Frank Ellermann - cp's OEIS frontend

A333772 a(n) = n * 2^n * (n!)^2.

2, 32, 864, 36864, 2304000, 199065600, 22759833600, 3329438515200, 606790169395200, 134842259865600000, 35895009576222720000, 11277559372311429120000, 4129466323494701629440000, 1743270091026070964797440000, 840505222458998500884480000000

Offset: 1

Frank Ellermann, Apr 05 2020

nonn

Sum_{n>=1} a(n) / (2*n)! = Pi + 3.

			a(2) = 2 * 2^2 * ( 2! )^2 = 2 * 4 * 4 = 32.
a(3) = 3 * 2^3 * ( 3! )^2 = 3 * 8 * 36 = 864.
Sum_{n=1..10} a(n) / ( 2n )! = 3 + 3.01310...
Sum_{n=1..12} a(n) / ( 2n )! = 3 + 3.10046...
Sum_{n=1..18} a(n) / ( 2n )! = 3 + 3.14046...
Sum_{n=1..20} a(n) / ( 2n )! = 3 + 3.14126...
Sum_{n=1..23} a(n) / ( 2n )! = 3 + 3.14154...

Simon Plouffe, On the computation of the n'th decimal digit of various transcendental numbers, arXiv:0912.0303 [math.NT], 2009.

Cf. A001044 ( (n!)^2 ), A010050 ( (2n)! ), A000796 (digits of Pi).

Table[n*2^n*(n!)^2,{n,20}] (* Harvey P. Dale, Jun 01 2024 *)

S = 2
do N = 2 while length( S ) < 255
   S = S || ', ' || N * ( 2 ** N ) * ( !( N ) ** 2 )
end N
say S                         ;  return S

A333436 Partition numbers A000041(k*x_n + y_n) are known to be divisible by prime(n); sequence gives the list of x_n.

Original entry on oeis.org

5, 7, 11, 17303, 206839, 1977147619

Offset: 3

Frank Ellermann, Mar 21 2020

Grime notes that Ramanujan's pattern for a(3), a(4), a(5) or prime(3), prime(4), prime(5) cannot be directly extended to prime(6) = 13, and shows solutions for 13, 17, 19.

			All {partition( 5k+4)} are divisible by prime(3) = 5, so a(3) = 5.
All {partition( 7k+5)} are divisible by prime(4) = 7, so a(4) = 7.
All {partition(11k+6)} are divisible by prime(5) = 11, so a(5) = 11.

James Grime and Brady Haran, Partitions, Numberphile video (2016).
Lasse Winquist, An elementary proof of p(11m+6) == 0 (mod 11), J. Combinatorial Theory 6 1969 56--59. MR0236136 (38 #4434).

Cf. A333435 (x_n), A000040 (primes), A000041 (partitions).
Cf. A071734 (p(5k+4)/5), A071746 (p(7k+5)/7), A076394 (p(11k+6)/11).
Cf. A213260 (p(5k+4)).

A333435 Partition numbers A000041(k*x_n + y_n) are known to be divisible by prime(n); sequence gives the list of y_n.

Original entry on oeis.org

4, 5, 6, 237, 2623, 815655

Offset: 3

Frank Ellermann, Mar 21 2020

Grime notes that Ramanujan's pattern for a(3), a(4), a(5) or prime(3), prime(4), prime(5) cannot be directly extended to prime(6) = 13, and shows solutions for 13, 17, 19.

			All {partition( 5k+4)} are divisible by prime(3) = 5, so a(3) = 4.
All {partition( 7k+5)} are divisible by prime(4) = 7, so a(4) = 5.
All {partition(11k+6)} are divisible by prime(5) = 11, so a(5) = 6.

James Grime and Brady Haran, Partitions, Numberphile video (2016).
Lasse Winquist, An elementary proof of p(11m+6) == 0 (mod 11), J. Combinatorial Theory 6 1969 56--59. MR0236136 (38 #4434).

Cf. A333436 (y_n), A000040 (primes), A000041 (partitions).
Cf. A071734 (p(5k+4)/5), A071746 (p(7k+5)/7), A076394 (p(11k+6)/11).
Cf. A213260 (p(5k+4)).

A332772 Numbers k > 0 such that 30k +- 7 is prime.

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 12, 13, 15, 19, 20, 25, 26, 29, 32, 33, 37, 41, 43, 48, 52, 53, 54, 58, 66, 67, 76, 78, 81, 85, 88, 89, 90, 92, 95, 97, 101, 107, 118, 120, 121, 128, 129, 134, 143, 150, 153, 155, 165, 166, 172, 178, 180, 194, 195, 202, 207, 209, 211, 212

Offset: 1

Frank Ellermann, Feb 25 2020

Looking for prime factors > 5=prime(3) in 8=A005867(3) candidates mod 30=A002110(3) two candidates in the form 30k +- 7 with k > 0 never belong to a twin prime pair. Twin primes can be (30k-13, 30k-11) A331840, (30k-1, 30k +1) A176114, or (30k+11, 30k+13) A089160.

			a(4)=4 for prime(30)=113=4*30-7 and prime(31)=127=4*30+7.
a(5)=9 for prime(56)=263=9*30-7 and prime(59)=277=9*30+7.

