A027861 -id:A027861 - cp's OEIS frontend

A309332 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the sum of two positive triangular numbers.

0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 2, 0, 3, 0, 0, 1, 1, 3, 0, 0, 1, 0, 1, 0, 0, 3, 1, 1, 0, 1, 3, 0, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 2, 1, 1, 1, 1, 2, 1, 0, 3, 1, 1, 1, 0, 3, 1, 1, 0, 0, 2, 0, 1, 1, 1, 1, 1, 5, 0, 1, 1, 0, 1, 0, 0, 3, 0, 3, 1, 0, 3, 1, 3, 1, 3, 3, 0, 1, 0, 0, 3, 0, 2, 0, 1

Offset: 1

Alois P. Heinz, Aug 01 2019

nonn

The order doesn't matter. 21 = 6+15 = 15+6 are not counted as distinct solutions. - N. J. A. Sloane, Feb 22 2020

			a(3) = 1: 2*3/2 + 2*3/2 = 3*4/2.
a(21) = 2: 6*7/2 + 20*21/2 = 12*13/2 + 17*18/2 = 21*22/2.
a(23) = 3: 9*10/2 + 21*22/2 = 11*12/2 + 20*21/2 = 14*15/2 + 18*19/2 = 23*24/2.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000217, A001652, A012132, A027861, A046080 (the same for squares), A053141, A062301 (the same for primes), A108769, A309507.

a:= proc(n) local h, j, r, w; h, r:= n*(n+1), 0;
      for j from n-1 by -1 do w:= j*(j+1);
        if 2*w

a[n_] := Module[{h = n(n+1), j, r = 0, w}, For[j = n-1, True, j--, w = j(j+1); If[2w < h, Break[]]; If[ IntegerQ[Sqrt[4(h-w)+1]], r++]]; r];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Nov 16 2022, after Alois P. Heinz *)

a(n) > 0 <=> n in { A012132 }.
a(n) = 0 <=> n in { A027861 }.
a(n) = 1 <=> n in { A108769 }.

A378898 a(n) is the least k > 0 such that (n+k)^2 + n^2 is prime.

Original entry on oeis.org

1, 1, 5, 1, 1, 5, 1, 5, 1, 3, 3, 1, 7, 1, 7, 3, 1, 5, 1, 3, 5, 1, 7, 1, 1, 5, 5, 5, 1, 1, 13, 1, 7, 1, 1, 13, 3, 7, 1, 3, 3, 1, 5, 5, 7, 3, 1, 5, 25, 1, 5, 5, 5, 5, 3, 5, 11, 5, 5, 1, 3, 3, 17, 7, 1, 5, 13, 27, 1, 1, 13, 1, 27, 5, 19, 9, 3, 5, 1, 9, 19, 1, 5, 1, 1, 9, 1, 15, 7, 1, 3, 3, 5, 5, 7

Offset: 1

Robert Israel, Dec 11 2024

nonn

			a(3) = 5 because (3+5)^2 + 3^2 = 73 is prime, and no smaller number works.

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

Cf. A027861 (a(n) = 1), A089489, A378945, A378946.

f:= proc(n) local k;
  for k from n+1 by 2 do
    if igcd(k,n) = 1 and isprime(k^2 + n^2) then return k-n fi
  od
end proc;
map(f, [$1..100]);

a(n) = my(k=1); while (!isprime((n+k)^2 + n^2), k++); k; \\ Michel Marcus, Dec 11 2024

a(n) = A089489(n) - n.

A089619 a(n) = greatest prime factor of n^2 + (n+1)^2 for n >= 1.

Original entry on oeis.org

5, 13, 5, 41, 61, 17, 113, 29, 181, 17, 53, 313, 73, 421, 37, 109, 613, 137, 761, 29, 37, 1013, 17, 1201, 1301, 281, 89, 13, 1741, 1861, 397, 2113, 449, 2381, 2521, 41, 97, 593, 3121, 193, 53, 3613, 757, 233, 101, 173, 4513, 941, 29, 5101, 1061, 149, 229, 457, 101

Offset: 1

Cino Hilliard, Dec 31 2003

			2*7^2 - 2*7 + 1 = 85 = 5*17, so a(7) = 17.

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

Cf. A001844, A006530, A027861, A027862.

a[n_] := FactorInteger[n^2 + (n+1)^2][[-1, 1]]; Array[a, 60] (* Amiram Eldar, Oct 29 2024 *)

xnpym1n(m) = { for(n=1,m, y = n^2+(n+1)^2; f = factor(y); l = length(component(f,1)); v = component(component(f,1),l); print1(v","); ) }

a(n) = A006530(A001844(n)).

Edited by Ray Chandler, Jan 03 2004

Offset corrected by Georg Fischer, May 27 2024

A108769 Numbers m such that m^2 + (m+1)^2 is a semiprime.

