A008837 -id:A008837 - cp's OEIS frontend

A273222 a(n) = p(p - 1)(73p^2 - 45p + 14)/24, where p = prime(n).

18, 134, 1345, 5733, 38280, 76479, 230588, 363546, 792649, 2033451, 2664915, 5454873, 8260270, 10012464, 14337303, 23275109, 35855716, 41007555, 59825238, 75546485, 84478374, 116064351, 141557994, 187394306, 264812328, 311476425, 336995709, 392705408, 423017991

Offset: 1

Vincenzo Librandi, May 19 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
F. V. Weinstein, Notes on Fibonacci partitions, arXiv:math/0307150 [math.NT], 2003-2015, page 22.

Cf. A006093, A008837, A179545, A273221.

[p*(p-1)*(73*p^2-45*p+14)/24: p in PrimesUpTo(200)];

Table[p = Prime[n]; p (p - 1) (73 p^2 - 45 p + 14) / 24, {n, 40}]
(#(#-1)(73#^2-45#+14))/24&/@Prime[Range[30]] (* Harvey P. Dale, Jan 17 2017 *)

A121057 Triangle read by rows: T(n,m) = Prime[m]^n*(Prime[m] - 1)/2.

Original entry on oeis.org

1, 3, 9, 10, 50, 250, 21, 147, 1029, 7203, 55, 605, 6655, 73205, 805255, 78, 1014, 13182, 171366, 2227758, 28960854, 136, 2312, 39304, 668168, 11358856, 193100552, 3282709384, 171, 3249, 61731, 1172889, 22284891, 423412929, 8044845651

Offset: 1

Roger L. Bagula, May 27 2007

Row sums are 1, 12, 310, 8400, 885775, 31374252, 3487878712, 161343848880, 20713255606813, 6100254882852900, ...

First column is A008837. - Michel Marcus, Apr 11 2013

			Triangle begins
1;
3, 9;
10, 50, 250;
21, 147, 1029, 7203;

t[n_, m_] = Prime[m]^n*(Prime[m] - 1)/2; a = Table[Table[t[n, m], {n, 1, m}], {m, 1, 10}] Flatten[a]

trga(nrows) = {for (n=1, nrows, for (m=1, n, print1(prime(n)^m*(prime(n) - 1)/2, ", ");); print(););}  \\ Michel Marcus, Apr 11 2013

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

A231808 Numerator of asymptotic density of Union{H_p: p is odd prime and p <= n-th prime}, where H_p is {Kp(p-1)/2 : K integer}.

Original entry on oeis.org

0, 1, 2, 2, 67, 67, 230, 230, 5317, 70307, 70307, 70307, 70307, 70307, 23158993, 58560723101, 10373287618037, 10373287618037, 10373287618037, 736719736564627, 736719736564627, 736719736564627, 119433196256360189, 1970856524120023, 1970856524120023, 1970856524120023

Offset: 1

José María Grau Ribas, Nov 13 2013

a(n)/A231809(n) is the asymptotic density of Union{H_p: p is odd prime and p <= n-th prime}, where H_p is {K*p*(p-1)/2 : K integer}; a(n) tends to 0.41.. (the asymptotic density of A229307 = Union{H_p: p odd prime}).

			0, 1/3, 2/5, 2/5, 67/165, 67/165, 230/561, 230/561, 5317/12903, 70307/170085, 70307/170085, 70307/170085, 70307/170085, 70307/170085, 23158993/55957965, 58560723101/141368472245, 10373287618037/25022219587365, ....

José María Grau Ribas, Table of n, a(n) for n = 1..40
Jose María Grau, A. M. Oller-Marcen, and Jonathan Sondow, On the congruence 1^n + 2^n +... + n^n = d (mod n), where d divides n, arXiv:1309.7941 [math.NT], 2013-2014.

Cf. A008837, A229307, A231809.

<< DiscreteMath`Combinatorica` (*ver 5.0*)
<< Combinatorica` (*ver 8.0*)
fa[n_] := FactorInteger[n]; lcm[lis_] := lcm[lis] = {aux = 1; Do[aux = LCM[aux, lis[[i]]], {i, 1, Length@lis}]; aux}[[1]]; inclusexclus[lis_] := inclusexclus[lis] =Sum[(-1)^(1 + Length[lis[[i]]])/lcm[lis[[i]]], {i, 1, Length@lis}]; densidad[lis_] := Sum[inclusexclus[KSubsets[lis, i]], {i, 1, Length[lis]}]; lista[n_] := Table[(Prime[i]^2 - Prime[i])/2, {i, 2, n}]; Table[Numerator@densidad[lista[i]], {i, 1, 15}]

A231809 Denominator of asymptotic density of Union{H_p: p is odd prime and p <= n-th prime}, where H_p is {Kp(p-1)/2 : K integer}.

