Pedro Caceres - cp's OEIS frontend

A332188 a(n) = (1/e^n) * Sum_{j>=2} j^n * n^j / (j-2)!.

0, 3, 72, 1557, 36928, 986550, 29641608, 994006209, 36887753216, 1502798312547, 66730937637400, 3209318261685690, 166242143849148864, 9229638177763268395, 546842961612529341032, 34443269219453881669425

Offset: 0

Pedro Caceres, Oct 30 2020

nonn

			a(3) = 1557 = (1/e^3) * Sum_{j>=2} j^3 * 3^j / factorial(j-2).

Cf. A005494, A143495, A242817, A299824, A338282.

a[n_] := Sum[n^k*(StirlingS2[n + 2, k] - StirlingS2[n + 1, k]), {k, 2, n + 2}]; Array[a, 16, 0] (* Amiram Eldar, Oct 30 2020 *)

a(n) = sum(k=0, n+2, n^k*(stirling(n+2,k,2) - stirling(n+1,k,2))); \\ Michel Marcus, Oct 30 2020

# Increase precision for larger n!
R = RealField(100)
t = 2
sol = [0]*18
for n in range(0, 18):
    suma = R(0)
    for j in range(t, 1000):
        suma += (j^n * n^j) / factorial(j - t)
    suma *= exp(-n)
    sol[n] = round(suma)
print(sol) # Thanks to Peter Luschny for his example in A338282.

a(n) = Sum_{k=0..n+2} n^k*(Stirling2(n+2,k) - Stirling2(n+1,k)). [Thanks to Andrew Howroyd for his example in A338282]

A338282 a(n) = (1/e^n) * Sum_{j>=3} j^n * n^j / (j-3)!.

Original entry on oeis.org

0, 4, 216, 7371, 239424, 8127875, 296315496, 11685617608, 498593804800, 22959117809685, 1137033860419000, 60338078785131785, 3418430599382500800, 206053517402599981504, 13172124530670958537160, 890361160360138336174875, 63463906792476058870550528, 4758276450884470061869230823

Offset: 0

Pedro Caceres, Oct 20 2020

nonn

			a(3) = 7371 = (1/e^3) * Sum_{j>=3} j^3 * 3^j / factorial(j-3).

Cf. A005494, A143495, A242817, A299824.

seq(add(n^(k+3)*A143495(n+3, k+3), k = 0..n), n = 0..17); # Peter Luschny, Oct 21 2020

a[n_] := Exp[-n] * Sum[j^n * n^j/(j - 3)!, {j, 3, Infinity}]; Array[a, 17, 0] (* Amiram Eldar, Oct 20 2020 *)

a(n)={sum(k=0, n+3, n^k*(stirling(n+3,k,2) - 3*stirling(n+2,k,2) + 2*stirling(n+1,k,2)))} \\ Andrew Howroyd, Oct 20 2020

# Increase precision for larger n!
R = RealField(100)
t = 3
sol = [0]*18
for n in range(0, 18):
    suma = R(0)
    for j in range(t, 1000):
        suma += (j^n * n^j) / factorial(j - t)
    suma *= exp(-n)
    sol[n] = round(suma)
print(sol) # Peter Luschny, Oct 20 2020

a(n) = Sum_{k=0..n+3} n^k*(Stirling2(n+3,k) - 3*Stirling2(n+2,k) + 2*Stirling2(n+1,k)). - Andrew Howroyd, Oct 20 2020

a(n) = Sum_{k=0..n} n^(k+3)*A143495(n+3, k+3). - Peter Luschny, Oct 21 2020

A333721 Numbers k such that k + 1, 2k + 1, 3k + 1, 4k + 1, and 6k + 1 are all prime.

Original entry on oeis.org

1530, 4260, 25410, 26040, 78540, 111720, 174990, 211050, 214830, 395430, 403260, 409290, 459690, 487830, 512820, 711120, 779790, 910560, 1023750, 1135950, 1280370, 1312350, 1451520, 1464810, 1487070, 1563510, 1623360, 1698060, 1824330, 1933680, 2006340, 2097480

Offset: 1

Pedro Caceres, May 04 2020

nonn

All terms are multiples of 6.

