A061397 -id:A061397 - cp's OEIS frontend

A282462 Integers but with the primes cubed.

0, 1, 8, 27, 4, 125, 6, 343, 8, 9, 10, 1331, 12, 2197, 14, 15, 16, 4913, 18, 6859, 20, 21, 22, 12167, 24, 25, 26, 27, 28, 24389, 30, 29791, 32, 33, 34, 35, 36, 50653, 38, 39, 40, 68921, 42, 79507, 44, 45, 46, 103823, 48, 49, 50, 51, 52, 148877, 54, 55, 56

Offset: 0

Vincenzo Librandi, Feb 17 2017

			a(4) = 4 because 4 is composite.
a(5) = 125 because 5 is prime and 5^3 = 125.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A061397, A103164, A143545.

[0] cat [IsPrime(n) select n^3 else n: n in [1..60]];

Mathematica
```
Join[{0},If[PrimeQ@#,#^3,#]&/@Range@80]
```

A351369 a(n) = Sum_{p|n, p prime} p * prime(p).

Original entry on oeis.org

0, 6, 15, 6, 55, 21, 119, 6, 15, 61, 341, 21, 533, 125, 70, 6, 1003, 21, 1273, 61, 134, 347, 1909, 21, 55, 539, 15, 125, 3161, 76, 3937, 6, 356, 1009, 174, 21, 5809, 1279, 548, 61, 7339, 140, 8213, 347, 70, 1915, 9917, 21, 119, 61, 1018, 539, 12773, 21, 396, 125, 1288, 3167

Offset: 1

Wesley Ivan Hurt, Feb 08 2022

nonn

Inverse Möbius transform of n * prime(n) * c(n), where c(n) is the characteristic function of primes (A010051). - Wesley Ivan Hurt, Apr 01 2025

			a(6) = 21; a(6) = Sum_{p|6} p * prime(p) = 2*3 + 3*5 = 21.

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

Cf. A000040, A010051, A033286, A061397, A351368.

Join[{0},Table[Total[# Prime[#]&/@FactorInteger[n][[;;,1]]],{n,2,80}]] (* Harvey P. Dale, Jan 28 2024 *)

a(n) = Sum_{d|n} d * prime(d) * c(d), where c = A010051. - Wesley Ivan Hurt, Apr 01 2025

a(p^k) = p * prime(p) for p prime and k>=1. - Wesley Ivan Hurt, Jul 16 2025

A359177 Product_{n>=1} (1 + a(n) * x^n) = 1 + Sum_{n>=1} prime(n) * x^prime(n).

Original entry on oeis.org

0, 2, 3, 0, -1, 0, 9, 3, -18, -27, 38, 63, 3, -132, -54, 200, 395, -258, -666, 336, 2111, 836, -3454, -4020, 9290, 14408, 1215, -43886, -13043, 92456, 155854, -162756, -396316, 43971, 1201705, 442641, -1922050, -3359287, 3887946, 9202219, 529036, -24727323, -11980753, 48090463

Offset: 1

Seiichi Manyama, Dec 28 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A061397, A348128.

A259182 a(n) = prime(n) if n prime otherwise 1.

Original entry on oeis.org

1, 3, 5, 1, 11, 1, 17, 1, 1, 1, 31, 1, 41, 1, 1, 1, 59, 1, 67, 1, 1, 1, 83, 1, 1, 1, 1, 1, 109, 1, 127, 1, 1, 1, 1, 1, 157, 1, 1, 1, 179, 1, 191, 1, 1, 1, 211, 1, 1, 1, 1, 1, 241, 1, 1, 1, 1, 1, 277, 1, 283, 1, 1, 1, 1, 1, 331, 1, 1, 1, 353, 1, 367, 1, 1, 1, 1

Offset: 1

Vincenzo Librandi, Jun 20 2015

The subsequence of prime terms is A006450.

			a(7) = 17 because prime(7) = 17 is prime.
a(8) = 1 because 8 is not prime.

Cf. A006450, A007821, A010051, A061397, A191558, A259175.

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

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

A277186 Sum of primes within 2n-wide closed interval centered upon prime(n).

Original entry on oeis.org

5, 10, 17, 26, 31, 67, 83, 83, 119, 139, 161, 228, 281, 281, 341, 408, 474, 553, 546, 635, 635, 780, 824, 1092, 954, 1008, 1008, 1139, 1197, 1336, 1621, 1687, 1650, 1823, 1854, 1854, 2238, 2634, 2507, 2587, 2450, 2673, 3223, 3223, 3403, 3403, 3591, 4054, 4054, 4331, 4535, 4535, 4828, 4444, 4666

Offset: 1

Walter Carlini, Oct 04 2016

a(n) is the sum of primes within the closed interval [prime(n)-n, prime(n)+n], where prime(n) is the n-th prime.

			a(3) = 2 + 3 + 5 + 7 = 17; starting at prime(3) = 5, subtract 3 and add 3 to obtain the interval 2 through 8, and then add up the primes within that interval, inclusive of the endpoints of the interval.

