A055469 -id:A055469 - cp's OEIS frontend

A159047 Primes which are triangular numbers plus 3.

3, 13, 31, 139, 193, 409, 499, 823, 1381, 1543, 2083, 2281, 3163, 3919, 6673, 7753, 9319, 9733, 17581, 19309, 21739, 22369, 24979, 27031, 27733, 30631, 39343, 40189, 51043, 53959, 54949, 57973, 62131, 67531, 70879, 81409, 85081, 86323, 91381

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

nonn

For n>1, a(n)== 1 (mod 6). [Proof: the triangular numbers are {0,1,3,4} (mod 6), see A104686. 3 plus triangular numbers in the same set, and only those == 1 (mod 6) can be primes.] - Zak Seidov, Oct 16 2015

			13=10+3, 31=28+3, 139=136+3, 193=190+3, 409=406+3, ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000217, A055469, A055472.

s=0;lst={};Do[s+=n;p=s+3;If[PrimeQ[p],AppendTo[lst,p]],{n,0,7!}];lst
Select[Table[n*(n + 1)/2 + 3, {n, 0, 250}], PrimeQ] (* G. C. Greubel, Jul 13 2017 *)
Select[Accumulate[Range[0,500]]+3,PrimeQ] (* Harvey P. Dale, Jul 30 2018 *)

for(n=0, 1e3, if(isprime(k=3+n*(n+1)/2), print1(k", "))) \\ Altug Alkan, Oct 16 2015

A285789 Primes equal to a pentagonal number plus 1.

Original entry on oeis.org

2, 13, 23, 71, 211, 331, 853, 1163, 1427, 2381, 2753, 3433, 3877, 5923, 6113, 6701, 7741, 8627, 9323, 11311, 12377, 14653, 14951, 17443, 18427, 23003, 27271, 31033, 32341, 32783, 34127, 38321, 43777, 52361, 55201, 56941, 57527, 62323, 64171, 64793, 69877

Offset: 1

Colin Barker, Apr 26 2017

nonn

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000040, A000326, A002496, A055469, A285790, A285791, A285792.

Select[PolygonalNumber[5,Range[300]]+1,PrimeQ] (* Harvey P. Dale, Dec 04 2024 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
maxk=300; L=List(); for(k=1, maxk, if(isprime(p=pg(5, k) + 1), listput(L, p))); Vec(L)

A285791 Primes equal to a heptagonal number plus 1.

Original entry on oeis.org

2, 19, 113, 149, 541, 617, 971, 1289, 1783, 2357, 3011, 3187, 5689, 6427, 7481, 7757, 9829, 12497, 12853, 14327, 15881, 17099, 18793, 21023, 24851, 28463, 30637, 31193, 45361, 50909, 54539, 60607, 63761, 66179, 69473, 70309, 83449, 88079, 90917, 91873, 94771

Offset: 1

Colin Barker, Apr 26 2017

nonn

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000040, A000566, A002496, A055469, A285789, A285790, A285792.

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
maxk=300; L=List(); for(k=1, maxk, if(isprime(p=pg(7, k) + 1), listput(L, p))); Vec(L)

from sympy import isprime
def heptagonal(n): return n*(5*n-3)//2
def aupto(limit):
  alst, n, hn = [], 1, heptagonal(1)
  while hn < limit:
    if isprime(hn+1): alst.append(hn+1)
    n, hn = n+1, heptagonal(n+1)
  return alst
print(aupto(94771)) # Michael S. Branicky, Feb 19 2021

A159048 Primes of the form m*(m+1)/2 + 4.

Original entry on oeis.org

5, 7, 19, 59, 109, 157, 257, 439, 599, 907, 1039, 1229, 1279, 1489, 1657, 3407, 3659, 4099, 5569, 6907, 7507, 7879, 8389, 9049, 10589, 11329, 11939, 14369, 16657, 17209, 17959, 18149, 18919, 19507, 20507, 22159, 26339, 30139, 31379, 34457, 36319

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

nonn

			7=3+4, 19=15+4, 59=55+4, 109=105+4, 157=153+4, 257=253+4, ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000217, A055469, A055472, A159047

Select[Accumulate[Range[500]]+4,PrimeQ]  (* Harvey P. Dale, Apr 23 2011 *)

for(n=1, 500, if(isprime(k=n*(n+1)/2 + 4), print1(k, ", "))) \\ G. C. Greubel, Jul 03 2017

Edited by N. J. A. Sloane, Apr 06 2009

A088579 Primes of the form nx^n + (n-1)x^(n-1) + . . . + x + 1 for x=2.