All terms are multiples of 30. - Robert Israel, Jun 17 2020

			25410 is in the sequence because 25411, 50821, 76231, 101641, 152461 are all prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Pedro Caceres, 30 Ideas about Prime Numbers

Cf. A006093, A005097, A024892, A087370, A005098, A005099, A024898, A024899.

select(t -> andmap(isprime, [t+1,2*t+1,3*t+1,4*t+1,6*t+1]), [seq(i,i=30..3*10^6,30)]); # Robert Israel, Jun 17 2020

isok(m)={for(i=1, 6, if(i<>5&&!isprime(i*m+1), return(0))); 1}
{ forstep(n=0, 3*10^6, 6, if(isok(n), print1(n, ", "))) } \\ Andrew Howroyd, May 04 2020

A308090 a(n) = gcd(2^n + n!, 3^n + n!, n+1).

Original entry on oeis.org

1, 1, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, 1, 1, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 1, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 1, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1, 1, 97, 1, 1, 1

Offset: 1

Pedro Caceres, May 11 2019

nonn

From observation: For n > 3, if n+1 is prime, then a(n) = n+1.
This implies that (2^n + n!)= 0 mod (n+1) iff (n+1) is prime, and (3^n + n!)= 0 mod (n+1) iff (n+1) is prime.
Conjecture: Conversely, if gcd(2^n + n!, 3^n + n!, n+1) = n+1, then n+1 is prime.
Appears to be the same as A090585(n) except at n=2. - R. J. Mathar, Jul 22 2021

			a(4) = gcd(2^4 + 4!, 3^4 + 4!, 5) = gcd(40, 105, 5) = 5.
a(5) = gcd(2^5 + 5!, 3^5 + 5!, 6) = gcd(152, 363, 6) = 1.

Cf. A007611, A249945.

Table[GCD[2^n+n!,3^n+n!,n+1],{n,100}] (* Harvey P. Dale, Aug 27 2020 *)

a(n) = gcd([2^n + n!, 3^n + n!, n+1]); \\ Michel Marcus, May 12 2019

a(n) = gcd(A007611(n), A249945(n), n+1).

A307642 a(n) = n!*Sum_{i=1..n} (Sum_{j=1..i} (i/j)).

Original entry on oeis.org

1, 8, 57, 428, 3510, 31644, 312984, 3380544, 39664080, 502927200, 6858181440, 100135491840, 1559197261440, 25797280723200, 452046655872000, 8364495012249600, 162994310248089600, 3336683369519001600, 71596721810396160000, 1606993396943155200000

Offset: 1

Pedro Caceres, Apr 19 2019

nonn

			a(3) = 57 because a(3) = 3!*Sum_{i=1..3} (Sum_{j=1..i} (i/j)).

Cf. A001008/A002805 (harmonic), A182541.

List([1..25], n-> n*Factorial(n+1)*(1+2*Sum([2..n+1], j-> 1/j))/4 ); # G. C. Greubel, Jul 15 2019

[n*Factorial(n+1)*(2*HarmonicNumber(n+1)-1)/4: n in [1..25]]; // G. C. Greubel, Jul 15 2019

Array[#!*Sum[Sum[i/j, {j, i}], {i, #}] &, 25] (* Michael De Vlieger, Apr 21 2019 *)
Table[n*(n+1)!*(2*HarmonicNumber[n+1] -1)/4, {n, 25}] (* G. C. Greubel, Jul 15 2019 *)

a(n)=n!*sum(i=1, n, sum(j=1, i, i/j)); \\ Michel Marcus, Apr 20 2019

[n*factorial(n+1)*(2*harmonic_number(n+1)-1)/4 for n in (1..25)] # G. C. Greubel, Jul 15 2019

a(n) = n! * Sum_{i=1..n} (Sum_{j=1..i} (i/j)).
a(n) = n * A182541(n+2).
a(n) = (1/4) * n * (n+1)! * (2*harmonic(n+1) - 1).

A307663 a(n) = (n-1)!(Sum_{i=1..n} Sum_{j=1..i} binomial(i,j)i/j).

Original entry on oeis.org

1, 6, 41, 329, 3090, 33654, 420792, 5981688, 95782320, 1712555280, 33909364800, 737868052800, 17521164259200, 451126883894400, 12522623670144000, 372847351488998400, 11853064556660275200, 400718191717647820800, 14354714544806716416000, 543129329390299739136000, 21642934280974058207232000

Offset: 1

Pedro Caceres, Apr 20 2019

nonn

			a(2) = 1! * (C(1,1)*1/1 + C(2,1)*2/1 + C(2,2)*2/2) = 6.

Sela Fried, A307663, 2024.

