A036689 -id:A036689 - cp's OEIS frontend

A349660 Numbers which are the sum of a prime and the square of the next prime.

11, 28, 54, 128, 180, 302, 378, 548, 864, 990, 1400, 1718, 1890, 2252, 2856, 3534, 3780, 4550, 5108, 5400, 6314, 6968, 8004, 9498, 10298, 10710, 11552, 11988, 12878, 16242, 17288, 18900, 19458, 22340, 22950, 24800, 26726, 28052, 30096, 32214, 32940, 36662

Offset: 1

Karl-Heinz Hofmann, Nov 24 2021

			a(2) = 3 + 5^2 = 28; a(3) = 5 + 7^2 = 54.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A140511, A000040, A001248, A036690, A001223, A001043, A036689, A349661, A124129.

nterms=100;Table[Prime[n]+Prime[n+1]^2,{n,nterms}] (* Paolo Xausa, Nov 24 2021 *)

a(n) = prime(n) + prime(n+1)^2; \\ Michel Marcus, Nov 24 2021

from sympy import sieve;
for n in range(1,10001): print(sieve[n] + sieve[n+1]**2)

a(n) = prime(n) + prime(n+1)^2.
a(n) = A000040(n) + A001248(n+1).
a(n) = A036690(n+1) - A001223(n).
a(n) = A001043(n) + A036689(n+1). - Michel Marcus, Nov 24 2021

A124827 Order of Galois groups of irreducible Chebyshev polynomials of order n.

Original entry on oeis.org

1, 2, 6, 8, 20, 12, 42, 16, 54, 40, 110, 48, 156, 84, 120, 64, 272, 108, 342, 160, 252

Offset: 1

Artur Jasinski, Nov 09 2006

All groups belonging to solvable Galois groups.
Very similar sequence is A002618 (disagreement occurred only for Chebyshev polynomials orders 8 and 16).
When the order of an irreducible Chebyshev polynomial is a prime number p, the Galois group is the Frobenius group of order p*(p-1) A036689.
In Magma classification the Galois groups are the following: T1_1, T2_1, T3_2, T4_3, T5_3, T6_3, T7_4, T8_8, T9_10, T11_4, T12_28, T13_6, T14_7, T15_11, T16_144, T17_5, T18_45, T19_6, T20_42, T21_15.
Is a(n) the order of Galois group of the polynomial x^n - 2? If so, then a(n) = n*phi(n) for n not divisible by 8, and n*phi(n)/2 otherwise (see the Math Overflow link below). Under this assumption, a(n) is multiplicative with a(p^e) = p^(2*e-1)*(p-1) for p being an odd prime; a(2) = 2, a(4) = 8, and a(2^e) = 2^(2*e-2) for e >= 3. - Jianing Song, Nov 22 2022

			a(5)=20 because the order of the Galois group of polynomial 16x^5-20x^3+5x-c is 20 (where c is an integer chosen so that the polynomial is irreducible). This transitive group is the Frobenius group F5 of order 20 (also called the metacyclic group M_5) T5_3(20) in Magma classification.

Math Overflow, Galois Group of x^n - 2

Cf. A001710, A000142, A036689, A002618, A036689.

Cf. A127835.

Zx:=PolynomialRing(Integers()); f:=16*x^5-20*x^3+5*x-7; G:=GaloisGroup(f:Old); "Order of group",#G; // Juergen Klueners klueners(AT)math.uni-duesseldorf.de

A124900 Largest order of any solvable transitive Galois group for an irreducible polynomial of degree n.

Original entry on oeis.org

1, 2, 6, 24, 20, 72, 42, 1152, 1296, 800, 110, 82944, 156, 3528, 155520, 7962624, 272, 2239488, 342, 159252480, 11757312, 225280, 506, 13759414272, 64000000, 1277952, 13060694016, 192631799808, 812, 48372940800

