A073046 -id:A073046 - cp's OEIS frontend

A102084 a(1) = 0; for n>0, write 2n=p+q (p, q prime), pq maximal; then a(n)=pq (see A073046).

0, 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

Offset: 1

Michael Taktikos, Feb 16 2005

nonn

For n>1, largest semiprime whose sum of prime factors = 2n. Assumes the Goldbach conjecture is true. Also the largest semiprime <= n^2.

Also the greatest integer x such that x' = 2*n, or 0 if there is no such x, where x' is the arithmetic derivative (A003415). Bisection of A099303. The only even number without an anti-derivative is 2. All terms are <= n^2, with equality only when n is prime. In fact a(n) = n^2 - k^2, where k is the least number such that both n-k and n+k are prime; k = A047160(n). It appears that the anti-derivatives of even numbers are overwhelmingly semiprimes of the form n^2 - k^2. For example, 1000 has 28 anti-derivatives, all of this form. Sequence A189763 lists the even numbers that have anti-derivatives not of this form. - T. D. Noe, Apr 27 2011

			n=13: 2n = 26; 26 = 23 + 3 = 19 + 7 = 13 + 13; 13*13 = maximal => p*q = 13*13 = 169.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A073046, A003415, A047160, A099303, A189762.

f[n_] := Block[{pf = FactorInteger[n]}, If[Plus @@ Last /@ pf == 2, If[ Length[pf] == 2, Plus @@ First /@ pf, 2pf[[1, 1]]], 0]]; t = Table[0, {51}]; Do[a = f[n]; If[ EvenQ[a] && 0 < a < 104, t[[a/2]] = n], {n, 2540}]; t (* Robert G. Wilson v, Jun 14 2005 *)
Table[k = 0; While[k < n && (! PrimeQ[n - k] || ! PrimeQ[n + k]), k++]; If[k == n, 0, (n - k)*(n + k)], {n, 100}] (* T. D. Noe, Apr 27 2011 *)

a(n) = n^2 - A047160(n)^2. - Jason Kimberley, Jun 26 2012

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A070015 Least m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 0, 4, 9, 0, 6, 8, 10, 15, 14, 21, 121, 27, 22, 16, 12, 39, 289, 65, 34, 18, 20, 57, 529, 95, 46, 69, 28, 115, 841, 32, 58, 45, 62, 93, 24, 155, 1369, 217, 44, 63, 30, 50, 82, 123, 52, 129, 2209, 75, 40, 141, 0, 235, 42, 36, 106, 99, 68, 265, 3481, 371, 118, 64, 56

Offset: 0

Labos Elemer, Apr 12 2002

For odd n >= 9, a(n) <= A073046((n-1)/2). - Max Alekseyev, Sep 01 2025

			For n=128: a(128)=16129, divisors={1,127,16129}, 1+127=sigma(n)-n=128 and 16129 is the smallest.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 9884 terms from Richard J. Mathar)
Mersenne Forum, Given sigma(n)-n, find the smallest possible n

Cf. A000203, A001065, A048050, A051444, A007369, A070016, A005114, A048995.

See A359132 for another version.

f[x_] := DivisorSigma[1, x]-x; t=Table[0, {128}]; Do[c=f[n]; If[c<129&&t[[c]]==0, t[[c]]=n], {n, 1000000}]; t

a(n) = min(x, A001065(x)=n) or a(n)=0 if n is an untouchable number (i.e., if from A005114).

a(0)=1 prepended by Max Alekseyev, Sep 01 2025

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

A210957 Prime pair (p, q), p<=q, such that p + q = 2n and pq is the minimal product.

Original entry on oeis.org

2, 2, 3, 3, 3, 5, 3, 7, 5, 7, 3, 11, 3, 13, 5, 13, 3, 17, 3, 19, 5, 19, 3, 23, 5, 23, 7, 23, 3, 29, 3, 31, 5, 31, 7, 31, 3, 37, 5, 37, 3, 41, 3, 43, 5, 43, 3, 47, 5, 47, 7, 47, 3, 53, 5, 53, 7, 53, 3, 59, 3, 61, 5, 61, 7, 61, 3, 67, 5, 67, 3, 71, 3, 73

Offset: 2

Omar E. Pol, Jun 29 2012

nonn

A020481 and A020482 interleaved.

