Giacomo Fecondo - cp's OEIS frontend

A177082 a(n) = 2nA071148(n).

6, 32, 90, 208, 390, 672, 1050, 1568, 2286, 3160, 4290, 5664, 7254, 9128, 11370, 14016, 16966, 20376, 24206, 28400, 33138, 38368, 44206, 50784, 57950, 65624, 73926, 82768, 92278, 103080, 114638, 127104, 140250, 154632, 169750, 185904, 203130, 221312, 240630

Offset: 1

Giacomo Fecondo, Dec 09 2010

nonn

a(n) is the sum of all elements of the n X n matrix M(i,j) = prime(i+1)+prime(j+1).

[For n<= 23, only five matrices (with n=1, n=2, n=3, n=5 and n=7) contain all the even numbers starting from 6 and ending with 2*prime(n+1), the maximum element. If the prime gap prime(n+1)-prime(n) is larger than 2, the even term 2*prime(n+1)-2 is missing in the matrix; the difference equal 2 between prime(n) and prime(n-1) is not a sufficient condition to have a complete set of even numbers in the range 6 .. 2*prime(n+1) in the matrix.]

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A071148, A167214.

seq(n)={2*Vec(deriv((Ser(primes(n+1))-2)/(1-x)))} \\ Andrew Howroyd, Jan 14 2020

A177082 = lambda n: 2*n*(sum(primes_first_n(n+1))-2) # D. S. McNeil, Dec 18 2010

Terms a(25) and beyond from Andrew Howroyd, Jan 14 2020

A178509 Smallest value of k for which 6k+1 divides the subset of centered hexagonal terms included in A177019 that admit only factors like 6k+1.

Original entry on oeis.org

1, 55, 26, 5, 50005000, 1, 1, 16, 1936, 500000000500000000, 15333927, 1, 1, 18316, 3, 7, 1526, 1, 1, 12, 73, 38, 47, 1, 1, 121, 43502, 12, 11, 1, 1, 18, 3, 5, 10, 1, 1, 481, 2043419605725853, 921, 3835, 1, 1, 12, 10, 13, 25, 1, 1, 18, 3, 12, 62, 1, 1, 76, 398, 7

Offset: 0

Giacomo Fecondo, May 29 2010, May 30 2010

nonn

The terms a(0), a(1), a(4) and a(9) confirm the primality of the terms included in A160432;

k assumes the value 1 when the value of n in a(n) is equal to 6*i or 6*i-1 where i is a positive integer.

			a(0)= 1 so 6*1+1 = 7 is the minimum factor dividing 7; a(1)= 55 so 6*55+1 = 331 the minimum factor dividing 331; a(2)= 26 so 6*26+1 = 157 the minimum factor dividing 30301; a(3)= 5 so 6*5+1 = 31 the minimum factor dividing 3003001; a(10)=15333927 so 6*15333927+1 = 92003563 the minimum factor dividing 3*10^20+3*10^10+1.

Cf. A177019, A160432.

A160432 Primes of the form 310^(2n) + 3*10^n + 1.

Original entry on oeis.org

7, 331, 300030001, 3000000003000000001

Offset: 1

Giacomo Fecondo, May 13 2009

nonn

Primes of the form (x^3-y^3)/(x-y) with x = y+1 (which gives A002407) and also y=10^k for some k.
These prime numbers (differences of consecutive cubes: A002407), for k>0, have only three digits different from zero. The first is 3, the middle digit is 3 and the final digit is 1. The other 2(k-1) digits are value 0.
If k=6*i or k=6*i-1 the number is always divisible by 7. [Giacomo Fecondo, May 22 2010]

			a(1) = 7 = (10^0+1)^3 -(10^0)^3 , 2^3-1^3.
a(2) = 331 =(10^1+1)^3 -(10^1)^3, 11^3-10^3.
a(3) = 300030001 = (10^4+1)^3 - (10^4)^3, 10001^3-10000^3.
a(1)= 3t(t+1)+1 with t=10^0; a(2)= 3t(t+1)+1 with t=10^1; a(3)= 3t(t+1)+1 with t=10^4.
For k=102 (k=6*17) the number (10^102+1)^3-(10^102)^3 is divisible by 7; for k=101 (k=6*17-1) the number (10^101+1)^3-(10^101)^3 is divisible by 7. [_Giacomo Fecondo_, May 22 2010]

Cf. A002407, A003215.

[a: n in [0..30] | IsPrime(a) where a is 3*10^(2*n) + 3*10^n + 1]; // Vincenzo Librandi, Jan 28 2013

Select[Table[3*10^(2 n) + 3*10^n + 1, {n, 0, 1000}], PrimeQ] (* Vincenzo Librandi, Jan 28 2013 *)

A160432(n,print_all=0,Start=0,Limit=9e9)={for(k=Start,Limit,ispseudoprime(p=3*100^k+3*10^k+1) & !(print_all & print1(p",")) & !n-- & return(p))} \\ - M. F. Hasler, Jan 28 2013

New name from Vincenzo Librandi, Jan 28 2013

A159961 Cuban composites: composite numbers equal to the difference of two consecutive cubes.

Original entry on oeis.org

91, 169, 217, 469, 721, 817, 1027, 1141, 1261, 1387, 1519, 2107, 2611, 2977, 3367, 3781, 3997, 4681, 4921, 5677, 5941, 6487, 6769, 7651, 7957, 8587, 8911, 9577, 9919, 10621, 10981, 11347, 12481, 12871, 14077, 14491, 14911, 15337, 15769, 16207, 17101, 17557

Offset: 1

Giacomo Fecondo, Apr 28 2009

nonn

Analogous to the cuban primes A002407, but select the composite numbers rather than the primes.
Cuban composites are a subset of hexagonal centered numbers.
A cuban composite has an integer divisor of the form 6*k+1 other than 1 and itself.
Also, composite numbers of the form (n^2 + nm + m^2) where n and m are consecutive numbers. - K. D. Bajpai, Jun 12 2014

			a(1) = 91 = 1+3t*(t+1) with t = 5 is the smallest cuban composite number. Note that 91 = 7*13, so its factors have the form 6k+1, in fact 7 = 6*1+1.

K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A003215, A002407, A221717.

nn = 200; Select[Table[3 x^2 + 3 x + 1, {x, nn}], ! PrimeQ[#] &] (* T. D. Noe, Jan 30 2013 *)
Select[Table[m=n+1;( n^2 + n m + m^2),{n,100}],!PrimeQ[#]&] (* K. D. Bajpai, Jun 12 2014 *)
Select[Differences[Range[80]^3],CompositeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 07 2018 *)

a(1)=1+3t*(t+1) with t=5, a(2)=1+3t*(t+1) with t=7.

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User: Giacomo Fecondo

A177082 a(n) = 2*n*A071148(n).

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A178509 Smallest value of k for which 6*k+1 divides the subset of centered hexagonal terms included in A177019 that admit only factors like 6*k+1.

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A160432 Primes of the form 3*10^(2*n) + 3*10^n + 1.

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Examples

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Magma

Mathematica

PARI

Extensions

A159961 Cuban composites: composite numbers equal to the difference of two consecutive cubes.

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Author

Keywords

Comments

Examples

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Programs

Mathematica

Formula

A177082 a(n) = 2nA071148(n).

A178509 Smallest value of k for which 6k+1 divides the subset of centered hexagonal terms included in A177019 that admit only factors like 6k+1.

A160432 Primes of the form 310^(2n) + 3*10^n + 1.