Original entry on oeis.org

3, 6, 8, 10, 11, 13, 15, 16, 18, 20, 26, 27, 31, 33, 37, 38, 40, 43, 44, 45, 48, 51, 52, 54, 55, 56, 57, 59, 62, 63, 64, 67, 68, 73, 74, 75, 76, 77, 80, 81, 83, 89, 92, 94, 98, 105, 107, 111, 112, 113, 114, 117, 120, 123, 124, 129, 131, 133, 134, 138, 140, 141, 142, 143

Offset: 1

Jason Earls, Jun 25 2005

Numbers m such that A099776(m) is a semiprime. - Michel Marcus, Nov 17 2022

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A001358, A027861, A099776.

a:= proc(n) local k; for k from 1+`if`(n=1, 0, a(n-1))
      while (t-> isprime(t) or numtheory[bigomega](t)
      >2)(2*k*(k+1)+1) do od: k
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Aug 01 2019

Select[Range[1000], PrimeOmega[#^2 + (#+1)^2] == 2&] (* Jean-François Alcover, Nov 17 2022 *)

isok(m) = bigomega(m^2 + (m+1)^2) == 2; \\ Michel Marcus, Nov 17 2022

A158526 n and (1 + 2n + 2n^2) are primes.

Original entry on oeis.org

2, 5, 7, 17, 19, 29, 47, 79, 97, 109, 137, 139, 149, 157, 167, 199, 229, 347, 349, 389, 409, 467, 479, 547, 577, 599, 709, 719, 757, 857, 929, 937, 967, 1039, 1069, 1087, 1187, 1229, 1259, 1399, 1409, 1447, 1559, 1579, 1597, 1607, 1657, 1697, 1699, 1709

Offset: 1

Zak Seidov, Mar 20 2009

Numbers n such that A048395(n) is semiprime, or A048395(n)/n is prime.

Or, primes in A027861. Also, (1+2*n+2*n^2) are in A027862. - Zak Seidov, Sep 19 2015

			A048395(2)=26=2*13, A048395(5)=305=5*61, A048395(7)=791=7*113.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A048395 (sum of consecutive nonsquares), A001358 (semiprimes).

Cf. A027861, A027862.

[p: p in PrimesUpTo(2000) | IsPrime(2*p^2+2*p+1)]; // Vincenzo Librandi, Jun 22 2014

f[n_]:=PrimeQ[n^2+(n+1)^2];lst={};Do[p=Prime[n];If[f[p],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)

forprime(p=2,1e4,if(isprime(2*p*(p+1)+1),print1(p","))) \\ Charles R Greathouse IV, Sep 09 2009

A218213 Number of n-digit primes representable as sums of consecutive squares.

Original entry on oeis.org

1, 4, 13, 30, 69, 187, 519, 1401, 3889, 10861, 31640, 90735

Offset: 1

Martin Renner, Oct 23 2012

There are no common representations of two, three or six squares for n < 13, so

a(n) = A218207(n) + A218209(n) + A218211(n); n < 13.

Cf. A027861, A027862, A027863, A027864, A027866, A027867, A218207, A218209, A218211.

nn = 8; nMax = 10^nn; t = Table[0, {nn}]; Do[k = n; s = 0; While[s = s + k^2; s <= nMax, If[PrimeQ[s], t[[Ceiling[Log[10, s]]]]++];  k++], {n, Sqrt[nMax]}]; t (* T. D. Noe, Oct 23 2012 *)

a(n) = A218214(n) - A218213(n-1).

A242927 Numbers m such that k^m + (k+1)^m + ... + (k+m-1)^m is prime for some k.

Original entry on oeis.org

1, 2, 6, 42, 1806

Offset: 1

Derek Orr, May 26 2014

a(5) > 500. For m values < 500 not listed above, k has been checked for k <= 5000.
For the first four terms, the least k that makes k^m + (k+1)^m + ... + (k+m-1)^m prime is {2, 1, 4, 99} respectively.
For a(5) = 1806, k = 3081 yields a strong PRP with 6663 digits. - Don Reble, Mar 23 2018
The known terms a(1..5) coincide with the finite sequence A014117. - M. F. Hasler, May 20 2019

			k^1 = k is prime for k = 2 or any other prime (cf. A000040), so 1 is a term of this sequence.
k^2 + (k+1)^2 is prime for some k (e.g., k = 2 yields 13, see A027861 for the full list), so 2 is a term of this sequence.
k^3 + (k+1)^3 + (k+2)^3 = 3*(k+1)*(k^2+2*k+3) is never prime, therefore 3 is not a term of this sequence.
Similarly, the corresponding expression for m = 4 and m = 5 is a multiple of 2 and 5, respectively, and for all m = 7, ..., 41, the expression also shares a factor with m (and thus is a multiple of m whenever m is prime).
Index m = 110 is the smallest m > 42 for which the expression is not algebraically composite (the polynomial in k has content 1 and is irreducible over Q), but it does factor as (k(k+1)(k+2)(k+3)(k+4))^10 over Z_5, so is always a multiple of 5. Index m = 210 is the next one which is a similar case.
Index m = 231 is much like m = 110, but with a factor 7 instead of 5.
Index m = 330 again yields an irreducible polynomial with content 1, but as before one can show that it is always divisible by 5. And so on.