Original entry on oeis.org

3, 11, 643, 425987, 1909526242005090307

Offset: 1

Cino Hilliard, Nov 20 2003

nonn

Sum of reciprocals = 0.4257999816852453227652727311..

Next term is too large to include.

			2*2^2 + 1*2 + 1 = 11 is prime.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10 (terms 1..9 from Michel Marcus)

Cf. A055469 (for x=1), A088584 (for x=3), A088583 (for x=4).

Select[Table[1+Sum[k 2^k, {k,n}], {n,1000}], PrimeQ] (* T. D. Noe, Nov 15 2006 *)

polypn2(n,p) = { my(x=n,y=1); for(m=1,p, y=y+m*x^m; ); return(y) }
trajpolyp(n1,k) = { my(s=0); for(x1=0,n1, y1 = polypn2(k,x1); if(isprime(y1),print1(y1, ","); s+=1.0/y1; ) ); }
trajpolyp(500,2)

Corrected by T. D. Noe and Don Reble, Nov 15 2006

A088584 Primes of the form nx^n + (n-1)x^(n-1) + . . . + x + 1 for x=3.

Original entry on oeis.org

103, 312088729, 9955641160957, 163142317702973500798039087, 327058383882861814163660125754017, 67973813526967723994124686175157751059, 545249446055539622797498212423248888694512551610580463

Offset: 1

Cino Hilliard, Nov 20 2003

nonn

Sum of reciprocals = 0.009708741068395080316898549713.. Are these primes infinite?

The next term (a(8)) has 148 digits. - Harvey P. Dale, Dec 15 2018

			3*3^3 + 2*3^2 + 3 + 1 = 103.

Michel Marcus, Table of n, a(n) for n = 1..8

Cf. A055469, A088579, A088583.

Select[Accumulate[Join[{1},Table[n*3^n,{n,200}]]],PrimeQ] (* Harvey P. Dale, Dec 15 2018 *)

trajpolyp(n1,k) = { s=0; for(x1=0,n1, y1 = polypn2(k,x1); if(isprime(y1),print1(y1","); s+=1.0/y1; ) ); print(); print(s) }
polypn2(n,p) = { x=n; y=1; for(m=0,p, y=y+m*x^m; ); return(y) }

More terms from Harvey P. Dale, Dec 15 2018

A159049 Primes of the form (5+ a triangular number A000217).

Original entry on oeis.org

5, 11, 41, 71, 83, 281, 383, 1181, 1601, 2351, 2633, 3491, 3833, 4283, 5783, 6221, 6791, 8783, 10301, 10883, 11633, 12251, 14033, 15581, 18341, 26111, 26801, 30881, 31883, 34721, 38231, 41333, 41621, 42491, 43961, 46061, 47591, 53633, 60383

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

			11 is in the list because it is A000217(3)+5 and a prime. 41=36+5= A000217(8)+5 is a prime. 71=66+5=A000217(11)+5 is a prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A055469, A055472, A159047, A159048

s=0;lst={};Do[s+=n;p=s+5;If[PrimeQ[p],AppendTo[lst,p]],{n,0,7!}];lst
Select[Accumulate[Range[0,500]]+5,PrimeQ] (* Harvey P. Dale, Jul 08 2017 *)

for(n=1,500, if(isprime(k=n*(n+1)/2 + 5), print1(k, ", "))) \\ G. C. Greubel, Jul 13 2017

Definition rephrased, R. J. Mathar, Apr 05 2009

A185510 Array of primes in the natural number array A000027, by antidiagonals.

Original entry on oeis.org

2, 7, 3, 11, 5, 13, 29, 17, 31, 19, 37, 23, 139, 59, 41, 67, 47, 193, 109, 71, 61, 79, 107, 409, 157, 83, 97, 43, 137, 173, 499, 257, 281, 331, 73, 53, 191, 233, 823, 439, 383, 601, 127, 113, 199, 211, 353, 1381, 599, 1181, 709, 197, 179, 829, 101, 277, 467, 1543, 907, 1601, 1087, 283, 239, 1549, 163, 89

Offset: 1

Clark Kimberling, Jan 29 2011

Start with the natural number array A000027:
...2.....4....7...11...16...22...29...
...5.....8...12...17...23...30...38...
...9....13...18...24...31...39...48...
..14...19...25...32...40...49...59...
..20...26...33...41...50...60...71...
..27...34...42...51...61...72...84...
..35...43...52...62...73...85...98...
Row n of A185510 shows the primes in row n of A000027:
...7....11...29...37....67....79...137...(A055469)
...5....17...23...47...107...173...233...(A055472)
.31...139..193..409...499...823..1381...(A159047)
.59...109..157..257...439...599...907...(A159048)
.71....83..281..383..1181..1601..2351...(A159049)
.97...331..601..709..1087..1231..2707...
.73...127..197..283..307...503...673...
Conjecture:  Every row contains infinitely many primes.
Every prime occurs exactly once; that is, every prime is uniquely expressible as (1/2)(n^2 + (2k-1)n + (k-2)(k-1)) for some positive integers n and k.  We assume as true the conjecture that each row is infinite. - Clark Kimberling, Mar 10 2020