Offset: 1

Artur Jasinski, Nov 12 2006

nonn

These transitive groups are in classification of MAGMA:
a(1)=1T1,a(2)=2T1,a(3)=3T2,a(4)=4T5,a(5)=5T3,a(6)=6T13,
a(7)=7T4,a(8)=8T47,a(9)=9T31,a(10)=10T33,a(11)=11T4,
a(12)=12T294,a(13)=13T6,a(14)=14T45,a(15)=15T87,
a(16)=16T1947,a(17)=17T5,a(18)=18T945,a(19)=19T6,
a(20)=20T1067,a(21)=21T142,a(22)=22T37,a(23)=23T5,
a(24)=24T24921,a(25)=25T179,a(26)=26T79,a(27)=27T2372,
a(28)=28T1773,a(29)=29T6,a(30)=30T5358.
Conjecture: The sequence a(prime(n)), which begins 2, 6, 20, 42, 110, 156, 272, 342, 506, 812, increases without bound. It appears that a(prime(n)) may equal prime(n)(prime(n)-1), which is A036689. - Artur Jasinski, Feb 26 2011

			a(9)=1296 because solvable Galois group T9_31 (in MAGMA's list) has order 1296

Cf. A099732, A124901, A186772.

a(11)-a(30) from Artur Jasinski, Feb 26 2011

A127921 1/12 of product of three numbers: n-th prime, previous and following number.

Original entry on oeis.org

2, 10, 28, 110, 182, 408, 570, 1012, 2030, 2480, 4218, 5740, 6622, 8648, 12402, 17110, 18910, 25058, 29820, 32412, 41080, 47642, 58740, 76048, 85850, 91052, 102078, 107910, 120232, 170688, 187330, 214268, 223790, 275650, 286900, 322478, 360882, 388108, 431462

Offset: 2

Artur Jasinski, Feb 06 2007

Summation of products of partitions into two parts of prime(n): a(6) = (1*12) + (2*11) + (3*10) + (4*9) + (5*8) + (6*7) = 182. - César Aguilera, Feb 20 2018

G. C. Greubel, Table of n, a(n) for n = 2..10000

Cf. A000040, A036689, A034953, A127917, A127918, A127919, A127920.

[(NthPrime(n) + 1)*NthPrime(n)*(NthPrime(n) - 1)/12: n in [2..50]]; // G. C. Greubel, Apr 30 2018

a:= n-> (p->p*(p^2-1)/12)(ithprime(n)):
seq(a(n), n=2..40);  # Alois P. Heinz, Mar 08 2022

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/12, {n, 2, 100}]
((#-1)#(#+1))/12&/@Prime[Range[2,40]] (* Harvey P. Dale, Mar 08 2022 *)

a(n,p=prime(n))=binomial(p+1,3)/2 \\ Charles R Greathouse IV, Feb 28 2018

a(n) ~ (n log n)^3/12. - Charles R Greathouse IV, Feb 28 2018

A174869 a(n) is 0 if n is a power of 2, otherwise the smallest k > 0 such that A006530(n+k) < A006530(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 7, 2, 1, 4, 1, 1, 1, 0, 1, 14, 1, 4, 3, 2, 1, 8, 2, 1, 5, 2, 1, 2, 1, 0, 2, 1, 1, 28, 1, 1, 1, 8, 1, 3, 1, 1, 3, 2, 1, 16, 1, 4, 1, 2, 1, 10, 1, 4, 3, 2, 1, 4, 1, 1, 1, 0, 1, 4, 1, 2, 1, 2, 1, 56, 1, 1, 6, 1, 3, 2, 1, 1, 47, 2, 1, 6, 3, 1, 1, 2, 1, 6, 5, 3, 2, 1, 1, 32, 1, 2, 1, 8, 1, 2, 1

Offset: 1

Vladimir Shevelev, Mar 31 2010

nonn

a(n)=1 if the index n is an odd prime.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A006530, A103667, A023503, A023509, A174857, A020639, A002618, A036689.

A006530 := proc(n) option remember; if n = 1 then 1; else max(op(numtheory[factorset](n)) ) ; end if; end proc:
A174869 := proc(n) if n <= 2 then 0; else gpf := A006530(n) ; if gpf = 2 then 0; else for k from 1 do if A006530(n+k) < gpf then return k; end if; end do: end if; end if; end proc:
seq(A174869(n),n=1..120) ; # R. J. Mathar, Aug 10 2010

Block[{s = Array[FactorInteger[#][[-1, 1]] &, 120]}, Array[If[IntegerQ@ Log2[#], 0, Block[{k = 1, n = s[[#]]}, While[n <= s[[# + k]], k++; If[# + k > Length[s], AppendTo[s, FactorInteger[# + k][[-1, 1]] ]] ]; k]] &, 102, 2]] (* Michael De Vlieger, Apr 06 2021 *)

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A174869(n) = if(!bitand(n,n-1), 0, my(gpf=A006530(n)); for(k=1,oo,if(A006530(n+k)Antti Karttunen, Apr 06 2021

More terms from R. J. Mathar, Aug 10 2010

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

A366362 Triangle read by rows: T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.