Array[(# - 1)!*Sum[Sum[Binomial[i, j] i/j, {j, i}], {i, #}] &, 21] (* Michael De Vlieger, Apr 21 2019 *)

a(n) = (n-1)!*sum(i=1, n, sum(j=1, i, binomial(i,j)*i/j)); \\ Michel Marcus, Apr 20 2019

Conjectures from Robert Israel, Oct 26 2020: (Start)
E.g.f. ((4*x^2 - 8*x + 5)*log(-x + 1))/(2*(x - 1)^2) - ((4*x^2 - 8*x + 5)*log(1 - 2*x))/(2*(x - 1)^2) + x*(-6 + 5*x)/(4*(x - 1)^2).
D-finite with recurrence 2*(n+3)*(n+2)*n*(n-2)*a(n) - (n+3)*(5*n^2-6*n-17)*a(n+1) + (4 n^2-n-29)* a(n+2) -(n-3)*a(n+3) = 0. (End)
The conjecture regarding the e.g.f. is true. See links. - Sela Fried, Jul 30 2024.

A322921 From Goldbach's conjecture: a(n) is the number of decompositions of 6n into a sum of two primes.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 8, 9, 7, 8, 8, 10, 12, 10, 9, 8, 11, 12, 11, 10, 13, 11, 14, 13, 11, 13, 14, 19, 13, 11, 12, 15, 18, 16, 16, 14, 16, 19, 16, 16, 17, 19, 21, 15, 17, 15, 20, 24, 19, 17, 16, 20, 22, 18, 18, 22, 19, 27, 21, 17, 20, 21, 30

Offset: 1

Pedro Caceres, Dec 30 2018

nonn

According to Goldbach's conjecture all even numbers can be decomposed into one or more sums of two prime numbers.
Each even number N belongs to one of the following sets: {N == 0 (mod 6)}, {(N + 2) == 0 (mod 6)}, and {(N - 2) == 0 (mod 6)}.
Conjecture: In any combination of three consecutive even numbers >= 48, the one of the form N == 0 (mod 6) will have the largest number of decompositions into 2 prime numbers. This sequence contains those local maxima for every set of three consecutive even numbers. This sequence forms the upper envelope of Goldbach's comet chart.

			a(1) = 1 because 6 * 1 = 6 can be decomposed as (3 + 3);
a(8) = 5 is the number of ways that 6 * 8 = 48 can be decomposed into sums of two prime numbers: 5 + 43, 11 + 37, 17 + 31, 29 + 19, 41 + 7.

Cf. A002375, A045917.

Table[Count[IntegerPartitions[6n, {2}], ?(AllTrue[#, PrimeQ] && FreeQ[#, 2]&)], {n, 100}] (* _Alonso del Arte, Dec 31 2018, just a tiny modification of Harvey P. Dale's for A002375 *)

a(n) = A002375(3*n).

A323139 Integers that are not multiples of 6 and are the sum of two consecutive primes.

Original entry on oeis.org

5, 8, 52, 68, 100, 112, 128, 152, 172, 268, 308, 320, 340, 352, 410, 434, 472, 508, 520, 532, 548, 668, 712, 740, 752, 772, 872, 892, 946, 1012, 1030, 1064, 1088, 1120, 1132, 1148, 1180, 1192, 1208, 1220, 1250, 1300, 1312, 1334, 1360, 1460, 1472, 1508, 1606, 1888, 1900, 1948, 1960, 1988, 2006, 2032, 2072, 2132, 2156

Offset: 1

Pedro Caceres, Jan 05 2019

nonn

All primes, except 2 and 3, are of the form 6k+1 or 6k-1 for k a positive integer. The converse statement is not true for all k, so the sum of two consecutive primes is not always a multiple of 6. This sequence lists the sums of two consecutive primes that are not multiple of 6.

			52 = 23 + 29 is not a multiple of 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001043, A047253, A323138.

p:= 2:
count:= 0: Res:= NULL:
while count < 100 do
  q:= nextprime(p);
  if p+q mod 6 <> 0 then
     count:= count+1; Res:= Res, p+q;
  fi;
  p:= q;
od:
Res; # Robert Israel, Jan 09 2019

Select[Total /@ Partition[Prime@ Range@ 180, 2, 1], Mod[#, 6] != 0 &] (* Michael De Vlieger, Jan 07 2019 *)

my(p=2); forprime(q=3, 1e3, if ((p+q) % 6, print1(p+q", ")); p=q); \\ Michel Marcus, Jan 05 2019

A323138 Multiples of 6 that are not the sum of two consecutive primes.

