Joshua S.M. Weiner - cp's OEIS frontend

A228093 Primes congruent to 5 (mod 504).

5, 509, 1013, 3533, 6053, 8069, 8573, 10589, 11093, 11597, 12101, 13109, 13613, 14621, 15629, 18149, 19157, 19661, 23189, 24197, 26717, 28229, 29741, 31253, 32261, 33773, 34781, 36293, 39317, 39821, 40829, 41333, 43853, 44357, 45869, 46877, 47381, 50909, 51413

Offset: 1

Joshua S.M. Weiner, Aug 09 2013

[p: p in PrimesUpTo(105500) | p mod 504 eq 5 ];

select(isprime, [5+504*i$i=0..200])[];  # Alois P. Heinz, Jan 01 2022

Select[Prime@Range[6000], MemberQ[{5}, Mod[#, 504]] &] (* Vincenzo Librandi, Apr 05 2015 *)

Missing a(30)=39821 inserted by Georg Fischer, Jan 01 2022

A218037 Decimal numbers with exactly two consecutive zeros.

Original entry on oeis.org

100, 200, 300, 400, 500, 600, 700, 800, 900, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2100, 2200, 2300, 2400, 2500, 2600, 2700, 2800

Offset: 1

Joshua S.M. Weiner, Oct 19 2012

The various "beatmap" sequences used in music, dance choreography and even juggling has been standardized with an often employed "00" subsequence. This "00" represents two empty beats, like a musical rest of 2 beats in a measure.

Wikipedia, Juggling Notation

Cf. A043490 (superset).

A218043 Base 3 numbers that have digits that sum to 3.

Original entry on oeis.org

12, 21, 102, 111, 120, 201, 210, 1002, 1011, 1020, 1101, 1110, 1200, 2001, 2010, 2100, 10002, 10011, 10020, 10101, 10110, 10200, 11001, 11010, 11100, 12000, 20001, 20010, 20100, 21000, 100002, 100011, 100020, 100101, 100110, 100200, 101001, 101010, 101100

Offset: 1

Joshua S.M. Weiner, Oct 19 2012

It is a representation of the "energy states" of "multiplex pair" notation of 3 quantum of objects in a juggling pattern.

0 = an empty site, or empty hand. 1 = one object resides in the site. 2 = two objects reside in the site.

Eric W. Weisstein, Mathworld: Siteswap

t = Select[Range[500], Total[IntegerDigits[#, 3]] == 3 &]; FromDigits /@ IntegerDigits[t, 3] (* T. D. Noe, Oct 19 2012 *)

A217863 a(n) = phi(lcm(1,2,3,...,n)), where phi is Euler's totient function.

Original entry on oeis.org

1, 1, 2, 4, 16, 16, 96, 192, 576, 576, 5760, 5760, 69120, 69120, 69120, 138240, 2211840, 2211840, 39813120, 39813120, 39813120, 39813120, 875888640, 875888640, 4379443200, 4379443200, 13138329600, 13138329600, 367873228800, 367873228800, 11036196864000

Offset: 1

Joshua S.M. Weiner, Oct 13 2012

This is a composition f(g(x)). g(x) = lcm(1...x) and f(x) = phi(x), Euler's totient function. The sequence generated is the number of prime congruence classes (prime spokes) for wheel factorization in mod g(x).

First column of A096180. - Eric Desbiaux, Apr 23 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000010 (Euler phi), A003418 (LCM), A072211, A173557.

a217863 = a000010 . a003418  -- Reinhard Zumkeller, Nov 24 2012

with(numtheory): a:=n->phi(lcm(seq(m,m=1..n))): seq(a(n),n=1..40); # Muniru A Asiru, Feb 20 2019

EulerPhi[Table[LCM @@ Range[n], {n, 35}]] (* T. D. Noe, Oct 16 2012 *)

a(n) = eulerphi(lcm(vector(n, k, k))); \\ Michel Marcus, Aug 25 2015

a(n) = A000010(A003418(n)). - Omar E. Pol, Nov 25 2012
From Peter Bala, Feb 19 2019: (Start)
a(n) = Product_{k = 1..n} A072211(k).
With p denoting a prime, a(n) = ( Product_{p <= n} (p - 1) ) * ( Product_{p^2 <= n} p ) * ( Product_{p^3 <= n} p ) * ... . For example, a(16) = ((2-1)*(3-1)*(5-1)*(7-1)*(11-1)*(13-1)) * (2*3) * 2 * 2 = 138240. (End)

A217862 Primes p of the form p = 1 + 840*k for some k.

