A060800 -id:A060800 - cp's OEIS frontend

A055021 Smallest number k such that n iterations of sigma() are required for the result to be >= 2k.

6, 2, 9, 81

Offset: 1

Jud McCranie, May 31 2000

These are the first terms of A023196, A107912, A107913, A107914. - Jud McCranie, May 28 2005
a(5) > 4*10^9, if it exists. - Jud McCranie, May 28 2005
There are no more terms: sigma(2*k) is never prime if k is not a power of 2, so an even number needs at most two steps; sigma(k) is odd iff k is a square or twice a square. So A107914 (four recursive steps) contains only odd squares. Assume p prime so sigma(p^2) = p^2 + p + 1 = m^2 never meets the condition with p + 2k = m that (p + 2k)^2 = m^2. This implies the impossibility of a solution for numbers of the form p^(2i) and numbers of the form p^(2i)q^(2i). - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jun 06 2005
If k is a power of 2 then sigma(sigma(2*k)) = sigma(4*k - 1) >= 4*k and so the number of iterations is exactly 2. - David A. Corneth, Mar 18 2024

			sigma(sigma(sigma(9))) = 24 >= 2*9, so a(3)=9.

Cf. A000203, A055020, A107912, A107913, A107914, A060800.

isok(k, n) = my(kk=k); for (i=1, n, k = sigma(k); if ((i=2*kk), return(0))); k >= 2*kk;
a(n) = my(k=2); while (!isok(k, n), k++); k; \\ Michel Marcus, Mar 18 2024

seq() = {
	my(todo = Set([1,2,3,4]), res = vector(4));
	for(i = 2, oo,
		t = 1;
		s = sigma(i);
		while(s < 2*i,				
			s = sigma(s);
			t++
		);
		if(res[t] == 0,
			res[t] = i;
			todo = setminus(todo, Set(t));
			if(#todo == 0,
				return(res)
			)
		);	
	)
} \\ David A. Corneth, Mar 18 2024

A083681 Sum of divisors of semiprimes.

Original entry on oeis.org

7, 12, 13, 18, 24, 24, 32, 36, 31, 42, 48, 54, 48, 60, 56, 72, 57, 72, 72, 80, 90, 96, 84, 96, 114, 96, 126, 108, 132, 120, 112, 128, 144, 120, 162, 152, 144, 180, 144, 133, 186, 168, 176, 160, 204, 192, 216, 168, 180, 222, 192, 240, 216, 192, 252, 183, 240, 270, 248

Offset: 1

Shyam Sunder Gupta, Jun 15 2003

Contains all products of distinct terms of A008864 contributed by the squarefree semiprimes and all terms of A060800 contributed by the squared primes: 7 = A060800(1), 12 = A008864(1)*A008864(2), 13 = A060800(2), 18=A008864(1)*A008864(3) etc. - R. J. Mathar, Mar 15 2018

			a(2) = 12 because the sum of divisors of the 2nd semiprime, i.e. 6, is 1+2+3+6 = 12.

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

Cf. A000203, A001358, A006530, A008864, A020639, A060800.

DivisorSigma[1, Select[Range[200], PrimeOmega[#] == 2 &]] (* Amiram Eldar, Nov 24 2020 *)

a(n) = A000203(A001358(n)) = 1 + A020639(n) + A057427(A006530(n)-A020639(n))*A006530(n) + A020639(n)*A006530(n). - Reinhard Zumkeller, Jun 16 2003

A138452 a(n) = ((n-th prime)^6-(n-th prime)^3)/2.

Original entry on oeis.org

28, 351, 7750, 58653, 885115, 2412306, 12066328, 23519511, 74011861, 297399466, 443736945, 1282837878, 2375017660, 3160641771, 5389555753, 11082106126, 21090164131, 25760073690, 45229040703, 64049963005, 75666918636

Offset: 1

Artur Jasinski, Mar 22 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000217, A006093, A030078, A060800.

