A339465 -id:A339465 - cp's OEIS frontend

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

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

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

nonn

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)

			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.

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

T. D. Noe, Table of n, a(n) for n = 1..2000
Graeme L. Cohen, On a conjecture of Makowski and Schinzel, Colloquium Mathematicae, Vol. 74, No. 1 (1997), pp. 1-8.

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

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

Select[Range[3, 1000], PrimeQ[#] && !CompositeQ[(# - 1)/2^IntegerExponent[(# - 1), 2]] &] (* Amiram Eldar, Dec 28 2018 *)

A339466 Primes p such that gpf((p - 1)/gpf(p - 1)) > 3, where gpf(m) is the greatest prime factor of m, A006530.

Original entry on oeis.org

71, 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, 631, 647, 659, 661, 677, 683, 691, 701, 727, 743, 751, 761, 821, 827, 859, 881, 883, 911, 941, 947, 953, 967, 971, 991, 1013, 1021

Offset: 1

Bernard Schott, Dec 10 2020

nonn

Paul Erdős asked if there are infinitely many primes p such that (p-1)/gpf(p-1) = 2^k or = 2^q * 3^r (see Richard K. Guy reference). This sequence lists the primes p that do not satisfy these two previous relations.

Replacing in the definition gpf by lpf (A020639) leads to A122259. In fact this sequence is a subsequence of A122259. - Peter Luschny, Dec 13 2020

			71 is prime, 70/7 = 10 = 2*5 hence 71 is a term.
101 is prime, 100/5 = 20 = 2^2*5 hence 101 is a term.
151 is prime, 150/5 = 30 = 2*3*5 hence 151 is a term.
The first few quotients obtained are: 10, 20, 10, 30, 10, 28, 30, 14, 50, 40, ...

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

Robert Israel, Table of n, a(n) for n = 1..10000
P. Erdős and C. Pomerance, On the largest prime factors of n and n+1, Aequationes Math. 17 (1978), p. 311-321. [alternate link]

Cf. A006093, A006530, A052126, A339464.
Cf. A074781 (ratio=2^k), A339465 (ratio=2^q*3^r), A339463 (ratio=2^q*5^r).
Cf. A122259.

s:=func; [p:p in PrimesInInterval(3,1100)|( not 3 in PrimeDivisors(a) and #PrimeDivisors(a) ge 2) or ( 3 in PrimeDivisors(a) and #PrimeDivisors(a) ge 3) where a is (p-1) div s(p-1)]; // Marius A. Burtea, Dec 10 2020

alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and gpf((n-1)/gpf(n-1)) > 3:
select(is_a, [$5..1021]); # Peter Luschny, Dec 13 2020

q[n_] := Module[{f = FactorInteger[n - 1]}, (Length[f] == 1 && f[[1, 1]] == 2) || (Length[f] == 2 && f[[1, 1]] == 2 && f[[2, 2]] == 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[3, 1000], PrimeQ[#] && ! q[#] &] (* Amiram Eldar, Dec 10 2020 *)

is(n) = {if(!isprime(n) || n==2, return(0)); my(pm1 = n-1, f = factor(pm1)[,1]); (pm1 /= (f[#f]*1<1} \\ David A. Corneth, Dec 13 2020

More terms from Amiram Eldar, Dec 11 2020

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

Bernard Schott, Dec 06 2020

nonn

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.

			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.

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

MacTutor History of Mathematics, Paul Erdős.

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).

f[n_] := n/FactorInteger[n][[-1, 1]]; f /@ (Select[Range[3, 400], PrimeQ] - 1) (* Amiram Eldar, Dec 07 2020 *)

gpf(n) = vecmax(factor(n)[, 1]); \\ A006530
a(n) = my(x=prime(n)-1); x/gpf(x); \\ Michel Marcus, Dec 07 2020

a(n) = A006093(n)/A006530(A006093(n)).

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

Bernard Schott, Dec 13 2020

nonn

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)

			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.

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

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

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.

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

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 *)

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

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A339466 Primes p such that gpf((p - 1)/gpf(p - 1)) > 3, where gpf(m) is the greatest prime factor of m, A006530.

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

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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.

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