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 1-5 of 5 results.

A074781 Primes of the form p*2^k + 1 for any k and any prime p.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 23, 29, 41, 47, 53, 59, 83, 89, 97, 107, 113, 137, 149, 167, 173, 179, 193, 227, 233, 257, 263, 269, 293, 317, 347, 353, 359, 383, 389, 449, 467, 479, 503, 509, 557, 563, 569, 587, 593, 641, 653, 719, 769, 773, 797, 809, 839, 857, 863, 887
Offset: 1

Views

Author

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

Keywords

Comments

From Bernard Schott, Dec 14 2020: (Start)
Equivalently, primes p such that the ratio (p-1)/gpf(p-1) = 2^k where gpf(m) is the greatest prime factor of m, A006530.
Paul Erdős asked if there are infinitely many primes p in this sequence (see R. K. Guy reference). (End)

Examples

			3 = 2*2^0+1 is a term and 2/2 = 1 = 2^0.
7 = 3*2^1+1 is a term and 6/3 = 2 = 2^1.
13 = 3*2^2+1 is a term and 12/3 = 4 = 2^2.
41 = 5*2^3+1 is a term and 40/5 = 8 = 2^3.
113 = 7*2^4+1 is a term and 112/7 = 16 = 2^4.

References

  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.

Links

Crossrefs

Cf. A000079, A006093, A006530, A052126, A339464.
Cf. other ratios : A339463, A339465, A339466.
Subsequences: A039687, A051900, A058500 (this sequence without the Fermat primes), A090866, A147545,

Programs

  • Maple
    alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2}:
3, op(select(is_a, [$3..919])); # Peter Luschny, Dec 14 2020
  • Mathematica
    Select[Range[3, 1000], PrimeQ[#] && !CompositeQ[(# - 1)/2^IntegerExponent[(# - 1), 2]] &] (* Amiram Eldar, Dec 28 2018 *)

A339464 a(n) = (prime(n)-1) / gpf(prime(n)-1) where gpf(m) is the greatest prime factor of m, A006530.

Original entry on oeis.org

1, 2, 2, 2, 4, 8, 6, 2, 4, 6, 12, 8, 6, 2, 4, 2, 12, 6, 10, 24, 6, 2, 8, 32, 20, 6, 2, 36, 16, 18, 10, 8, 6, 4, 30, 12, 54, 2, 4, 2, 36, 10, 64, 28, 18, 30, 6, 2, 12, 8, 14, 48, 50, 128, 2, 4, 54, 12, 40, 6, 4, 18, 10, 24, 4, 30, 48, 2, 12, 32, 2, 6, 12, 54, 2
Offset: 2

Views

Author

Bernard Schott, Dec 06 2020

Keywords

Comments

Paul Erdős asked if there are infinitely many primes p such that (p-1)/A006530(p-1) = 2^k or = 2^q*3^r (see Richard K. Guy reference).
A074781 is the sequence of primes p such that (p-1)/A006530(p-1) = 2^k.
A339465 is the sequence of primes p such that (p-1)/A006530(p-1) = 2^q*3^r with q, r >=1.
It is not known if these two sequences are infinite.

Examples

			Prime(6) = 13 and a(6) = 12/3 = 4 = 2^2.
Prime(11) = 31 and a(11) = 30/5 = 6 = 2*3.
Prime(20) = 71 and a(20) = 70/7 =10 = 2*5.
Prime(36) = 151 and a(36) = 150/5 = 30 = 2*3*5.

References

  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.

Links

Crossrefs

Cf. A006093 (prime(n)-1), A006530, A052126, A074781 (ratio = 2^k), A339465 (ratio = 2^q*3^r), A339466 (ratio <> 2^k and <> 2^q*3^r).

Programs

  • Mathematica
    f[n_] := n/FactorInteger[n][[-1, 1]]; f /@ (Select[Range[3, 400], PrimeQ] - 1) (* Amiram Eldar, Dec 07 2020 *)
  • PARI
    gpf(n) = vecmax(factor(n)[, 1]); \\ A006530
a(n) = my(x=prime(n)-1); x/gpf(x); \\ Michel Marcus, Dec 07 2020

Formula

a(n) = A006093(n)/A006530(A006093(n)).
a(n) = A052126(A006093(n)). - Michel Marcus, Dec 07 2020

A339465 Primes p such that (p-1)/gpf(p-1) = 2^q * 3^r with q, r >= 1, where gpf(m) is the greatest prime factor of m, A006530.

Original entry on oeis.org

19, 31, 37, 43, 61, 67, 73, 79, 103, 109, 127, 139, 157, 163, 181, 199, 223, 229, 241, 271, 277, 283, 307, 313, 337, 349, 367, 373, 379, 397, 409, 433, 439, 457, 487, 499, 523, 541, 577, 607, 613, 619, 643, 673, 709, 733, 739, 757, 787, 811, 823, 829, 853, 877, 907, 919
Offset: 1

Views

Author

Bernard Schott, Dec 09 2020

Keywords

Comments

Paul Erdős asked if there are infinitely many primes p such that (p-1)/A006530(p-1) = 2^k or = 2^q*3^r (see Richard K. Guy reference).
It is not known if this sequence is infinite.
Proposition: if prime p is a term, then p is of the form 6*m+1 (A002476).