Cf. A000040, A014688, A014689, A061397.

Table[Total@ Select[Range[Prime@ n - n, Prime@ n + n], PrimeQ], {n, 55}] (* Michael De Vlieger, Oct 04 2016 *)

a(n) = sum(k=prime(n)-n, prime(n)+n, isprime(k)*k); \\ Michel Marcus, Nov 01 2016

More terms from Michael De Vlieger, Oct 04 2016

A300692 Primes that are the sum of all primes up to some power of 2.

Original entry on oeis.org

2, 5, 17, 41, 202288087, 4394533064208947008756469709307

Offset: 1

Christoph Zurnieden, Apr 03 2018

Elements in the sequence are certified primes.

The corresponding exponents of 2 are 1, 2, 3, 4, 16 and 54.

			17 is a term because the sum of all primes below 2^3 is 2+3+5+7 = 17 which is prime.

Kim Walisch, List of sums of primes up to 2^n with n<=80 (at the end).

Cf. A130739, A061397.

Select[Array[Total@ Prime@ Range@ PrimePi[2^#] &, 27, 0], PrimeQ] (* Michael De Vlieger, Apr 10 2018 *)

lista(nn) = {for (n=0, nn, s = 0; forprime(k=0, 2^n, s+=k); if (isprime(s), print1(s, ", ")));}

Numbers of the form Sum_{i=2..2^n-1} A061397(i) that are prime.

A346180 a(n) = prime(n) + n if n is prime, a(n) = prime(n) otherwise.

Original entry on oeis.org

2, 5, 8, 7, 16, 13, 24, 19, 23, 29, 42, 37, 54, 43, 47, 53, 76, 61, 86, 71, 73, 79, 106, 89, 97, 101, 103, 107, 138, 113, 158, 131, 137, 139, 149, 151, 194, 163, 167, 173, 220, 181, 234, 193, 197, 199, 258, 223, 227, 229, 233, 239, 294, 251, 257, 263, 269, 271

Offset: 1

Samuel Vilz, Jul 09 2021

nonn

For n > 0, take the n-th prime and only add n to it if n is prime itself.

			a(1) = 2 = 2+0; 2 is the first prime. 1 is not prime and thus is not added.
a(2) = 5 = 3+2; 3 is the second prime, and since 2 is also prime, add 2.
a(3) = 8 = 5+3; 5 is the third prime, and since 3 is also prime, add 3.

Cf. A000040, A061397.

Table[Prime@n+Boole@PrimeQ[n]n,{n,60}] (* Giorgos Kalogeropoulos, Jul 29 2021 *)
Table[If[PrimeQ[n],Prime[n]+n,Prime[n]],{n,60}] (* Harvey P. Dale, Nov 20 2022 *)

a(n) = prime(n) + isprime(n)*n; \\ Michel Marcus, Jul 12 2021

from sympy import isprime, prime
def a(n): return prime(n) + n*isprime(n)
print([a(n) for n in range(1, 59)]) # Michael S. Branicky, Jul 29 2021

# faster version for initial segment of sequence
from sympy import isprime, prime, primerange
def aupton(nn): return [p + n*isprime(n) for n, p in enumerate(primerange(1, prime(nn)+1), start=1)]
print(aupton(10000)) # Michael S. Branicky, Jul 29 2021

a(n) = A000040(n) + A061397(n).

A352167 a(n) is the sum of the prime factors of n (with multiplicity) that are less than n.

Original entry on oeis.org

0, 0, 0, 4, 0, 5, 0, 6, 6, 7, 0, 7, 0, 9, 8, 8, 0, 8, 0, 9, 10, 13, 0, 9, 10, 15, 9, 11, 0, 10, 0, 10, 14, 19, 12, 10, 0, 21, 16, 11, 0, 12, 0, 15, 11, 25, 0, 11, 14, 12, 20, 17, 0, 11, 16, 13, 22, 31, 0, 12, 0, 33, 13, 12, 18, 16, 0, 21, 26, 14, 0, 12, 0, 39, 13, 23, 18, 18, 0, 13

Offset: 1

Ilya Gutkovskiy, Mar 06 2022

nonn

Eric Weisstein's World of Mathematics, Sum of Prime Factors

Cf. A001414, A061397, A105221, A144494.

a[n_] := If[n == 1 || PrimeQ[n], 0, Plus @@ Times @@@ FactorInteger@n]; Table[a[n], {n, 80}]

a(n) = if (isprime(n), 0, my(f=factor(n)); sum(k=1, #f~, f[k,1]*f[k,2])); \\ Michel Marcus, Mar 07 2022

a(n) = 0 if n is prime, A001414(n) otherwise.

a(n) = A001414(n) - A061397(n).

