A061397 -id:A061397 - cp's OEIS frontend

A143536 Triangle read by rows, T(n,k) = 1 if n is prime, 0 otherwise.

0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Aug 23 2008

Row sums = A061397: (0, 2, 3, 0, 5, 0, 7,...).

			First few rows of the triangle =
0;
1, 1;
1, 1, 1;
0, 0, 0, 0;
1, 1, 1, 1, 1;
0, 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1;
...

Reinhard Zumkeller, Rows n = 1..125 of table, flattened

Cf. A143535, A061397.

Cf. A010051.

a143536 n k = a143536_tabl !! (n-1) !! (k-1)
a143536_row n = a143536_tabl !! (n-1)
a143536_tabl = zipWith take [1..] $ map repeat a010051_list
-- Reinhard Zumkeller, Mar 21 2014

Triangle read by rows, T(n,k) = 1 if n is prime, 0 otherwise. Mobius transform (A054525) of triangle A143535.

A143545 a(n) = n unless n is a prime, in which case a(n) = 2n.

Original entry on oeis.org

1, 4, 6, 4, 10, 6, 14, 8, 9, 10, 22, 12, 26, 14, 15, 16, 34, 18, 38, 20, 21, 22, 46, 24, 25, 26, 27, 28, 58, 30, 62, 32, 33, 34, 35, 36, 74, 38, 39, 40, 82, 42, 86, 44, 45, 46, 94, 48, 49, 50, 51, 52, 106, 54, 55, 56, 57, 58, 118, 60, 122, 62, 63, 64, 65, 66, 134, 68, 69, 70, 142

Offset: 1

Gary W. Adamson, Aug 23 2008

nonn

			a(3) = 6 = 2*3 since 3 is prime.
a(4) = 4 since 4 is a nonprime.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A066560, A143544, A061397.

See A282028 for another version.

a(n) = n + A061397(n), where A061397 = (0, 2, 3, 0, 5, 0, 7,...).
Equals row sums of triangle A143544.
a(n)=A066560(n), n>1. [From R. J. Mathar, Sep 05 2008]

Extended beyond a(14) by R. J. Mathar, Sep 05 2008

A347104 Dirichlet g.f.: primezeta(s-1) * zeta(s-1) / zeta(s).

Original entry on oeis.org

0, 2, 3, 2, 5, 7, 7, 4, 6, 13, 11, 10, 13, 19, 22, 8, 17, 18, 19, 18, 32, 31, 23, 20, 20, 37, 18, 26, 29, 38, 31, 16, 52, 49, 58, 24, 37, 55, 62, 36, 41, 56, 43, 42, 54, 67, 47, 40, 42, 60, 82, 50, 53, 54, 94, 52, 92, 85, 59, 60, 61, 91, 78, 32, 112, 92, 67, 66, 112, 106, 71, 48, 73, 109, 100

Offset: 1

Ilya Gutkovskiy, Aug 18 2021

nonn

a(n) is the sum of the prime terms in row n of A050873.

Moebius transform of A328260.

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

Cf. A000010, A001221, A008472, A010051, A018804, A050873, A061397, A078439, A117494, A122411, A255655, A328260, A329354.

Table[DivisorSum[n, MoebiusMu[n/#] # PrimeNu[#] &], {n, 1, 75}]
Table[DivisorSum[n, # EulerPhi[n/#] &, PrimeQ[#] &], {n, 1, 75}]
Table[Sum[Boole[PrimeQ[GCD[n, k]]] GCD[n, k], {k, 1, n}], {n, 1, 75}]

a(n) = sumdiv(n, d, moebius(n/d)*d*omega(d)); \\ Michel Marcus, Aug 18 2021

a(n) = Sum_{d|n} mu(n/d) * d * omega(d).
a(n) = Sum_{p|n, p prime} p * phi(n/p).
a(n) = Sum_{k=1..n} A010051(gcd(n,k)) * gcd(n,k).

A143544 Triangle read by rows, T(n,k) = 2 if n is prime, 1 otherwise; 1<=k<=n.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Gary W. Adamson, Aug 23 2008

Row sums = A143545: (1, 4, 6, 4, 10, 6, 14,...) = componentwise addition of (1, 2, 3, 4, 5,...) and A061397: (0, 2, 3, 0, 5, 0, 7,...).

			First few rows of the triangle =
1;
2, 2;
2, 2, 2;
1, 1, 1, 1;
2, 2, 2, 2, 2;
1, 1, 1, 1, 1, 1;
2, 2, 2, 2, 2, 2, 2;
...

Michael De Vlieger, Table of n, a(n) for n = 1..11325

Cf. A010051, A061397, A143536, A143545.

Table[1 + Boole[PrimeQ[n]], {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Oct 31 2021 *)

Triangle read by rows, T(n,k) = 2 if n is prime, 1 otherwise; 1<=k<=n.