Cf. A000027, A000040, A185511, A057060, A057061.

f[n_,k_]:=n+(k+n-2)(k+n-1)/2;
TableForm[Map[Select[#,PrimeQ]&, Table[f[n,k],{n,1,20}, {k,1,100}]]]

A360944 Numbers m such that phi(m) is a triangular number, where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 7, 9, 11, 14, 18, 22, 29, 37, 57, 58, 63, 67, 74, 76, 79, 108, 114, 126, 134, 137, 143, 155, 158, 175, 183, 191, 211, 225, 231, 244, 248, 274, 277, 286, 308, 310, 329, 341, 350, 366, 372, 379, 382, 396, 417, 422, 423, 450, 453, 462, 554, 556, 604, 623, 631, 658, 682

Offset: 1

Bernard Schott, Feb 26 2023

nonn

Subsequence of primes is A055469 because in this case phi(k(k+1)/2+1) = k(k+1)/2.

Subsequence of triangular numbers is A287472.

			phi(57) = 36 = 8*9/2, a triangular number; so 57 is a term of the sequence.

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

Cf. A000010, A000217, A055469, A287472.

Similar, but with phi(m) is: A039770 (square), A078164 (biquadrate), A096503 (repdigit), A117296 (palindrome), A236386 (oblong).

filter := m ->  issqr(1 + 8*numtheory:-phi(m)) : select(filter, [$(1 .. 700)]);

Select[Range[700], IntegerQ[Sqrt[8 * EulerPhi[#] + 1]] &] (* Amiram Eldar, Feb 27 2023 *)

isok(m) = ispolygonal(eulerphi(m), 3); \\ Michel Marcus, Feb 27 2023

from itertools import islice, count
from sympy.ntheory.primetest import is_square
from sympy import totient
def A360944_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:is_square((totient(n)<<3)+1), count(max(1,startvalue)))
A360944_list = list(islice(A360944_gen(),20)) # Chai Wah Wu, Feb 28 2023

A169583 n-th prime*8-7 is the square of a prime.

Original entry on oeis.org

1, 4, 12, 19, 47, 59, 115, 167, 217, 251, 306, 348, 414, 618, 630, 662, 809, 1077, 1138, 1218, 1670, 1876, 2272, 2680, 2869, 3402, 3633, 4242, 4353, 4661, 5255, 6463, 6596, 6986, 8543, 8870, 8992, 9340, 9444, 10265, 11544, 11921, 12449, 13887, 14031

Offset: 1

Juri-Stepan Gerasimov, Mar 01 2010

nonn

			1 is in the sequence because 1st prime*8-7=(2nd prime)^2;
4 is in the sequence because 4th prime*8-7=(4th prime)^2;
12 is in the sequence because 12th prime*8-7=(7th prime)^2.

Harvey P. Dale, Table of n, a(n) for n = 1..350

Cf. A000040, A001248, A055469, A118940.

Select[Range[15000],PrimeQ[Sqrt[8Prime[#]-7]]&] (* Harvey P. Dale, Nov 10 2017 *)

isok(n) = issquare(sq=prime(n)*8-7) && isprime(sqrtint(sq)); \\ Michel Marcus, Apr 15 2014

Corrected (1077 inserted) and extended by R. J. Mathar, Jun 04 2010

cp's OEIS Frontend

A159047 Primes which are triangular numbers plus 3.

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A285789 Primes equal to a pentagonal number plus 1.

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A285791 Primes equal to a heptagonal number plus 1.

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A159048 Primes of the form m*(m+1)/2 + 4.

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A088579 Primes of the form n*x^n + (n-1)*x^(n-1) + . . . + x + 1 for x=2.

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A088584 Primes of the form n*x^n + (n-1)*x^(n-1) + . . . + x + 1 for x=3.

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A159049 Primes of the form (5+ a triangular number A000217).

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A185510 Array of primes in the natural number array A000027, by antidiagonals.

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A088579 Primes of the form nx^n + (n-1)x^(n-1) + . . . + x + 1 for x=2.

A088584 Primes of the form nx^n + (n-1)x^(n-1) + . . . + x + 1 for x=3.