Frank Ellermann, Illustration for A002110, A005867, A038110, A060753

Cf. A000040, A002110, A005867, A089160, A176114, A331840.

Subsequence of A158573. Prime pairs 30k +- 7 in A329262.

Select[Range@ 215, AllTrue[30 # + {-7, 7}, PrimeQ] &] (* Michael De Vlieger, Feb 25 2020 *)

S = 1
do N = 2 while length( S ) < 255
   if NOPRIME( N * 30 + 7 )   then  iterate N
   if NOPRIME( N * 30 - 7 )   then  iterate N
   S = S || ',' N
end N
say S

A331840 Numbers k such that 30k-13, 30k-11 are twin primes.

Original entry on oeis.org

1, 4, 5, 7, 8, 12, 21, 28, 29, 43, 48, 50, 54, 56, 57, 60, 63, 67, 68, 70, 75, 76, 89, 90, 106, 109, 116, 118, 119, 126, 131, 138, 139, 141, 145, 151, 152, 155, 160, 166, 181, 183, 189, 196, 207, 228, 232, 238, 244, 249, 250, 252, 259, 263, 270, 280, 285, 287

Offset: 1

Frank Ellermann, Feb 26 2020

All twin primes > 7 have the form 30*k-{13,11}, or 30*k +-1 (A176114), or 30*k+{11,13} (A089160).
All twin primes > 7 with least significant decimal digit 7 have the form 30*k-13.
All twin primes > 7 with least significant decimal digit 3 have the form 30*k+13.

			1 is a term because 1*30 - 13 =  17 = prime(6)  and 1*30 - 11 =  19 = prime(7).
4 is a term because 4*30 - 13 = 107 = prime(28) and 4*30 - 11 = 109 = prime(29).
5 is a term because 5*30 - 13 = 137 = prime(33) and 5*30 - 11 = 139 = prime(34).

Cf. A089160, A089161, A176114, A332772.

Cf. A000040, A001097, A002822, A132242, A282323, A282324.

Select[Range[300], And @@ PrimeQ[30*# - {11, 13}] &] (* Amiram Eldar, Feb 29 2020 *)

isok(k) = isprime(30*k-13) && isprime(30*k-11); \\ Michel Marcus, Feb 29 2020

S = 1
do N = 2 while length( S ) < 255
   if NOPRIME( N*30 -13 )  then  iterate N
   if NOPRIME( N*30 -11 )  then  iterate N
   S = S || ',' N
end N
say S

a(n) = A089161(n)+1.

A333058 0, 1, or 2 primes at primorial(n) +- 1.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0

Offset: 0

Frank Ellermann, Mar 06 2020

nonn

a(n) = 0 marks a prime gap size of at least 2*prime(n+1)-1, e.g., primorial(8) +- prime(9) = {9699667,9699713} are primes, gap 2*23-1.
Mathworld reports that it is not known if there are an infinite number of prime Euclid numbers.
The tables in Ondrejka's collection contain no further primorial twin primes after {2309,2311} = primorial(13) +- 1 up to primorial(15877) +- 1 with 6845 digits.

			a(2) = a(3) = a(5) = 2: 2*3 +-1 = {5,7}, 6*5 +-1 = {29,31} and 210*11 +-1 = {2309,2311} are twin primes.
a(1) = a(4) = a(6) = 1: 1, 30*7 - 1 = 209 and 2310*13 + 1 = 30031 are not primes.
a(7) = 0: 510509 = 61 * 8369 and 510511 = 19 * 26869 are not primes.

H. Dubner, A new primorial prime, J. Rec. Math., 21 (No. 4, 1989), 276.

Chris K. Caldwell, the top 20: Primorial, 2012.
H. Dubner & N. J. A. Sloane, Correspondence, 1991, on A005234.
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 30029.
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 9699667.
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations, 2001, tables 20, 20A, 20B.
Eric Weisstein's World of Mathematics, Primorial Prime.
Eric Weisstein's World of Mathematics, Euclid Number.