[(p^6 - p^3) / 2: p in PrimesUpTo(100)]; // Vincenzo Librandi, Nov 14 2018

a = {}; Do[p = Prime[n]; AppendTo[a, (p^6 - p^3)/2], {n, 1, 24}]; a

forprime(p=2,1e3,print1((p^6-p^3)/2", ")) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = binomial(prime(n)^3, 2) = A000217(A030078(n)). - Jon Maiga, Nov 14 2018

a(n) = (A030078(n) * A006093(n) * A060800(n))/2. - Vincenzo Librandi, Nov 14 2018

A237360 Numbers n of the form p^2+p+1 (for prime p) such that n^2+n+1 is also prime.

Original entry on oeis.org

57, 381, 993, 4557, 16257, 32943, 49953, 58323, 109893, 135057, 167691, 214833, 237657, 453603, 503391, 564753, 658533, 678153, 780573, 995007, 1248807, 1516593, 1746363, 2218611, 2400951, 3465183, 3738423, 4340973, 4750221, 5232657, 6118203

Offset: 1

Derek Orr, Feb 06 2014

nonn

			57 = 7^2+7+1 (7 is prime) and 57^2+57+1 = 3307 is also prime. Thus, 57 is a member of this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A060800, A002383.

for k from 1 do
    p := ithprime(k) ;
    n := numtheory[cyclotomic](3,p) ;
    pn := numtheory[cyclotomic](3,n) ;
    if isprime( pn) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Feb 07 2014

Select[Table[p^2+p+1,{p,Prime[Range[500]]}],PrimeQ[#^2+#+1]&] (* Harvey P. Dale, Feb 09 2014 *)

s=[]; forprime(p=2, 4000, if(isprime(p^4+2*p^3+4*p^2+3*p+3), s=concat(s, p^2+p+1))); s \\ Colin Barker, Feb 07 2014

import sympy
from sympy import isprime
{print(n**2+n+1) for n in range(10**4) if isprime(n) and isprime((n**2+n+1)**2+(n**2+n+1)+1)}

A253257 Least positive integer k such that prime(k*n) has the form p^2 - 2 with p prime, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 3, 1, 3, 12, 47, 9, 1, 100, 502, 6, 3, 1817, 1, 362, 3141, 4, 104, 50, 14157, 251, 222, 3, 27, 76, 25, 5423, 416, 73, 28764, 181, 488, 3860, 1249, 2, 138, 52, 1, 25, 8734, 65719, 7089, 214, 15, 111, 7, 990, 6254, 20, 1047, 38, 367, 880, 435, 3712, 3287, 208, 5194, 598

Offset: 1

Zhi-Wei Sun, Jul 05 2015

nonn

Conjecture: a(n) > 0 for all n > 0.
This is stronger than the conjecture that there are infinitely many primes of the form p^2-2 with p prime.
I also conjecture that for any positive integer n there is a positive integer k such that prime(k*n) has the form 2*p^2-1 (or 4*p^2+1, or p^2+p+1) with p prime.

			a(1) = 1 since prime(1*1) = 2 = 2^2-2 with 2 prime.
a(6) = 12 since prime(12*6) = 359 = 19^2-2 with 19 prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..800
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A049002, A060429, A060800, A062326, A179262, A237367, A237413, A259731.

SQ[n_]:=IntegerQ[Sqrt[n]]&&PrimeQ[Sqrt[n]]
Do[k=0;Label[bb];k=k+1;If[SQ[Prime[k*n]+2],Goto[aa],Goto[bb]];Label[aa];Print[n, " ", k];Continue,{n,1,60}]

use ntheory ":all"; use Math::Prime::Util::PrimeArray qw/$probj/; my %v; forprimes { undef $v{$*$-2} } 4e7; for my $n (1..800) { my $k=1; $k++ until exists $v{$probj->FETCH($k*$n-1)}; say "$n $k"; } # Dana Jacobsen, Dec 15 2015

A306190 a(n) = p^2 - p - 1 where p = prime(n), the n-th prime.