Original entry on oeis.org

6, 48, 54, 66, 72, 96, 102, 108, 114, 126, 132, 150, 156, 168, 174, 180, 192, 228, 234, 246, 252, 264, 270, 282, 294, 306, 312, 318, 324, 336, 342, 348, 354, 366, 378, 402, 408, 414, 420, 426, 432, 438, 444, 468, 474, 486, 498, 504, 510, 516, 522, 528, 534, 546, 552, 570, 582

Offset: 1

Pedro Caceres, Jan 05 2019

nonn

All primes, except 2 and 3, are of the form 6k+1 or 6k-1 for k a positive integer. The converse statement is not true for all k, so the sum of two consecutive primes is not always a multiple of 6. This sequence lists the multiples of 6 that cannot be expressed as a sum of two consecutive primes.

			6 belongs to the sequence because there are no two consecutive primes adding up to 6. 12 is not in the sequence because 12 = 5 + 7.

Cf. A001043. Subsequence of A008588.

Complement[6 Range[Last[#]/6], #] &@ Select[Total /@ Partition[Prime@ Range@ 63, 2, 1], Mod[#, 6] == 0 &] (* Michael De Vlieger, Jan 07 2019 *)

isok(n) = !(n % 6) && (precprime((n-1)/2) + nextprime(n/2) != n); \\ Michel Marcus, Jan 05 2019

A305825 Number of different ways that a number between two members of a twin prime pair can be expressed as a sum of two smaller such numbers.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 3, 1, 4, 3, 3, 3, 2, 6, 3, 5, 3, 3, 3, 3, 3, 8, 4, 2, 3, 3, 6, 4, 4, 6, 7, 8, 3, 6, 3, 9, 8, 6, 7, 5, 8, 4, 1, 5, 6, 3, 7, 1, 6, 6, 4, 8, 1, 5, 5, 8, 9, 11, 10, 6, 8, 16, 13, 9, 12, 6, 7, 8, 4, 16, 9, 6, 13, 10, 9, 5, 6, 6

Offset: 1

Pedro Caceres, Jun 10 2018

nonn

Number of pairs i, j such that A014574(i) + A014574(j) = A014574(n) where 1 <= i <= j < n. - David A. Corneth, Aug 05 2018

			a(8)=2 because the 8th isolated composite number is 72 = 60 + 12 and 42 + 30 with (12,30,42,60) all isolated composite numbers.

Cf. A014574, A134143.

lista(nn) = {my(vc = select(x->(isprime(x-1) && isprime(x+1)), [1..nn])); for (n=1, #vc, nb = 0; for (j=1, n, for (k=j+1, n, if (vc[j]+vc[k] == vc[n], nb++));); print1(nb, ", "););} \\ Michel Marcus, Jul 05 2018

first(n) = {my(isolated = List(), isomap = Map, res = vector(n), k, q = 3); forprime(p = 5, , if(p - q == 2, listput(isolated, q+1); mapput(isomap, q+1, #isolated); if(#isolated == n, break)); q = p); for(i = 1, #isolated, for(j = 1, i - 1, diff = isolated[i] - isolated[j]; if(diff < isolated[j], if( mapisdefined(isomap, diff, &k), res[i]++), next(1)))); res} \\ David A. Corneth, Aug 05 2018

Name changed, extended data by David A. Corneth, Aug 05 2018

cp's OEIS Frontend

User: Pedro Caceres

A332188 a(n) = (1/e^n) * Sum_{j>=2} j^n * n^j / (j-2)!.

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A338282 a(n) = (1/e^n) * Sum_{j>=3} j^n * n^j / (j-3)!.

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A333721 Numbers k such that k + 1, 2k + 1, 3k + 1, 4k + 1, and 6k + 1 are all prime.

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A308090 a(n) = gcd(2^n + n!, 3^n + n!, n+1).

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A307642 a(n) = n!*Sum_{i=1..n} (Sum_{j=1..i} (i/j)).

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A307663 a(n) = (n-1)!*(Sum_{i=1..n} Sum_{j=1..i} binomial(i,j)*i/j).

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A322921 From Goldbach's conjecture: a(n) is the number of decompositions of 6n into a sum of two primes.

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A307663 a(n) = (n-1)!(Sum_{i=1..n} Sum_{j=1..i} binomial(i,j)i/j).