T(n, k) = A143536(n, k) + 1. - Georg Fischer, Oct 31 2021

A193315 Write 2n=j+q (j,q positive noncomposite numbers); jq maximal; then a(n)=jq.

Original entry on oeis.org

1, 4, 9, 15, 25, 35, 49, 55, 77, 91, 121, 143, 169, 187, 221, 247, 289, 323, 361, 391, 437, 403, 529, 551, 589, 667, 713, 703, 841, 899, 961, 943, 1073, 1147, 1189, 1271, 1369, 1363, 1517, 1591, 1681, 1763, 1849, 1927, 2021, 1891, 2209, 2279, 2257, 2491, 2537, 2623, 2809, 2867, 2881

Offset: 1

Juri-Stepan Gerasimov, Aug 26 2011

nonn

a(n) = A102084(n) for n > 0. [Reinhard Zumkeller, Aug 28 2011]

			At n=6, 2n=12; 12 = 1 + 11 = 7 + 5; 7*5 = maximal => j*q = 7*5 = 35.

Cf. A073046, A008578, A061397.

a193315 1 = 1
a193315 n = maximum $ zipWith (*) prims $ map (a061397 . (2*n -)) prims
   where prims = takeWhile (<= n) a008578_list
-- Reinhard Zumkeller, Aug 28 2011

isA008578 := proc(n) if n = 1 then true ; elif isprime(n) then true; else false; end if; end proc:
A193315 := proc(n) local mx,j,q ; mx := 0 ; for j from 1 to 2*n-1 do if isA008578(j) then q := 2*n-j ; if isA008578(q) then mx := max(mx,j*q) ; end if ; end if; end do: mx ; end proc:
seq(A193315(n),n=1..60) ; # R. J. Mathar, Aug 28 2011

def is_A008578(n): return n == 1 or is_prime(n)
def A193315(n): return max((j*(2*n-j)) for j in [1]+prime_range(n+1) if is_A008578(2*n-j))
[A193315(i) for i in range(1,15)]
# D. S. McNeil, Aug 27 2011

A349712 a(n) = Sum_{d|n} sopf(d) * sopf(n/d).

Original entry on oeis.org

0, 0, 0, 4, 0, 12, 0, 8, 9, 20, 0, 32, 0, 28, 30, 12, 0, 42, 0, 48, 42, 44, 0, 52, 25, 52, 18, 64, 0, 124, 0, 16, 66, 68, 70, 87, 0, 76, 78, 76, 0, 164, 0, 96, 78, 92, 0, 72, 49, 90, 102, 112, 0, 72, 110, 100, 114, 116, 0, 234, 0, 124, 102, 20, 130, 244, 0, 144, 138, 236, 0, 132, 0, 148, 110

Offset: 1

Ilya Gutkovskiy, Nov 26 2021

nonn

Dirichlet convolution of A008472 with itself.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A008472, A034761, A061397, A070288, A319131, A349711.

sopf[n_] := DivisorSum[n, # &, PrimeQ[#] &]; a[n_] := Sum[sopf[d] sopf[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 75}]

sopf(n) = vecsum(factor(n)[, 1]); \\ A008472
a(n) = sumdiv(n, d, sopf(d)*sopf(n/d)); \\ Michel Marcus, Nov 26 2021

from sympy import divisors, factorint
def sopf(n): return sum(factorint(n))
def a(n): return sum(sopf(d)*sopf(n//d) for d in divisors(n))
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Nov 26 2021

Dirichlet g.f.: ( zeta(s) * primezeta(s-1) )^2.
a(n) = Sum_{d|n} A061397(d) * A319131(n/d).
a(p) = 0 for p prime. - Michael S. Branicky, Nov 26 2021
a(p^k) = (k-1)*p^2 for p prime and k > 0. - Chai Wah Wu, Nov 28 2021

A002123 a(1) = 0, a(2) = 0; for n > 2, a(n) - a(n-3) - a(n-5) - ... - a(n-p) = n if n is prime, otherwise = 0, where p = largest prime < n.

Original entry on oeis.org

0, 0, 3, 0, 5, -3, 7, -8, 3, -15, 22, -15, 39, -35, 38, -72, 85, -111, 152, -175, 241, -308, 414, -551, 655, -897, 1164, -1463, 2001, -2538, 3286, -4296, 5503, -7259, 9357, -12147, 15910, -20406, 26640, -34703, 44854, -58481, 75809, -98340

Offset: 1

N. J. A. Sloane

sign

Arises in studying the Goldbach conjecture.

P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, Vol. II, pp. 354-382] [The sequence f_n]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380.
Index entries for sequences related to Goldbach conjecture

Cf. A065091, A061397.

import Data.List (genericIndex)
a002123 n = genericIndex a002123_list (n - 1)
a002123_list = 0 : 0 : f 3 where
   f x = y : f (x + 1) where
     y = a061397 x -
         sum (map (a002123 . (x -)) $ takeWhile (< x) a065091_list)
-- Reinhard Zumkeller, Mar 21 2014

Extended with signs by T. D. Noe, Dec 05 2006

A063084 a(n) = pi(n-1)n - pi(n)(n-1), where pi() = A000720().