Original entry on oeis.org

1, 5, 19, 41, 109, 155, 271, 341, 505, 811, 929, 1331, 1639, 1805, 2161, 2755, 3421, 3659, 4421, 4969, 5255, 6161, 6805, 7831, 9311, 10099, 10505, 11341, 11771, 12655, 16001, 17029, 18631, 19181, 22051, 22649, 24491, 26405, 27721, 29755, 31861, 32579, 36289

Offset: 1

Kritsada Moomuang, Jan 28 2019

nonn

Terms are divisible by 5 iff p is of the form 10*m + 3 (A030431).

			a(3) = 19 because 5^2 - 5 - 1 = 19.

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

Supersequence of A091568.
Subsequence of A028387 or A165900.
Second column of A378979.
Cf. A000040, A001248, A030431, A033879, A036689, A036690, A060800, A065479, A065488, A072055, A089241.
A039914 is an essentially identical sequence.

map(p -> p^2-p-1, [seq(ithprime(i),i=1..100)]); # Robert Israel, Mar 11 2019

Table[Prime[n]^2-Prime[n]-1, {n, 1, 100}] (* Jinyuan Wang, Feb 02 2019 *)

a(n) = {p=prime(n);p^2-p-1;} \\ Jinyuan Wang, Feb 02 2019

a(n) = A036689(n) - 1.
a(n) = A036690(n) - A072055(n).
a(n) = A060800(n) - A089241(n).
From Amiram Eldar, Nov 07 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065488.
Product_{n>=2} (1 - 1/a(n)) = A065479. (End)
a(n) = A033879(A001248(n)). [Deficiency of squares of primes] - Antti Karttunen, Dec 13 2024

A181150 a(n) = prime(n)^3 + prime(n) + 1.

Original entry on oeis.org

11, 31, 131, 351, 1343, 2211, 4931, 6879, 12191, 24419, 29823, 50691, 68963, 79551, 103871, 148931, 205439, 227043, 300831, 357983, 389091, 493119, 571871, 705059, 912771, 1030403, 1092831, 1225151, 1295139, 1443011, 2048511, 2248223, 2571491, 2685759, 3308099, 3443103, 3870051, 4330911, 4657631, 5177891, 5735519

Offset: 1

Jani Melik, Jan 24 2011

			a(4)=351 because the 4th prime is 7, and 7^3 + 7 + 1 = 351.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A060800.

[p^3+p+1: p in PrimesUpTo(700)]; // Vincenzo Librandi, Jan 26 2011

A181150 := n -> map (p -> p^(3)+p+1, ithprime(n)):
seq (A181150(n), n=1..41);

#^3+#+1&/@Prime[Range[50]]  (* Harvey P. Dale, Jan 25 2011 *)

A319076 Square array T(n,k) read by antidiagonal upwards in which column k lists the partial sums of the powers of the k-th prime, n >= 0, k >= 1.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 15, 13, 6, 1, 31, 40, 31, 8, 1, 63, 121, 156, 57, 12, 1, 127, 364, 781, 400, 133, 14, 1, 255, 1093, 3906, 2801, 1464, 183, 18, 1, 511, 3280, 19531, 19608, 16105, 2380, 307, 20, 1, 1023, 9841, 97656, 137257, 177156, 30941, 5220, 381, 24, 1, 2047, 29524, 488281, 960800, 1948717

Offset: 0

Omar E. Pol, Sep 09 2018

T(n,k) is also the sum of the divisors of the n-th nonnegative power of the k-th prime, n >= 0, k >= 1.

			The corner of the square array is as follows:
         A126646 A003462 A003463  A023000    A016123    A091030     A091045
A000012        1,      1,      1,       1,         1,         1,          1, ...
A008864        3,      4,      6,       8,        12,        14,         18, ...
A060800        7,     13,     31,      57,       133,       183,        307, ...
A131991       15,     40,    156,     400,      1464,      2380,       5220, ...
A131992       31,    121,    781,    2801,     16105,     30941,      88741, ...
A131993       63,    364,   3906,   19608,    177156,    402234,    1508598, ...
.......      127,   1093,  19531,  137257,   1948717,   5229043,   25646167, ...
.......      255,   3280,  97656,  960800,  21435888,  67977560,  435984840, ...
.......      511,   9841, 488281, 6725601, 235794769, 883708281, 7411742281, ...
...