Examples

			31 is prime, 30/5 = 6 = 2*3 hence 31 is a term.
37 is prime, 36/3 = 12 = 2^2*3 hence 37 is a term.
127 is prime, 126/7 = 18 = 2*3^2 hence 127 is a term.

References

  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.

Links

Crossrefs

Cf. A006093, A006530, A033845, A052126, A339464.
Cf. A074781 (ratio=2^k), A339466 (ratio <> 2^k and <> 2^q*3^r).
Subsequence of A002476.

Programs

  • Magma
    s:=func; [p:p in PrimesInInterval(3,1000)|PrimeDivisors(a) eq [2,3] where a is (p-1) div s(p-1)]; // Marius A. Burtea, Dec 09 2020
  • Maple
    alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2, 3}:
select(is_a, [$3..919]); # Peter Luschny, Dec 13 2020
  • Mathematica
    q[n_] := PrimeQ[n] && Module[{f = FactorInteger[n - 1]}, (Length[f] == 2 && f[[2, 1]] == 3 && f[[2, 2]] > 1) || (Length[f] == 3 && f[[2, 1]] == 3 && f[[3, 2]] == 1)]; Select[Range[1000], q] (* Amiram Eldar, Dec 09 2020 *)

Extensions

More terms from Marius A. Burtea, Dec 09 2020

A339463 Primes p such that (p-1)/gpf(p-1) = 2^q * 5^r with q, r >= 1, where gpf(m) is the greatest prime factor of m, A006530.

Original entry on oeis.org

71, 101, 131, 191, 251, 281, 311, 401, 431, 461, 521, 701, 761, 821, 881, 941, 971, 1031, 1061, 1091, 1151, 1181, 1301, 1361, 1451, 1481, 1511, 1571, 1601, 1721, 1811, 1901, 1931, 2081, 2111, 2141, 2351, 2411, 2441, 2621, 2711, 2741, 2801, 3041, 3251, 3371
Offset: 1

Views

Author

Bernard Schott, Dec 13 2020

Keywords

Comments

These primes that are all congruent to 11 (mod 30) form a subsequence of A132232. The first terms of A132232 that are not terms here are 11, 41, 491, ... (see examples)

Examples

			41 is prime, 40/5 = 8 = 2^3, hence 41 is not a term.
101 is prime, 100/5 = 20 = 2^2 * 5, hence 101 is a term.
491 is prime, 490/7 = 70 = 2 * 5 * 7, hence 491 is not a term.
521 is prime, 520/13 = 40 = 2^3 * 5, hence 521 is a term.

References

  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.

Links

Crossrefs

Cf. A006093 (prime(n)-1), A006530, A052126, A339464.
Cf. A074781 (ratio=2^k), A339465 (ratio=2^q*3^r).
Subsequence of A132232 and of A339466.

Programs

  • Maple
    alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2, 5}:
select(is_a, [$5..3371]); # Peter Luschny, Dec 13 2020
  • Mathematica
    q[n_] := Divisible[n, 10] && ((PrimeQ[(r = n/2^IntegerExponent[n, 2]/5^(e = IntegerExponent[n, 5]))] && r > 5) || (r == 1 && e > 1)); Select[Range[3500], PrimeQ[#] && q[# - 1] &] (* Amiram Eldar, Dec 13 2020 *)

A339347 Primes p such that p < (gpf((p - 1)/gpf(p - 1)))^4, where gpf(k) is the greatest prime factor of k, A006530.

Original entry on oeis.org

5, 7, 11, 13, 19, 31, 37, 43, 61, 67, 71, 73, 79, 101, 131, 151, 191, 197, 211, 239, 251, 281, 311, 331, 401, 419, 421, 431, 443, 461, 463, 491, 521, 547, 571, 599, 601, 617, 647, 659, 677, 683, 727, 743, 827, 859, 883, 911, 947, 953, 967, 1013, 1093, 1103
Offset: 1

Views

Author

Peter Luschny, Dec 13 2020

Keywords

Comments

Inspired by A339466. See the references there.

Links

Crossrefs

Cf. A006530, A339466.

Programs

  • Maple
    alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and n < (gpf((n-1)/gpf(n-1)))^4:
select(is_a, [$5..1150]);
  • PARI
    gpf(n) = if (n==1, 1, vecmax(factor(n)[, 1])); \\ A006530
isok(p) = isprime(p) && (p < (gpf((p - 1)/gpf(p - 1)))^4); \\ Michel Marcus, Dec 14 2020
Showing 1-5 of 5 results.

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