Cf. A096831, A002110 (primorials, p#), A057706.
Cf. A006862 (Euclid, p#+1), A005234 (prime p#+1), A014545 (index prime p#+1).
Cf. A057588 (Kummer, p#-1), A006794 (prime p#-1), A057704 (index prime p#-1).
Cf. A010051, A088411 (where a(n) is positive), A088257.

p:= proc(n) option remember; `if`(n<1, 1, ithprime(n)*p(n-1)) end:
a:= n-> add(`if`(isprime(p(n)+i), 1, 0), i=[-1, 1]):
seq(a(n), n=0..120);  # Alois P. Heinz, Mar 18 2020

primorial[n_] := primorial[n] = Times @@ Prime[Range[n]];
a[n_] := Boole@PrimeQ[primorial[n] - 1] + Boole@PrimeQ[primorial[n] + 1];
a /@ Range[0, 105] (* Jean-François Alcover, Nov 30 2020 *)

S = ''                     ;  Q = 1
do N = 1 to 27
   Q = Q * PRIME( N )
   T = ISPRIME( Q - 1 ) + ISPRIME( Q + 1 )
   S = S || ',' T
end N
S = substr( S, 3 )
say S                      ;  return S

a(n) = [ isprime(primorial(n) - 1) ] + [ isprime(primorial(n) + 1) ].

a(n) = Sum_{i in {-1,1}} A010051(primorial(n) + i).

A329269 Integers k such that 8*k + 1 is a prime or a square.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 9, 10, 11, 12, 14, 15, 17, 21, 24, 28, 29, 30, 32, 35, 36, 39, 42, 44, 45, 50, 51, 54, 55, 56, 57, 65, 66, 71, 72, 74, 75, 77, 78, 80, 84, 91, 95, 96, 101, 105, 107, 110, 116, 117, 119, 120, 122, 126, 129, 131, 136, 137, 141, 144, 149, 150

Offset: 1

Frank Ellermann, Feb 23 2020

All odd squares have the form 8*n + 1.

			8*0 + 1 =  1 = 1^2, so 0 is a term;
8*1 + 1 =  9 = 3^2, so 1 is a term;
8*2 + 1 = 17 = prime(7), so 2 is a term;
8*3 + 1 = 25 = 5^2, so 3 is a term;
8*4 + 1 = 33 is neither prime nor square, so 4 is not a term;
8*5 + 1 = 41 = prime(13), so 5 is a term.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, theorem 14 and ch. 4.5

Union of the triangular numbers A000217 and A005123.

Cf. A000040, A016754 (odd squares).

q:= k-> (t-> isprime(t) or issqr(t))(8*k+1):
select(q, [$0..200])[];  # Alois P. Heinz, Feb 25 2020

Select[Range[0, 150], PrimeQ[(m = 8*# + 1)] || IntegerQ @ Sqrt[m] &] (* Amiram Eldar, Feb 29 2020 *)

isok(k) = my(x=8*k+1); isprime(x) || issquare(x); \\ Michel Marcus, Feb 27 2020

S = 0 ;  U = 1 ;  P = 1
do N = 1 while length( S ) < 256
   C = 8 * N + 1
   do I = U by 2
      K = I * I      ;  if K > C then  leave I
      U = I          ;  if K < C then  iterate I
      S = S || ',' N ;  iterate N
   end I
   do I = P
      K = PRIME( I ) ;  if K > C then  leave I
      P = I          ;  if K < C then  iterate I
      S = S || ',' N ;  iterate N
   end I
end N
say S ;  return S

A328525 Numbers k such that (k-1)k(k+1) = (k-1)(1+u) = k(1+v) = (k+1)*(1+w) with primes u, v, w.

Original entry on oeis.org

3, 5, 9, 21, 55, 131, 145, 155, 231, 259, 265, 449, 495, 561, 595, 1045, 1051, 1365, 1409, 1491, 1549, 1849, 1989, 2001, 2101, 2469, 2785, 3365, 3621, 3641, 3669, 3845, 3911, 4285, 4951, 5181, 5465, 6049, 6699, 7189, 7229, 8219, 8629, 9175, 9521, 9539, 9631, 9729

Offset: 1

Frank Ellermann, Feb 24 2020

nonn

			3 is a term because 2*3*4 = 2*(1+11) = 3*(1+7) = 4*(1+5) with primes 11, 7, 5.
9 is a term because 8*9*10 = 8*(1+89) = 9*(1+79) = 10*(1+71) with primes 89, 79, 71.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040.