A354836 Triangle T(n,k) where, if n-k and n+k are prime, T(n,k) = n+k is the greater term of a Goldbach partition of 2n into two odd primes, or zero otherwise.

Original entry on oeis.org

3, 0, 5, 5, 0, 7, 0, 7, 0, 0, 7, 0, 0, 0, 11, 0, 0, 0, 11, 0, 13, 0, 0, 11, 0, 13, 0, 0, 0, 0, 0, 13, 0, 0, 0, 17, 11, 0, 0, 0, 0, 0, 17, 0, 19, 0, 13, 0, 0, 0, 17, 0, 19, 0, 0, 13, 0, 0, 0, 0, 0, 19, 0, 0, 0, 23, 0, 0, 0, 17, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 17, 0, 19, 0, 0, 0, 23, 0, 0, 0, 0

Offset: 3

Jean-François Alcover, Jun 12 2022

This sequence has the same structure as A354805, which could be considered as sort of its characteristic function.

			Triangle begins:
    3;
    0, 5;
    5, 0, 7;
    0, 7, 0, 0;
    7, 0, 0, 0,11;
    0, 0, 0,11, 0,13;
    0, 0,11, 0,13, 0, 0;
    0, 0, 0,13, 0, 0, 0,17;
   11, 0, 0, 0, 0, 0,17, 0,19;
   ...
Example: for n=11, row {11,0,0,0,0,0,17,0,19}, when stripped of its zeros and subtracted from 2n=22, gives the partitions {{11,11},{17,5},{19,3}}.

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture

Cf. A085090 (main diagonal), A061397 (column k=0 prepended with (0,2)), A145091 (column k=1 prepended with (0,2,3,0)), A354805.

nmin = 3; nmax = 16;
T[n_ /; n >= nmin, k_ /; k >= 0] := If[PrimeQ[n-k] && PrimeQ[n+k], n+k, 0];
Table[T[n, k], {n, nmin, nmax}, {k, 0, n - nmin}] // Flatten

A368118 a(n) = ceiling(1/p(n)) if p(n) > 0 otherwise 0, where p(n) = 2sin(Pi Gamma(n) / n).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 3, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 5, 0, 5, 0, 0, 0, 0, 0, 6, 0, 0, 0, 7, 0, 7, 0, 0, 0, 8, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 10, 0, 10, 0, 0, 0, 0, 0, 11, 0, 0, 0, 12, 0, 12, 0, 0, 0, 0, 0, 13, 0, 0, 0, 14, 0

Offset: 1

Peter Luschny, Dec 17 2023

nonn

Replacing in the definition '2' by 'n', i.e., defining q(n) = n * sin(Pi * Gamma(n) / n), would make sequence a coincide with the characteristic function of the primes, A010051, since 0 < 1/q(n) < 1 and 1/q(n) -> 1/Pi for prime n -> oo.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 427.

Underwood Dudley, Formulas for primes, Math. Mag., 56 (1983), 17-22.
Jeffrey Shallit, No Formula for the Prime Numbers?, blog post, Jan 2013.
C. P. Willans, On Formulae for the Nth Prime Number, The Mathematical Gazette, Volume 48, Issue 366, December 1964, pp. 413 - 415.

Cf. A000040, A010051, A061397.

a[n_] := If[(p = 2Sin[Pi*Gamma[n]/n]) > 0, Ceiling[1/p], 0]; Array[a, 84]
(* Stefano Spezia, Dec 17 2023 *)

p = lambda s: 2*sin(pi*gamma(s)/s)
IsPrime = lambda n: p(n).n() > 0
def a(n): return ceil(1/p(n).n()) if IsPrime(n) else 0
print([a(n) for n in range(1, 85)])

a(n) > 0 if and only if n is prime. If n is not prime then a(n) = 0.

cp's OEIS Frontend

A282462 Integers but with the primes cubed.

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Magma

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A351369 a(n) = Sum_{p|n, p prime} p * prime(p).

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A359177 Product_{n>=1} (1 + a(n) * x^n) = 1 + Sum_{n>=1} prime(n) * x^prime(n).

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A259182 a(n) = prime(n) if n prime otherwise 1.

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Magma

Mathematica

A277186 Sum of primes within 2n-wide closed interval centered upon prime(n).

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Mathematica

PARI

Extensions

A300692 Primes that are the sum of all primes up to some power of 2.

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Mathematica

PARI

Formula

A346180 a(n) = prime(n) + n if n is prime, a(n) = prime(n) otherwise.

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A352167 a(n) is the sum of the prime factors of n (with multiplicity) that are less than n.

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A368118 a(n) = ceiling(1/p(n)) if p(n) > 0 otherwise 0, where p(n) = 2sin(Pi Gamma(n) / n).