Original entry on oeis.org

1, 3, 5, 5, 165, 165, 561, 561, 12903, 170085, 170085, 170085, 170085, 170085, 55957965, 141368472245, 25022219587365, 25022219587365, 25022219587365, 1776577590702915, 1776577590702915, 1776577590702915, 287890168626762845, 4749253940274679, 4749253940274679

Offset: 1

José María Grau Ribas, Nov 13 2013

See A231808.

			0, 1/3, 2/5, 2/5, 67/165, 67/165, 230/561, 230/561, 5317/12903, 70307/170085, 70307/170085, 70307/170085, 70307/170085, 70307/170085, 23158993/55957965, 58560723101/141368472245, 10373287618037/25022219587365

José María Grau Ribas, Table of n, a(n) for n = 1..40
Jose María Grau, A. M. Oller-Marcen, and Jonathan Sondow, On the congruence 1^n + 2^n +... + n^n = d (mod n), where d divides n, arXiv:1309.7941 [math.NT], 2013-2014.

Cf. A008837, A229307, A231808.

<< DiscreteMath`Combinatorica` (*ver 5.0*)
<< Combinatorica` (*ver 8.0*)
fa[n_] := FactorInteger[n]; lcm[lis_] := lcm[lis] = {aux = 1; Do[aux = LCM[aux, lis[[i]]], {i, 1, Length@lis}]; aux}[[1]]; inclusexclus[lis_] := inclusexclus[lis] =Sum[(-1)^(1 + Length[lis[[i]]])/lcm[lis[[i]]], {i, 1, Length@lis}]; densidad[lis_] := Sum[inclusexclus[KSubsets[lis, i]], {i, 1, Length[lis]}]; lista[n_] := Table[(Prime[i]^2 - Prime[i])/2, {i, 2, n}]; Table[Denominator@densidad[lista[i]], {i, 1, 15}]

A273223 a(n) = p(p - 1)(501p^3 - 414p^2 + 111*p - 54)/120, where p = prime(n).

Original entry on oeis.org

42, 504, 8796, 53298, 566412, 1341756, 5312160, 9373536, 24790458, 80346588, 112613886, 275440284, 462452448, 588037212, 920759046, 1686448764, 2893307844, 3421602972, 5484429720, 7340452434, 8440231968, 12551864598, 16086117120, 22838112000, 35181089856

Offset: 1

Vincenzo Librandi, May 19 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
F. V. Weinstein, Notes on Fibonacci partitions, arXiv:math/0307150 [math.NT], 2003-2015, page 22.

Cf. A006093, A008837, A179545, A273221, A273222.

[p*(p-1)*(501*p^3-414*p^2+111*p-54)/120: p in PrimesUpTo(200)];

Table[p = Prime[n]; p (p - 1) (501 p^3 - 414 p^2 + 111 p - 54) / 120, {n, 40}]

A061591 a(n) = 2^x, where x = p*(p-1)/2 and p is the n-th prime.

Original entry on oeis.org

2, 8, 1024, 2097152, 36028797018963968, 302231454903657293676544, 87112285931760246646623899502532662132736, 2993155353253689176481146537402947624255349848014848

Offset: 1

Labos Elemer, May 22 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..22

Cf. A007955, A006308. The exponents x are in A008837.

Table[2^((p(p-1))/2),{p,Prime[Range[10]]}] (* Harvey P. Dale, Sep 03 2024 *)

{ n=0; forprime (p=2, prime(22), write("b061591.txt", n++, " ", 2^(p*(p - 1)/2)) ) } \\ Harry J. Smith, Jul 25 2009

Offset changed from 2 to 1 by Harry J. Smith, Jul 25 2009

A164623 Primes p such that p(p-1)/2-5 and p(p-1)/2+5 are also prime numbers.

Original entry on oeis.org

13, 157, 673, 1069, 1117, 1153, 1213, 1597, 2029, 2089, 2437, 2713, 2833, 3613, 4057, 4909, 5653, 6337, 6529, 7549, 7993, 8053, 9613, 10789, 11497, 11689, 12073, 12373, 13309, 13669, 13789, 14173, 15289, 15937, 16249, 18097, 18637, 19249, 19993

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

Primes A000040(k) such that A008837(k)+-5 are also prime numbers.

			13 is in the sequence because 13*6-5=73 and 13*6+5=83 are both prime.

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

Cf. A008846, A020882, A068228, A068229, A082539, A086519, A107159, A158708, A139494, A164620, A164621, A164622, A023203.

Select[Prime[Range[2300]], PrimeQ[# (# - 1)/2 - 5] && PrimeQ[# (# - 1)/2 + 5] &]

forprime(p=2,10^6,my(b=binomial(p,2));if(isprime(b-5)&isprime(b+5),print1(p,", "))); /* Joerg Arndt, Apr 10 2013 */

Edited by R. J. Mathar, Aug 20 2009

Mathematica code adapted to the definition by Bruno Berselli, Apr 10 2013

A253655 Number of monic irreducible polynomials of degree 6 over GF(prime(n)).

