A115882 -id:A115882 - cp's OEIS frontend

A115883 The n-th prime minus n gives a triangular number.

1, 2, 4, 5, 7, 13, 34, 37, 46, 62, 104, 111, 210, 259, 274, 296, 306, 439, 488, 502, 513, 751, 763, 817, 969, 998, 1132, 1231, 1405, 1586, 1849, 1982, 2107, 2488, 2578, 2695, 2732, 2752, 2989, 3008, 3079, 3322, 3958, 4201, 4628, 5035, 5594, 5722, 5929

Offset: 1

Giovanni Resta, Feb 06 2006

nonn

			prime(13)-13 = 41-13 = 28 = T(7).

Harvey P. Dale and T. D. Noe, Table of n, a(n) for n = 1..1000 (first 500 terms from Harvey P. Dale)

Cf. A064370, A115882, A115887.

Flatten[Position[Table[Prime[n]-n,{n,6000}],?(IntegerQ[(Sqrt[ 8#+1]- 1)/2]&)]] (* _Harvey P. Dale, Oct 08 2012 *)
pntQ[{a_,b_}]:=OddQ[Sqrt[8(a-b)+1]]; Module[{nn=6000},Select[Thread[ {Prime[ Range[nn]],Range[nn]}],pntQ]][[All,2]] (* Harvey P. Dale, Dec 20 2019 *)

isok(n) = ispolygonal(prime(n) - n, 3); \\ Michel Marcus, Jan 25 2014

A115907 Numbers k such that sigma(k) plus the k-th prime is a triangular number.

Original entry on oeis.org

1, 2, 9, 13, 21, 80, 112, 129, 147, 225, 308, 349, 1063, 1282, 1300, 1635, 1880, 2686, 2956, 3517, 3759, 3907, 4058, 5025, 5196, 6742, 7606, 7708, 9244, 10154, 10596, 11259, 11719, 12687, 13167, 13599, 14081, 14624, 15937, 16437, 16766, 18289

Offset: 1

Giovanni Resta, Feb 06 2006

nonn

			sigma(1880) + prime(1880) = 20503 = T(202).

David A. Corneth, Table of n, a(n) for n = 1..11241 (first 1000 from Robert Israel)

Cf. A000040, A000203, A000217, A115882, A115908.

filter:= n -> issqr(1+8*(numtheory:-sigma(n)+ithprime(n))):
select(filter, [$1..10^5]); # Robert Israel, Nov 07 2018

Select[Range[25000],IntegerQ[(Sqrt[8(DivisorSigma[1,#]+Prime[#])+ 1]-1)/2]&] (* Harvey P. Dale, Jun 07 2011 *)

upto(n) = my(t = 0, res = List(), u = prime(n)); forprime(p = 2, u, t++; c = p + sigma(t); if(ispolygonal(c, 3), listput(res, t))); res \\ David A. Corneth, Nov 08 2018

A115908 Numbers k such that phi(k) + prime(k) is a triangular number.

Original entry on oeis.org

1, 5, 6, 15, 23, 141, 150, 258, 317, 340, 359, 378, 471, 503, 768, 1363, 2175, 2516, 3285, 3342, 3815, 3905, 4385, 4748, 5385, 5976, 6026, 6900, 7048, 7151, 7411, 7698, 8251, 8310, 8828, 10182, 10461, 12022, 12508, 13722, 13840, 15456, 16582

Offset: 1

Giovanni Resta, Feb 06 2006

nonn

			phi(503) + prime(503) = 4095 = T(90).

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

Cf. A000010, A115882, A115907.

With[{trnos = Accumulate[Range[5000]]}, Select[Range[20000], MemberQ[trnos, (EulerPhi[#] + Prime[#])] &]] (* Harvey P. Dale, Jan 12 2011 *)
Select[Range[20000],OddQ[Sqrt[8(EulerPhi[#]+Prime[#])+1]]&] (* Harvey P. Dale, Dec 08 2024 *)

isok(n) = ispolygonal(prime(n) + eulerphi(n), 3); \\ Michel Marcus, Jan 25 2014

A115886 Triangular numbers equal to the sum of a prime number with its index.

Original entry on oeis.org

3, 91, 276, 378, 666, 1225, 2080, 2145, 2278, 3486, 4186, 4560, 5565, 8646, 11628, 17020, 18528, 21115, 22791, 23220, 25425, 27730, 32385, 34716, 41905, 42195, 43365, 56616, 69751, 74691, 79003, 93528, 93961, 94395, 98790, 105570, 111628

Offset: 1

Giovanni Resta, Feb 06 2006

nonn

			666=T(36)=103+p(103).

Cf. A115882.

A236431 Numbers n such that both prime(n)+n and prime(n)-n give a triangular number.

Original entry on oeis.org

1, 513, 213796

Offset: 1

Michel Marcus, Jan 25 2014

Intersection of A115882 and A115883.
The corresponding primes are 2, 3673, 2955107.
No more terms up to 10^12. - Giovanni Resta, Jan 26 2014

			prime(1) is 2, and both 2+1=3 and 2-1=1 are triangular.

Cf. A115882, A115883.

Select[Range[214000],AllTrue[{Sqrt[8(Prime[#]-#)+1],Sqrt[8(Prime[#]+#)+ 1]},OddQ]&] (* Harvey P. Dale, Jul 22 2022 *)

lista(nn) = {p = primes(nn); for (n=1, #p, pn = p[n]; if (ispolygonal(pn - n, 3) && ispolygonal(pn + n, 3), print1(n, ", ")););}

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A115883 The n-th prime minus n gives a triangular number.

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A115907 Numbers k such that sigma(k) plus the k-th prime is a triangular number.

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A115908 Numbers k such that phi(k) + prime(k) is a triangular number.

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A115886 Triangular numbers equal to the sum of a prime number with its index.

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A236431 Numbers n such that both prime(n)+n and prime(n)-n give a triangular number.

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