Original entry on oeis.org

1, 0, 4, 5, 0, 4, 0, 8, 0, 8, 21, 0, 0, 0, 4, 0, 20, 0, 0, 0, 16, 40, 0, 0, 0, 0, 0, 9, 0, 32, 0, 16, 0, 0, 0, 16, 45, 0, 24, 0, 0, 0, 0, 0, 12, 0, 84, 0, 0, 0, 0, 0, 0, 0, 16, 111, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 40, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32

Offset: 1

Mats Granvik, Oct 08 2023

Row n appears to have sum n^2. T(prime(m),1) = A366346(m). The number of nonzero terms in row n appears to be A320111(n).

			{
{1}, = 1^2
{0, 4}, = 2^2
{5, 0, 4}, = 3^2
{0, 8, 0, 8}, = 4^2
{21, 0, 0, 0, 4}, = 5^2
{0, 20, 0, 0, 0, 16}, = 6^2
{40, 0, 0, 0, 0, 0, 9}, = 7^2
{0, 32, 0, 16, 0, 0, 0, 16}, = 8^2
{45, 0, 24, 0, 0, 0, 0, 0, 12}, = 9^2
{0, 84, 0, 0, 0, 0, 0, 0, 0, 16}, = 10^2
{111, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10}, = 11^2
{0, 40, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32} = 12^2
}

Cf. A127649, A366346, A000290, A060457, A002070, A036689, A001248, A272196.

f = x^3 - x^2 - y^2 - y; nn = 12; Flatten[Table[Table[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}]]

T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.

Conjecture: T(n,n) = A060457(n).

A053198 Totients of consecutive pure powers of primes.

Original entry on oeis.org

2, 4, 6, 8, 20, 18, 16, 42, 32, 54, 110, 100, 64, 156, 162, 128, 272, 294, 342, 256, 506, 500, 486, 812, 930, 512, 1210, 1332, 1640, 1806, 1024, 1458, 2028, 2162, 2058, 2756, 2500, 3422, 3660, 2048, 4422, 4624, 4970, 5256, 6162, 4374, 6498, 6806, 7832, 4096

Offset: 1

Labos Elemer, Mar 03 2000

nonn

Totients of prime powers are prime powers only for powers of 2.

			The 10th pure power of prime (but not a prime) is 81, so a(10) = EulerPhi(81) = 54.

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

Cf. A000010, A051953, A001248, A002618, A036689, A053650, A053191, A053192, A086242.

EulerPhi[Select[Range[2^13], CompositeQ[#] && PrimePowerQ[#] &]] (* Amiram Eldar, Dec 21 2020 *)

a(n) = A000010(A025475(n+1)).
Numbers of the form phi(p^k) = (p-1)*p^(k-1), where p is prime and k > 1.
Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p-1)^2 = A086242 = 1.3750649947... - Amiram Eldar, Dec 21 2020

A053211 Cototients of consecutive pure powers of primes.

Original entry on oeis.org

2, 4, 3, 8, 5, 9, 16, 7, 32, 27, 11, 25, 64, 13, 81, 128, 17, 49, 19, 256, 23, 125, 243, 29, 31, 512, 121, 37, 41, 43, 1024, 729, 169, 47, 343, 53, 625, 59, 61, 2048, 67, 289, 71, 73, 79, 2187, 361, 83, 89, 4096, 97, 101, 103, 107, 109, 529, 113, 1331, 3125, 127

Offset: 1

Labos Elemer, Mar 03 2000

nonn

Cototients of prime powers do not remain always prime powers, but are primes if their exponent is 2.

			The 10th pure power of prime (but not a prime) is 81, so a(10) = 81 - EulerPhi(81) = 81 - 54 = 27. For n=p^2, a(n)=p.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^20, showing even a(n) in blue, 3 | a(n) in green, and prime a(n) in red, else black.

Cf. A000010, A051953, A001248, A002618, A036689, A053650, A053191, A053192, A246547.

Map[# - EulerPhi@ # &, Select[Range[16200], And[! PrimeQ@ #, PrimePowerQ@ #] &]] (* Michael De Vlieger, Jun 11 2018 *)
With[{nn = 2^14}, Map[Times @@ Map[#1^(#2 - 1) & @@ FactorInteger[#][[1]]] &, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], PrimePowerQ] ] ] (* Michael De Vlieger, Mar 11 2023 *)

a(n) = A051953(A025475(n+1)) = cototient(p^k) = p^(k-1).