Original entry on oeis.org

9, 116, 2580, 19544, 295020, 804076, 4022064, 7839780, 24670536, 99133020, 147912160, 427612404, 791672280, 1053546956, 1796518224, 3694034916, 7030054140, 8586690620, 15076346164, 21349986840, 25222305336, 40514492720, 54489965796, 82830096360, 138828513824, 176919851700

Offset: 1

Robert Israel, Jan 07 2015

nonn

			For n=1 the a(1) = 9 irreducible monic polynomials of degree 6 over GF(2) are
x^6+x^5+1, x^6+x^3+1, x^6+x^5+x^4+x^2+1, x^6+x^5+x^3+x^2+1, x^6+x+1, x^6+x^5+x^4+x+1, x^6+x^4+x^3+x+1, x^6+x^5+x^2+x+1, x^6+x^4+x^2+x+1.

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

Cf. A008837, A127919, A138420, A138426.

[(p^6 - p^3 - p^2 + p) div 6: p in PrimesUpTo(110)]; // Vincenzo Librandi, Jan 08 2015

f:= p-> (p^6 - p^3 - p^2 + p)/6:
seq(f(ithprime(i)), i=1..100); # Robert Israel, Jan 07 2015

Table[(Prime[n]^6 - Prime[n]^3 - Prime[n]^2 + Prime[n]) / 6, {n, 1, 30}] (* Vincenzo Librandi, Jan 08 2015 *)

a(n) = (p^6 - p^3 - p^2 + p)/6, where p = prime(n).

A261365 Prime-numbered rows of Pascal's triangle.

Original entry on oeis.org

1, 2, 1, 1, 3, 3, 1, 1, 5, 10, 10, 5, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1, 1, 17, 136, 680, 2380, 6188, 12376, 19448, 24310, 24310, 19448, 12376, 6188, 2380, 680, 136, 17, 1, 1, 19, 171, 969, 3876, 11628, 27132, 50388, 75582, 92378, 92378, 75582, 50388, 27132, 11628, 3876, 969, 171, 19, 1

Offset: 1

Maghraoui Abdelkader, Aug 16 2015

			1,2,1;
1,3,3,1;
1,5,10,10,5,1;
1,7,21,35,35,21,7,1;
1,11,55,165,330,462,462,330,165,55,11,1;

Maghraoui Abdelkader, Table of n, a(n) for n = 1..4273

Cf. A007318, A034870, A034871.

Cf. A000040 (2nd column), A008837 (3rd column).

Table[Binomial[Prime@ n, k], {n, 8}, {k, 0, Prime@ n}] // Flatten (* Michael De Vlieger, Aug 20 2015 *)

forprime(n=2, 20, for(k=0,n,print1(binomial(n,k),", ")))

T(n,k) = binomial(prime(n), k).

cp's OEIS Frontend

A273222 a(n) = p*(p - 1)*(73*p^2 - 45*p + 14)/24, where p = prime(n).

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A121057 Triangle read by rows: T(n,m) = Prime[m]^n*(Prime[m] - 1)/2.

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

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A231808 Numerator of asymptotic density of Union{H_p: p is odd prime and p <= n-th prime}, where H_p is {K*p*(p-1)/2 : K integer}.

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A231809 Denominator of asymptotic density of Union{H_p: p is odd prime and p <= n-th prime}, where H_p is {K*p*(p-1)/2 : K integer}.

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A273223 a(n) = p*(p - 1)*(501*p^3 - 414*p^2 + 111*p - 54)/120, where p = prime(n).

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Magma

Mathematica

A061591 a(n) = 2^x, where x = p*(p-1)/2 and p is the n-th prime.

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Extensions

A164623 Primes p such that p*(p-1)/2-5 and p*(p-1)/2+5 are also prime numbers.

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Extensions

A253655 Number of monic irreducible polynomials of degree 6 over GF(prime(n)).

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A273222 a(n) = p(p - 1)(73p^2 - 45p + 14)/24, where p = prime(n).

A231808 Numerator of asymptotic density of Union{H_p: p is odd prime and p <= n-th prime}, where H_p is {Kp(p-1)/2 : K integer}.

A231809 Denominator of asymptotic density of Union{H_p: p is odd prime and p <= n-th prime}, where H_p is {Kp(p-1)/2 : K integer}.

A273223 a(n) = p(p - 1)(501p^3 - 414p^2 + 111*p - 54)/120, where p = prime(n).

A164623 Primes p such that p(p-1)/2-5 and p(p-1)/2+5 are also prime numbers.