Rows 0-5: A000012, A008864, A060800, A131991, A131992, A131993.
Columns 1-15: A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.
Main diagonal gives A319074.
Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Cf. A000040, A000203, A006093, A319075.

T(n, k) = sigma(prime(k)^n); \\ Michel Marcus, Sep 13 2018

T(n,k) = A000203(A000040(k)^n).
T(n,k) = Sum_{j=0..n} A000040(k)^j.
T(n,k) = Sum_{j=0..n} A319075(j,k).
T(n,k) = (A000040(k)^(n+1) - 1)/(A000040(k) - 1).
T(n,k) = (A000040(k)^(n+1) - 1)/A006093(k).

A181151 a(n) = prime(n)^3 + prime(n)^2 + 1.

Original entry on oeis.org

13, 37, 151, 393, 1453, 2367, 5203, 7221, 12697, 25231, 30753, 52023, 70603, 81357, 106033, 151687, 208861, 230703, 305253, 362953, 394347, 499281, 578677, 712891, 922083, 1040503, 1103337, 1236493, 1306911, 1455667, 2064513, 2265253, 2590123, 2704941, 3330151, 3465753, 3894543, 4357317, 4685353, 5207647, 5767381

Offset: 1

Jani Melik, Jan 24 2011

			a(4)=393 because the 4th prime is 7, 7^3=343, 7^2=49, and 343+49+1=393.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A001248, A030078, A060800, A135178.

[p^3+p^2+1: p in PrimesUpTo(700)]; // Vincenzo Librandi, Jan 26 2011

A181151 := n -> map (p -> p^(3)+p^(2)+1, ithprime(n)):
seq (A181151(n), n=1..41);

#^3+#^2+1&/@Prime[Range[45]]  (* Harvey P. Dale, Jan 25 2011 *)

a(n) = A135178(n) + 1. - Elmo R. Oliveira, Jan 27 2023

A121208 Cyclotomic polynomials evaluated at prime number abscissae.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 31, 32, 36, 38, 40, 42, 43, 44, 46, 48, 50, 52, 54, 57, 58, 60, 61, 62, 66, 68, 70, 72, 73, 74, 78, 80, 82, 84, 88, 90, 96, 98, 100, 102, 104, 106, 108, 110, 111, 112, 114, 121, 122, 126, 127

Offset: 1

R. J. Mathar, Aug 20 2006

nonn

Contains as subsets: all A006093 which are Phi_1(p), all A008864 which are Phi_2(p), all A060800 which are Phi_3(p), all A066872 which are Phi_4(p) and all A119959 which are Phi_6(p).

			Contains 50, which is the fourth cyclotomic polynomial evaluated at the prime 7, 50=Phi_4(7).

Eric Weisstein's World of Mathematics, Cyclotomic Polynomials.

Cf. A000040, A006093, A008864, A060800, A066872, A119959.

a(n) = Phi_j(p), p in A000040, j=1,2,3,4...

cp's OEIS Frontend

A055021 Smallest number k such that n iterations of sigma() are required for the result to be >= 2k.

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A083681 Sum of divisors of semiprimes.

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A138452 a(n) = ((n-th prime)^6-(n-th prime)^3)/2.

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A237360 Numbers n of the form p^2+p+1 (for prime p) such that n^2+n+1 is also prime.

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A253257 Least positive integer k such that prime(k*n) has the form p^2 - 2 with p prime, or 0 if no such k exists.

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A306190 a(n) = p^2 - p - 1 where p = prime(n), the n-th prime.

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A181150 a(n) = prime(n)^3 + prime(n) + 1.

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