Intersection of A002328, A028870 and A045546.

q:= k-> andmap(isprime, (t-> [t-1, t-k, t+k])(k^2-1)):
select(q, [$1..10000])[];  # Alois P. Heinz, Feb 25 2020

Select[Range[2, 10^4], AllTrue[{(# - 1)*#, #*(# + 1), (# + 1)*(# - 1)} - 1, PrimeQ] &] (* Amiram Eldar, Feb 24 2020 *)

isok(k) = isprime(k*(k+1)-1) && isprime((k+1)*(k-1)-1) && isprime(k*(k-1)-1); \\ Michel Marcus, Feb 25 2020

S = 3
do N = 5 to 595 by 2
   if NOPRIME( N*N +N -1 ) then  iterate N
   if NOPRIME( N*N    -2 ) then  iterate N
   if NOPRIME( N*N -N -1 ) then  iterate N
   S = S || ',' N
end N
say S

More terms from Amiram Eldar, Feb 24 2020

A333355 Number of bits in binary expansion of n minus the number of digits of n when written in base 3.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Frank Ellermann, Mar 15 2020

Record highs are at n = 2^A054414. All n=2^k >= 2 are increases, all n=3^j are decreases, and there is either one or none 3^j between 2^(k-1) and 2^k. When one, a(2^k) = a(2^(k-1)) so not a record high. When none, a(2^k) = a(2^(k-1)) + 1 which is a record high. If 2^k and 2^(k-1) are the same length in ternary then there is no 3^j between them. This is when 2^k has most significant ternary digit 2 since 2^(k-1) >= 3^j is 2^k >= 2*3^j. These k are A054414. Non-record increases are at its complement n = 2^A020914 >= 2. - Kevin Ryde, Apr 04 2020

			a(8) = 2 = 4 - 2 for binary 1000 and ternary 22.
a(64) = 3 = 7 - 4 for binary 1000000 and ternary 2101.

Cf. A007088 ( binary), A000523 (floor(log_2(n))), A029837.

Cf. A007089 (ternary), A062153 (floor(log_3(n))), A117966.

a:= n-> ilog[2](n)-ilog[3](n):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 15 2020

a[n_]: = Floor @ Log[2, n] - Floor @ Log[3, n]; Array[a, 100] (* Amiram Eldar, Mar 16 2020 *)

a(n) = logint(n,2) - logint(n,3); \\ Kevin Ryde, May 15 2020

L = 1 ;  M = 1 ;  B = 2 ;  T = 3       ;  S = 0
do N = 2 while length( S ) < 258
   if B = N then  do    ;  B = B * 2   ;  L = L + 1   ;  end
   if T = N then  do    ;  T = T * 3   ;  M = M + 1   ;  end
   S = S || ',' L - M
end N
say S                   ;  return S

a(n) = A000523(n) - A062153(n) = floor(log_2(n)) - floor(log_3(n)).

a(n) = length(A007088(n)) - length(A007089(n)).

A332799 Numbers whose smallest prime factor is 17.

Original entry on oeis.org

17, 289, 323, 391, 493, 527, 629, 697, 731, 799, 901, 1003, 1037, 1139, 1207, 1241, 1343, 1411, 1513, 1649, 1717, 1751, 1819, 1853, 1921, 2159, 2227, 2329, 2363, 2533, 2567, 2669, 2771, 2839, 2941, 3043, 3077, 3247, 3281, 3349, 3383, 3587, 3791, 3859, 3893

Offset: 1

Frank Ellermann, Feb 24 2020

The asymptotic density of this sequence is 192/17017. - Amiram Eldar, Dec 06 2020

			a(2) = 17*17, a(3) = 17*19.

Emmanuel Desurvire, Classical and Quantum Information Theory: An Introduction for the Telecom Scientist, Cambridge University Press, 2009, table 20.5 p. 421.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A084967 (5), A084968 (7), A084969 (11), A084970 (13), A332798 (19), A332797 (23), A008366 (17-rough numbers).

17 * Select[Range[230], CoprimeQ[#, 30030] &] (* Amiram Eldar, Feb 24 2020 *)

P = 17         ;  S = P
do N = P by 2 while length( S ) < 255
   do I = 1 until P = X
      X = PRIME( I )
      if P = X       then  leave I
      if N // X = 0  then  iterate N
   end I
   S = S || ',' P*N
end N
say S          ;  return S

a(n) = 17*A008366(n).

cp's OEIS Frontend

User: Frank Ellermann

A333772 a(n) = n * 2^n * (n!)^2.

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A333436 Partition numbers A000041(k*x_n + y_n) are known to be divisible by prime(n); sequence gives the list of x_n.

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A333435 Partition numbers A000041(k*x_n + y_n) are known to be divisible by prime(n); sequence gives the list of y_n.

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A332772 Numbers k > 0 such that 30k +- 7 is prime.

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A331840 Numbers k such that 30*k-13, 30*k-11 are twin primes.

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A333058 0, 1, or 2 primes at primorial(n) +- 1.

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A329269 Integers k such that 8*k + 1 is a prime or a square.

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A328525 Numbers k such that (k-1)*k*(k+1) = (k-1)*(1+u) = k*(1+v) = (k+1)*(1+w) with primes u, v, w.

A331840 Numbers k such that 30k-13, 30k-11 are twin primes.

A328525 Numbers k such that (k-1)k(k+1) = (k-1)(1+u) = k(1+v) = (k+1)*(1+w) with primes u, v, w.