Cf. A000040, A027861, A014117.

k(n)=for(k=1,5000,if(ispseudoprime(sum(i=0,n-1,(k+i)^n)),return(k)))
for(n=1,500,if(k(n),print(n)))  \\ Edited by M. F. Hasler, Mar 23 2018

a(5) from Don Reble, Mar 23 2018

Example corrected and extended by M. F. Hasler, Apr 05 2018

A108809 Numbers n such that both n+(n-1)^2 and n+(n+1)^2 are primes.

Original entry on oeis.org

2, 3, 4, 7, 9, 15, 18, 25, 34, 55, 58, 63, 67, 100, 102, 139, 144, 148, 154, 162, 163, 168, 190, 195, 219, 232, 247, 267, 280, 289, 330, 349, 379, 384, 417, 427, 448, 454, 477, 568, 580, 643, 645, 669, 672, 727, 762, 793, 802, 813, 837, 847, 900, 975, 988, 993

Offset: 1

Walter Kehowski, Jul 04 2005

			34 is in the sequence because 34 + 33^2 = 1123 and 34 + 35^2 = 1259 are both prime.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A027861.

Intersection of A055494 and A094210. - Michel Marcus, Feb 08 2017

L:=[]; for k from 1 to 1000 do if isprime(k+(k-1)^2) and isprime(k+(k+1)^2) then L:=[op(L),k] fi od;

Select[Range@1000, PrimeQ[#^2 - # + 1] && PrimeQ[#^2 + 3 # + 1] &] (* Ivan Neretin, Feb 08 2017 *)

isok(n) = isprime(n+(n-1)^2) && isprime(n+(n+1)^2); \\ Michel Marcus, Feb 08 2017

A134002 Positive integers n such that n(n+5)=a(a+5)+b(b+5) is solvable in positive integers.

Original entry on oeis.org

5, 10, 11, 13, 15, 16, 20, 23, 24, 25, 30, 31, 33, 35, 36, 37, 38, 40, 42, 45, 46, 47, 49, 50, 55, 57, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 75, 76, 80, 81, 84, 85, 86, 88, 89, 90, 92, 95, 98, 99, 100, 101, 102, 105, 108, 110, 111, 112, 114, 115, 118, 120, 124, 125

Offset: 1

John W. Layman, Oct 01 2007

nonn

Conjecture. If n a positive integer not a term of this sequence, then n^2+(n+5)^2 is prime. (This has been verified up to n=500.) Examples. For n=1,2,3,4,6,7, n^2+(n+5)^2 is 37, 53,73, 97, 157 and 193, each of which is prime. See A134003 for the complement of this sequence.

			5(5+5)=50=14+36=2(2+5)+4(4+5), so 5 is a term of the sequence.

Cf. A012132, A027861, A134003.

A134003 Positive integers n for which n^2+(n+5)^2 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 17, 18, 19, 21, 22, 26, 27, 28, 29, 32, 34, 39, 41, 43, 44, 48, 51, 52, 53, 54, 56, 58, 59, 66, 72, 74, 77, 78, 79, 82, 83, 87, 91, 93, 94, 96, 97, 103, 104, 106, 107, 109, 113, 116, 117, 119, 121, 122, 123, 126, 134, 136, 137, 144, 151, 157

Offset: 1

John W. Layman, Oct 01 2007

nonn

Conjecture. If n is in this sequence then n(n+5)=a(a+5)+b(b+5) is not solvable in integers. (This has been verified up to n=500.) See A134002 for the complement of this sequence (in the positive integers).

Cf. A012132, A027861, A134002.

Select[Range[1,200],PrimeQ[#^2+(#+5)^2]&] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

cp's OEIS Frontend

A309332 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the sum of two positive triangular numbers.

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A378898 a(n) is the least k > 0 such that (n+k)^2 + n^2 is prime.

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A089619 a(n) = greatest prime factor of n^2 + (n+1)^2 for n >= 1.

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A108769 Numbers m such that m^2 + (m+1)^2 is a semiprime.

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A158526 n and (1 + 2*n + 2*n^2) are primes.

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A218213 Number of n-digit primes representable as sums of consecutive squares.

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A242927 Numbers m such that k^m + (k+1)^m + ... + (k+m-1)^m is prime for some k.

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A108809 Numbers n such that both n+(n-1)^2 and n+(n+1)^2 are primes.

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A158526 n and (1 + 2n + 2n^2) are primes.