cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A339883 Values of Euler's totient phi for A050498.

Original entry on oeis.org

24, 72, 25440, 33840, 38880, 48960, 99360, 123120, 208320, 458640, 510720, 519360, 532800, 583440, 596400, 622080, 663840, 838800, 846960, 873600, 893760, 942480, 1050480, 1065600, 1078800, 1163040, 1201200, 1327200, 1408320, 1419840, 1567440, 1734000, 1809600
Offset: 1

Views

Author

Wolfdieter Lang, Jan 09 2021

Keywords

Comments

For the shown values the arithmetic progression with difference 6 has exactly 4 terms. Conjecture 1: this holds for all A050498 entries.
Conjecture 2: All a(n) are divisible by 24, starting with [1, 3, 1060, 1410, 1620, 2040, 4140, 5130, 8680, 19110, 21280, 21640, 22200, 24310, 24850, 25920, 27660, 34950, 35290, 36400, 37240, 39270, 43770, 44400, 44950, 48460, ...].
In the Lal and Gillard link only the first three A050498 values (with n <= 10^5) and their corresponding phi values are given.

References

  • David Wells, Curious and interesting numbers, Penguin Books, Revised edition, 1997 p. 112. [Gives under the number 72 the first three values of A050498 but with 76236 instead of 76326]

Links

Crossrefs

Cf. A000010, A050498, A163573.

Formula

a(n) = A000010(A050498(n)), n >= 1.

A163573 Primes p such that (p+1)/2, (p+2)/3 and (p+3)/4 are also primes.

Original entry on oeis.org

12721, 16921, 19441, 24481, 49681, 61561, 104161, 229321, 255361, 259681, 266401, 291721, 298201, 311041, 331921, 419401, 423481, 436801, 446881, 471241, 525241, 532801, 539401, 581521, 600601, 663601, 704161, 709921, 783721, 867001, 904801
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 31 2009

Keywords

Comments

Are all terms == 1 (mod 10)?
Subsequence of A005383, of A091180 and of A036570. - R. J. Mathar, Aug 01 2009
Since (p+2)/3 and (p+3)/4 must be integer, the Chinese remainder theorem shows that all terms are == 1 (mod 12). - R. J. Mathar, Aug 01 2009
All terms are of the form 120k+1: a(n)=120*A163625(n)+1. - Zak Seidov, Aug 01 2009
Each term is congruent to 1 mod 120, so the last digits are always '1': For all four values to be integers it must be that p = 1 (mod 12). As p is prime, it must be that p = 1, 13, 37, 49, 61, 73, 97, or 109 (mod 120). In all but the first case either (p+3)/4 is even or one of the three expressions gives a value divisible by 5 (or both, and possibly the same expression). - Rick L. Shepherd, Aug 01 2009
{6*a(n)}A050498.%20Proof:%20with%20p%20=%20a(n)%20the%20arithmetic%20progression%20with%20four%20terms%20of%20difference%206%20and%20constant%20value%20of%20Euler's%20phi,%20namely%202*(p-1),%20is%206*(p,%202*(p+1)/2,%203*(p+2)/3,%204*(p+3)/4).%20Use%20phi(n,%20prime)%20=%20phi(n)*(prime-1)%20if%20gcd(n,%20prime)%20=%201.%20Here%20n%20=%206,%2012,%2018,%2024%20and%20prime%20%3E%203%20for%20p%20%3E=%20a(1).%20Thanks%20to%20_Hugo%20Pfoertner">{n >= 1} is a subsequence of A050498. Proof: with p = a(n) the arithmetic progression with four terms of difference 6 and constant value of Euler's phi, namely 2*(p-1), is 6*(p, 2*(p+1)/2, 3*(p+2)/3, 4*(p+3)/4). Use phi(n, prime) = phi(n)*(prime-1) if gcd(n, prime) = 1. Here n = 6, 12, 18, 24 and prime > 3 for p >= a(1). Thanks to _Hugo Pfoertner for a link to the present sequence in connection with A339883. - Wolfdieter Lang, Jan 11 2021

Links

Crossrefs

Cf. A005383, A091180, A036570, A050498, A163623, A163624, A163625, A278583, A278585, A339883.

Programs

  • Magma
    [p: p in PrimesInInterval(6, 1200000) | IsPrime((p+1) div 2) and IsPrime((p+2) div 3) and IsPrime((p+3) div 4)]; // Vincenzo Librandi, Apr 09 2013
  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[(p+1)/2]&&PrimeQ[(p+2)/3]&&PrimeQ[(p+3)/ 4],AppendTo[lst,p]],{n,2*9!}];lst
  • PARI
    is(n)=n%120==1 && isprime(n) && isprime(n\2+1) && isprime(n\3+1) && isprime(n\4+1) \\ Charles R Greathouse IV, Nov 30 2016
  • Python
    from sympy import prime, isprime
A163573_list = [4*q-3 for q in (prime(i) for i in range(1,10000)) if isprime(4*q-3) and isprime(2*q-1) and (not (4*q-1) % 3) and isprime((4*q-1)//3)] # Chai Wah Wu, Nov 30 2016

Extensions

Slightly edited by R. J. Mathar, Aug 01 2009

A039670 Sets of 4 numbers in arithmetic progression with common difference 6 and whose phi values are equal.

Original entry on oeis.org

72, 78, 84, 90, 216, 222, 228, 234, 76326, 76332, 76338, 76344, 101526, 101532, 101538, 101544, 116646, 116652, 116658, 116664, 146886, 146892, 146898, 146904, 298086, 298092, 298098, 298104, 369366, 369372, 369378, 369384, 624966, 624972
Offset: 1

Views

Author

Felice Russo

Keywords

Comments

This is really four sequences, not one! A050498 is a better version. - N. J. A. Sloane, Jan 01 2005

References

  • D. Wells, Curious and interesting numbers, Penguin Books, p. 112 (but beware errors).

Programs

  • Maple
    with(numtheory):for n from 1 to 1000000 do if(phi(n)=phi(n+6) and phi(n+6)=phi(n+12) and phi(n+12)=phi(n+18)) then printf("%d, %d, %d, %d, ",n,n+6,n+12,n+18) fi od: # C. Ronaldo

Extensions

Corrected by Jud McCranie, Dec 26 1999
More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 29 2004
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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