A138459 a(n) = ((n-th prime)^6-(n-th prime)^4)/12.

Original entry on oeis.org

4, 54, 1250, 9604, 146410, 399854, 2004504, 3909630, 12313004, 49509670, 73881680, 213654354, 395606540, 526495354, 897861304, 1846372554, 3514034690, 4292210710, 7536519254, 10672906020, 12608819004, 20254042120, 27241076254

Offset: 1

Artur Jasinski, Mar 22 2008

Differences (p^k-p^m)/q such that k > m:
p^2-p is given in A036689
(p^2-p)/2 is given in A008837
p^3-p is given in A127917
(p^3-p)/2 is given in A127918
(p^3-p)/3 is given in A127919
(p^3-p)/6 is given in A127920
p^3-p^2 is given in A135177
(p^3-p^2)/2 is given in A138416
p^4-p is given in A138401
(p^4-p)/2 is given in A138417
p^4-p^2 is given in A138402
(p^4-p^2)/2 is given in A138418
(p^4-p^2)/3 is given in A138419
(p^4-p^2)/4 is given in A138420
(p^4-p^2)/6 is given in A138421
(p^4-p^2)/12 is given in A138422
p^4-p^3 is given in A138403
(p^4-p^3)/2 is given in A138423
p^5-p is given in A138404
(p^5-p)/2 is given in A138424
(p^5-p)/3 is given in A138425
(p^5-p)/5 is given in A138426
(p^5-p)/6 is given in A138427
(p^5-p)/10 is given in A138428
(p^5-p)/15 is given in A138429
(p^5-p)/30 is given in A138430
p^5-p^2 is given in A138405
(p^5-p^2)/2 is given in A138431
p^5-p^3 is given in A138406
(p^5-p^3)/2 is given in A138432
(p^5-p^3)/3 is given in A138433
(p^5-p^3)/4 is given in A138434
(p^5-p^3)/6 is given in A138435
(p^5-p^3)/8 is given in A138436
(p^5-p^3)/12 is given in A138437
(p^5-p^3)/24 is given in A138438
p^5-p^4 is given in A138407
(p^5-p^4)/2 is given in A138439
p^6-p is given in A138408
(p^6-p)/2 is given in A138440
p^6-p^2 is given in A138409
(p^6-p^2)/2 is given in A138441
(p^6-p^2)/3 is given in A138442
(p^6-p^2)/4 is given in A138443
(p^6-p^2)/5 is given in A138444
(p^6-p^2)/6 is given in A138445
(p^6-p^2)/10 is given in A138446
(p^6-p^2)/12 is given in A138447
(p^6-p^2)/15 is given in A138448
(p^6-p^2)/20 is given in A122220
(p^6-p^2)/30 is given in A138450
(p^6-p^2)/60 is given in A138451
p^6-p^3 is given in A138410
(p^6-p^3)/2 is given in A138452
p^6-p^4 is given in A138411
(p^6-p^4)/2 is given in A138453
(p^6-p^4)/3 is given in A138454
(p^6-p^4)/4 is given in A138455
(p^6-p^4)/6 is given in A138456
(p^6-p^4)/8 is given in A138457
(p^6-p^4)/12 is given in A138458
(p^6-p^4)/24 is given in A138459
p^6-p^5 is given in A138412
(p^6-p^5)/2 is given in A138460

a = {}; Do[p = Prime[n]; AppendTo[a, (p^6 - p^4)/12], {n, 1, 24}]; a

forprime(p=2,1e3,print1((p^6-p^4)/12", ")) \\ Charles R Greathouse IV, Jul 15 2011

cp's OEIS Frontend

A349660 Numbers which are the sum of a prime and the square of the next prime.

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A124827 Order of Galois groups of irreducible Chebyshev polynomials of order n.

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A124900 Largest order of any solvable transitive Galois group for an irreducible polynomial of degree n.

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A127921 1/12 of product of three numbers: n-th prime, previous and following number.

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A174869 a(n) is 0 if n is a power of 2, otherwise the smallest k > 0 such that A006530(n+k) < A006530(n).

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

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A366362 Triangle read by rows: T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.

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A053198 Totients of consecutive pure powers of primes.