037168(n). a(2n) = A113935(n).

a(n) = ( (3 - (-1)^n)*prime(floor((n+1)/2)) + (1 + 5*(-1)^n) )/2. - G. C. Greubel, May 20 2019

A173064 a(n) = prime(n) - 5.

Original entry on oeis.org

0, 2, 6, 8, 12, 14, 18, 24, 26, 32, 36, 38, 42, 48, 54, 56, 62, 66, 68, 74, 78, 84, 92, 96, 98, 102, 104, 108, 122, 126, 132, 134, 144, 146, 152, 158, 162, 168, 174, 176, 186, 188, 192, 194, 206, 218, 222, 224, 228, 234, 236, 246, 252, 258, 264, 266, 272, 276, 278, 288, 302, 306, 308, 312, 326, 332, 342, 344, 348, 354, 362, 368, 374, 378, 384, 392, 396, 404

Offset: 3

Vladimir Joseph Stephan Orlovsky, Nov 21 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 3..10000

Cf. A086801, A113935, A175222.

[NthPrime(n) - 5: n in [3..100]]; // G. C. Greubel, May 19 2019

Mathematica
```
Prime[Range[3,120]] - 5
```

{a(n) = prime(n) - 5}; \\ G. C. Greubel, May 19 2019

[nth_prime(n) - 5 for n in (3..100)] # G. C. Greubel, May 19 2019

A023578 Least odd prime divisor of prime(n)+3, or 1 if prime(n)+3 is a power of 2.

Original entry on oeis.org

5, 3, 1, 5, 7, 1, 5, 11, 13, 1, 17, 5, 11, 23, 5, 7, 31, 1, 5, 37, 19, 41, 43, 23, 5, 13, 53, 5, 7, 29, 5, 67, 5, 71, 19, 7, 5, 83, 5, 11, 7, 23, 97, 7, 5, 101, 107, 113, 5, 29, 59, 11, 61, 127, 5, 7, 17, 137, 5, 71, 11, 37, 5, 157, 79, 5, 167, 5, 5, 11, 89, 181, 5, 47, 191

Offset: 1

Clark Kimberling

nonn

Cf. A023505, A078701, A113935.

a(n) = my(p = prime(n)+3, v = p/(2^valuation(p, 2))) ; if (v == 1, 1, factor(v)[1, 1]); \\ Michel Marcus, Aug 05 2021

from sympy import factorint, prime
def A023578(n): return min((p for p in factorint(prime(n)+3) if p > 2), default=1) # Chai Wah Wu, Feb 03 2022

a(n) = A078701(A113935(n)). - Michel Marcus, Aug 05 2021

A023579 Exponent of 2 in prime factorization of prime(n)+3.

Original entry on oeis.org

0, 1, 3, 1, 1, 4, 2, 1, 1, 5, 1, 3, 2, 1, 1, 3, 1, 6, 1, 1, 2, 1, 1, 2, 2, 3, 1, 1, 4, 2, 1, 1, 2, 1, 3, 1, 5, 1, 1, 4, 1, 3, 1, 2, 3, 1, 1, 1, 1, 3, 2, 1, 2, 1, 2, 1, 4, 1, 3, 2, 1, 3, 1, 1, 2, 6, 1, 2, 1, 5, 2, 1, 1, 3, 1, 1, 3, 4, 2, 2, 1, 3, 1, 2, 1, 1, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 9, 2, 1

Offset: 1

Clark Kimberling

nonn

			For n=1, prime(1)+3=5, and 5 is odd, so a(1)=0.

G. C. Greubel, Table of n, a(n) for n = 1..10000

[Valuation(3+NthPrime(n), 2): n in [1..100]]; // G. C. Greubel, May 21 2019

Table[IntegerExponent[3 + Prime[n], 2], {n, 100}] (* G. C. Greubel, May 21 2019 *)

a(n) = valuation(prime(n)+3, 2); \\ Michel Marcus, Sep 30 2013

from sympy import prime
def A023579(n): return (~(m:=prime(n)+3)& m-1).bit_length() # Chai Wah Wu, Jul 07 2022

[(nth_prime(n)+3).valuation(2) for n in (1..100)] # G. C. Greubel, May 21 2019

a(n) = A007814(A113935(n)). - Michel Marcus, Sep 30 2013

a(1) corrected by Michel Marcus, Sep 30 2013

Name corrected by Charles R Greathouse IV, Mar 31 2014

cp's OEIS Frontend

A098090 Numbers k such that 2k-3 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A216624 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} gcd(c,d) for n>=1, k>=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A175222 a(n) = prime(n) + 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Extensions

A116366 Triangle read by rows: even numbers as sums of two odd primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A298252 Even integers n such that n-3 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A023576 Greatest prime divisor of prime(n)+3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica