A138666 -id:A138666 - cp's OEIS frontend

A076768 Positive integers not expressible as the sum of a prime and a triangular number.

1, 36, 105, 171, 210, 216, 325, 351, 406, 528, 561, 630, 741, 780, 990, 1081, 1176, 1275, 1596, 1711, 1830, 1953, 2016, 2145, 2346, 2628, 2775, 3003, 3081, 3240, 3321, 3655, 3741, 3916, 4278, 4371, 4465, 4560, 4851, 5253, 5460, 5565, 5886, 6105, 6216, 6786, 7021, 7140, 7503, 7626, 7750, 7875, 8256, 8515, 8911, 9045, 9591, 9870

Offset: 1

Jason Earls, Nov 14 2002

nonn

It appears that 1,2,3,8 are the only positive integers that cannot be partitioned as the sum of a semiprime and a triangular number. Here triangular numbers include t(0)=0 and t(1)=1. - Jonathan Vos Post and Ray Chandler, Nov 28 2004

This sequence contains 216 (and possibly other nontriangular numbers) together with an infinite number of triangular numbers. The indices of the triangular numbers are in A138666. This is related to the Sun's conjecture (see A132399) that every number except 216 is the sum of a triangular number and a prime or 0. - T. D. Noe, Mar 26 2008

			a(2) = 36 is an element of this sequence because 36 cannot be written as a sum of one of the primes <= 36 {2,3,5,7,11,13,17,19,23,29,31} and one of the triangular numbers <= 36 {1,3,6,10,15,21,28,36}. - corrected (added 28) by _Gionata Neri_, May 02 2015

T. D. Noe, Table of n, a(n) for n=1..1001

Cf. A000040, A000217, A046903.

Complement[Range[9871],Total/@Tuples[{Prime[Range[1220]],Accumulate[ Range[ 0,140]]}]] (* Harvey P. Dale, Jul 30 2019 *)

Added the terms 6786 through 9870 and conjecture that there are no further terms - Jonathan Vos Post and Ray Chandler, Nov 28 2004

Added "positive" to the name - Alex Ratushnyak, Apr 04 2013

A087401 Triangle of n*r-binomial(r+1,2).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 2, 3, 3, 0, 3, 5, 6, 6, 0, 4, 7, 9, 10, 10, 0, 5, 9, 12, 14, 15, 15, 0, 6, 11, 15, 18, 20, 21, 21, 0, 7, 13, 18, 22, 25, 27, 28, 28, 0, 8, 15, 21, 26, 30, 33, 35, 36, 36, 0, 9, 17, 24, 30, 35, 39, 42, 44, 45, 45, 0, 10, 19, 27, 34, 40, 45, 49, 52, 54, 55, 55, 0, 11

Offset: 0

Paul Boddington, Oct 21 2003

There is a curious connection with the character tables of cyclic groups of prime power order. Let G be a cyclic group of order p^n where p is prime and n is nonnegative. Construct an (n+1)x(n+1) matrix A whose rows and columns are indexed by the set 0,1,...,n as follows. The ij entry is obtained by taking any element of order p^(n-j) in G and summing its character values over all characters of order p^i in the dual group of G. Remarkably, all coefficients of the characteristic polynomial of A are powers of p (with alternating signs) and these powers can be read off from the appropriate row of our triangle. For example if n=2 then the characteristic polynomial is X^3 - p^2*X^2 + p^3*X - p^3.

			  0
  0   0
  0   1   1
  0   2   3   3
  0   3   5   6   6
  0   4   7   9  10  10
  0   5   9  12  14  15  15
  0   6  11  15  18  20  21  21
  0   7  13  18  22  25  27  28  28
  0   8  15  21  26  30  33  35  36  36
  0   9  17  24  30  35  39  42  44  45  45

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Cf. A138666.

a087401 n k = a087401_tabl !! n !! k
a087401_row n = a087401_tabl !! n
a087401_tabl = iterate f [0] where
   f row = row' ++ [last row'] where row' = zipWith (+) row [0..]
-- Reinhard Zumkeller, Oct 03 2012

A087401 := proc(n,k)
    n*k-binomial(k+1,2) ;
end proc:
seq(seq( A087401(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jan 21 2015

Table[n*r-Binomial[r+1,2],{n,0,20},{r,0,n}]//Flatten (* Harvey P. Dale, Jul 10 2020 *)

T(0,0)=0 and for n>0: T(n,k)=T(n-1,k)+k for kReinhard Zumkeller, Oct 03 2012

A241884 Numbers n such that n - n^2/k^2 is not prime for all k < n.

Original entry on oeis.org

1, 2, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 99, 100

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 30 2014

nonn

Conjecture: numbers n such that n-1 is not a prime number. - Vincenzo Librandi, Jul 16 2016

Counterexamples to conjecture: 56, 306, 552, 870, ... are not in the sequence. These are p^2+p where p is prime but p^2+p-1 is not prime. - Robert Israel, Jul 04 2017

			56 is not in the sequence because 56 - 56^2/8^2 = 7 is prime for k = 8.

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

Cf. A008864, A036690, A138666 (numbers n such that 2n - n/k is not prime for all k), A242221.

filter:= n -> andmap(t -> not isprime(n - t^2), numtheory:-divisors(n)):
select(filter, [$1..1000]); # Robert Israel, Jul 04 2017

isOK(n) = fordiv(n, k, if(isprime(n-(n/k)^2), return(0))); 1
s=[]; for(n=1, 100, if(isOK(n), s=concat(s, n))); s \\ Colin Barker, May 16 2014

A242221 Numbers n such that n - n^2/m^2 and 2n - n/m are not prime for all m.

Original entry on oeis.org

1, 25, 26, 28, 33, 35, 39, 46, 50, 58, 63, 65, 77, 78, 81, 85, 86, 88, 92, 93, 94, 95, 105, 111, 116, 118, 119, 122, 123, 124, 125, 130, 133, 134, 143, 144, 145, 146, 148, 153, 155, 160, 161, 162, 165, 170, 171, 172, 176, 178, 183, 185, 186, 188, 189, 196, 202

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, May 07 2014

nonn

Intersection of A241884 and A138666.

			26 is in this sequence because:
1) 26 - 26^2/1^2 = -650 and 2*26 - 26/1 = 26 both not prime for m = 1,
2) 26 - 26^2/2^2 = -143 and 2*26 - 26/2 = 39 both not prime for m = 2,
3) 26 - 26^2/13^2 = 22 and 2*26 - 26/13 = 50 both not prime for m = 13,
4) 26 - 26^2/26^2 = 25 and 2*26 - 26/26 = 51 both not prime for m = 26.

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

Cf. A138666, A241884.

filter:= proc(n) andmap(t -> not isprime(n - n^2/t^2) and not isprime(2*n - n/t), numtheory:-divisors(n)) end proc:
select(filter, [$1..200]); # Robert Israel, Jul 03 2017

filterQ[n_] := AllTrue[Divisors[n], !PrimeQ[n - n^2/#^2] && !PrimeQ[2n - n/#]&];
Select[Range[200], filterQ] (* Jean-François Alcover, Jul 27 2020, after Maple *)

f(n)=fordiv(n, m, if(isprime(n-n^2/m^2), return(0))); 1
g(n)=fordiv(n, m, if(isprime(2*n-n/m), return(0))); 1
for(n=1, 200, if(f(n) && g(n), print1(n, ", "))) \\ Colin Barker, May 08 2014

More terms from Colin Barker, May 08 2014

Example corrected by Colin Barker, May 09 2014

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A076768 Positive integers not expressible as the sum of a prime and a triangular number.

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A087401 Triangle of n*r-binomial(r+1,2).

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A241884 Numbers n such that n - n^2/k^2 is not prime for all k < n.

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A242221 Numbers n such that n - n^2/m^2 and 2n - n/m are not prime for all m.

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