Original entry on oeis.org

2521, 3361, 4201, 5881, 7561, 9241, 12601, 13441, 14281, 15121, 18481, 20161, 21001, 21841, 26041, 26881, 29401, 30241, 31081, 33601, 35281, 41161, 42841, 45361, 47041, 47881, 51241, 52081, 54601, 55441, 63841, 65521, 66361, 68041, 68881, 74761, 76441, 78121

Offset: 1

Joshua S.M. Weiner, Oct 13 2012

This is a prime sequence based on the wheel factorization of 840. There are 192 congruence classes that form prime wheel spokes mod 840.

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

Select[Prime[Range[5000]], Mod[#, 840] == 1 &]
Select[840*Range[0,100]+1,PrimeQ] (* Harvey P. Dale, Mar 03 2018 *)

A217656 Primes p such that p = 361 + 420*k for some k.

Original entry on oeis.org

1201, 1621, 3301, 4561, 5821, 6661, 8761, 9181, 9601, 10861, 11701, 12541, 13381, 14221, 15061, 15901, 16741, 17581, 19681, 20101, 20521, 22621, 23041, 25561, 25981, 26821, 27241, 28081, 28921, 29761, 30181, 33961, 34381, 35221, 36061, 36901, 37321

Offset: 1

Joshua S.M. Weiner, Oct 09 2012

nonn

This is one prime congruence class of a wheel factorization of mod 420. The wheel has 96 spokes.

Select[Prime[Range[50000]], Mod[#, 420]== 361 &]

A217692 Primes p such that p = 1 + 27720*k for some k.

Original entry on oeis.org

55441, 110881, 332641, 388081, 415801, 471241, 498961, 526681, 748441, 859321, 970201, 1025641, 1053361, 1108801, 1247401, 1275121, 1302841, 1358281, 1469161, 1580041, 1912681, 1940401, 1995841, 2051281, 2189881, 2273041, 2300761, 2383921, 2411641, 2855161

Offset: 1

Joshua S.M. Weiner, Oct 11 2012

This is a congruence class of a prime wheel factorization mod 27720. Note that 27720 is the LCM of {1,...,11}.

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A002476, A068228, A088955, A217587, A217588.

[p: p in PrimesUpTo(3*10^6) | IsOne(p mod 27720)]; // Bruno Berselli, Oct 12 2012

Select[Table[1 + 27720*k, {k, 200}], PrimeQ] (* T. D. Noe, Oct 11 2012 *)

A217588 Primes of the form 2520k + 1 for some k.

Original entry on oeis.org

2521, 7561, 12601, 15121, 20161, 30241, 35281, 42841, 45361, 47881, 55441, 65521, 68041, 78121, 93241, 100801, 110881, 126001, 128521, 131041, 141121, 146161, 151201, 156241, 158761, 161281, 176401, 178921, 186481, 196561, 199081, 206641, 211681, 229321

Offset: 1

Joshua S.M. Weiner, Oct 07 2012

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

Subsequence of A217587.

[p: p in PrimesInInterval(2521,260000) | IsOne(p mod 2520)]; // Bruno Berselli, Oct 10 2012

Select[1 + 2520*Range[100], PrimeQ] (* T. D. Noe, Oct 10 2012 *)

A217587 Primes p of the form 420k + 1 for some k.

Original entry on oeis.org

421, 2521, 3361, 4201, 4621, 5881, 6301, 7561, 8821, 9241, 9661, 10501, 12601, 13441, 14281, 15121, 15541, 16381, 18061, 18481, 20161, 21001, 21841, 24781, 25621, 26041, 26881, 29401, 30241, 30661, 31081, 32341, 33181, 33601, 35281, 36541, 39901, 41161

Offset: 1

Joshua S.M. Weiner, Oct 07 2012

nonn

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

Subsequence of A073102.

Select[Prime[Range[5000]], Mod[#, 420] == 1 &] (* T. D. Noe, Oct 08 2012 *)
Select[420*Range[100]+1,PrimeQ] (* Harvey P. Dale, Jun 06 2013 *)

cp's OEIS Frontend

User: Joshua S.M. Weiner

A228093 Primes congruent to 5 (mod 504).

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A218037 Decimal numbers with exactly two consecutive zeros.

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A218043 Base 3 numbers that have digits that sum to 3.

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A217863 a(n) = phi(lcm(1,2,3,...,n)), where phi is Euler's totient function.

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A217862 Primes p of the form p = 1 + 840*k for some k.

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A217656 Primes p such that p = 361 + 420*k for some k.

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A217692 Primes p such that p = 1 + 27720*k for some k.

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Magma

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A217588 Primes of the form 2520k + 1 for some k.

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Magma

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A217587 Primes p of the form 420k + 1 for some k.

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