			-----------------------------------
                 2*n    A073046(n)
       Pair       =         =
n     (p, q)     p+q       p*q
-----------------------------------
2     (2, 2)      4          4
3     (3, 3)      6          9
4     (3, 5)      8         15
5     (3, 7)     10         21
6     (5, 7)     12         35
7     (3, 11)    14         33
8     (3, 13)    16         39
9     (5, 13)    18         65
10    (3, 17)    20         51
11    (3, 19)    22         57
12    (5, 19)    24         95

Cf. A000040, A002373, A073046, A210967.

p_n = A020481(n), n >= 2.
q_n = A020482(n), n >= 2.
p_n + q_n = 2*n, n >= 2.
p_n * q_n = A073046(n), n >= 2.

A350455 T(n,k) is the k-th semiprime whose sum of prime factors equals 2n, triangle T(n,k), n>=2, 1<=k<=A045917(n), read by rows.

Original entry on oeis.org

4, 9, 15, 21, 25, 35, 33, 49, 39, 55, 65, 77, 51, 91, 57, 85, 121, 95, 119, 143, 69, 133, 169, 115, 187, 161, 209, 221, 87, 247, 93, 145, 253, 289, 155, 203, 299, 323, 217, 361, 111, 319, 391, 185, 341, 377, 437, 123, 259, 403, 129, 205, 493, 529, 215, 287, 407

Offset: 2

Alois P. Heinz, Dec 31 2021

Assuming Goldbach's conjecture, no row is empty.

			Triangle T(n,k) begins:
    4;
    9;
   15;
   21,  25;
   35     ;
   33,  49;
   39,  55;
   65,  77;
   51,  91;
   57,  85, 121;
   95, 119, 143;
   69, 133, 169;
  115, 187     ;
  161, 209, 221;
   87, 247     ;
   93, 145, 253, 289;
  155, 203, 299, 323;
  ...

Alois P. Heinz, Rows n = 2..1000, flattened
Wikipedia, Goldbach's conjecture

Column k=1 gives A073046.
Last elements of rows give A102084.
Row sums give A228553.
Row products give A337568.
Row lengths give A045917.
Cf. A000040, A001358, A046315, A350419.

T:= n-> seq(`if`(andmap(isprime, [h, 2*n-h]), h*(2*n-h), [][]), h=2..n):
seq(T(n), n=2..30);

A193216 Least semiprime whose sum of prime factors equals n!

Original entry on oeis.org

9, 95, 791, 7799, 85391, 1248959, 4717271, 39916679, 518918231, 6227020631, 143221477871, 1482030950111, 61460695293791, 1108907864061191, 20985558257660519, 262497321934846319, 12286155141292021799, 75419962253475839039

Offset: 3

Michel Lagneau, Jul 18 2011

nonn

Write n! = p+q (p,q prime), p*q minimal; then a(n)=p*q. Subset of A073046.

			a(4) = 95 because,for n=4, 4! = 24; 24 = 5 + 19 = 7 + 17 = 11 + 13; 5*19 is minimal => p*q = 5*19 = 95.

Cf. A073046.

 with(numtheory):for n from 3 to 20 do:x:=n!:id:=0:for m from 2 to 10000 while(id=0) do:p:=ithprime(m):y:=x-p:if type(y,prime)=true then z:=y*p: id:=1:printf(`%d, `, z): else fi:od:od:

cp's OEIS Frontend

A102084 a(1) = 0; for n>0, write 2n=p+q (p, q prime), p*q maximal; then a(n)=p*q (see A073046).

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A070015 Least m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists.

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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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A210957 Prime pair (p, q), p<=q, such that p + q = 2*n and p*q is the minimal product.

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A350455 T(n,k) is the k-th semiprime whose sum of prime factors equals 2n, triangle T(n,k), n>=2, 1<=k<=A045917(n), read by rows.

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Maple

A193216 Least semiprime whose sum of prime factors equals n!

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Maple

A102084 a(1) = 0; for n>0, write 2n=p+q (p, q prime), pq maximal; then a(n)=pq (see A073046).

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

A210957 Prime pair (p, q), p<=q, such that p + q = 2n and pq is the minimal product.