Original entry on oeis.org

0, -1, -1, 2, -2, 3, -3, 4, 4, 4, -6, 5, -7, 6, 6, 6, -10, 7, -11, 8, 8, 8, -14, 9, 9, 9, 9, 9, -19, 10, -20, 11, 11, 11, 11, 11, -25, 12, 12, 12, -28, 13, -29, 14, 14, 14, -32, 15, 15, 15, 15, 15, -37, 16, 16, 16, 16, 16, -42, 17, -43, 18, 18, 18, 18, 18, -48, 19, 19, 19, -51, 20, -52, 21, 21, 21, 21, 21, -57, 22, 22, 22, -60, 23, 23

Offset: 1

Labos Elemer, Aug 06 2001

sign

To define as positive sequence let C(n)= A062298; f(a) = pi(a) if a is nonprime, f(a)= C(a) if a is prime. - Daniel Tisdale, Nov 07 2008

			The function is positive for composite and negative for prime numbers. It is zero at n=1.

G. A. Kudrevatow, (1970): Exercises in Number Theory. Problem 488; page 56; Prosveshenie, Moscow [in Russian].

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000720, A000027, A010051, A061397, A000040, A002808.

a(n)={if(n>1, primepi(n-1)*n - primepi(n)*(n-1), 0)} \\ Harry J. Smith, Aug 17 2009

A259175 a(n) = 1 if n prime, otherwise prime(n).

Original entry on oeis.org

2, 1, 1, 7, 1, 13, 1, 19, 23, 29, 1, 37, 1, 43, 47, 53, 1, 61, 1, 71, 73, 79, 1, 89, 97, 101, 103, 107, 1, 113, 1, 131, 137, 139, 149, 151, 1, 163, 167, 173, 1, 181, 1, 193, 197, 199, 1, 223, 227, 229, 233, 239, 1, 251, 257, 263, 269, 271, 1, 281, 1, 293, 307, 311

Offset: 1

Vincenzo Librandi, Jun 20 2015

The subsequence of prime terms is A007821. - Michel Marcus, Jun 20 2015

			a(7) = 1 because 7 is prime.
a(8) = 19 because 8 is not prime and prime(8) = 19.

Cf. A007821, A010051, A061397, A191558.

[IsPrime(n) select 1 else NthPrime(n): n in [1..100]];

Table[If[PrimeQ[n], 1, Prime[n]], {n, 100}]

A272476 a(n) = n if n is prime, a(n) = 2*n+3 otherwise.

Original entry on oeis.org

3, 5, 2, 3, 11, 5, 15, 7, 19, 21, 23, 11, 27, 13, 31, 33, 35, 17, 39, 19, 43, 45, 47, 23, 51, 53, 55, 57, 59, 29, 63, 31, 67, 69, 71, 73, 75, 37, 79, 81, 83, 41, 87, 43, 91, 93, 95, 47, 99, 101, 103, 105, 107, 53, 111, 113, 115, 117, 119, 59, 123, 61, 127

Offset: 0

Vincenzo Librandi, May 02 2016

Prime numbers repeated: 3, 5, 11, 19, 23, 31, 43, 47, 53, 59, 67, 71, ..., that is A065091 without A089531.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000040, A018252.

Cf. A061397, A191558.

[IsPrime(n) select n else 2*n+3: n in [0..80]];

Table[If[PrimeQ[n], n, 2 n + 3], {n, 0, 150}]

cp's OEIS Frontend

A143536 Triangle read by rows, T(n,k) = 1 if n is prime, 0 otherwise.

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A143545 a(n) = n unless n is a prime, in which case a(n) = 2n.

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A347104 Dirichlet g.f.: primezeta(s-1) * zeta(s-1) / zeta(s).

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A143544 Triangle read by rows, T(n,k) = 2 if n is prime, 1 otherwise; 1<=k<=n.

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A193315 Write 2n=j+q (j,q positive noncomposite numbers); j*q maximal; then a(n)=j*q.

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A349712 a(n) = Sum_{d|n} sopf(d) * sopf(n/d).

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A002123 a(1) = 0, a(2) = 0; for n > 2, a(n) - a(n-3) - a(n-5) - ... - a(n-p) = n if n is prime, otherwise = 0, where p = largest prime < n.

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A063084 a(n) = pi(n-1)*n - pi(n)*(n-1), where pi() = A000720().

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A193315 Write 2n=j+q (j,q positive noncomposite numbers); jq maximal; then a(n)=jq.

A063084 a(n) = pi(n-1)n - pi(n)(n-1), where pi() = A000720().