A078701 -id:A078701 - cp's OEIS frontend

A079602 Greatest of smallest odd prime factors of all composite numbers between n-th prime and next prime.

3, 5, 3, 7, 3, 11, 13, 3, 17, 19, 3, 23, 13, 29, 3, 31, 17, 3, 37, 41, 43, 47, 7, 3, 53, 3, 7, 61, 5, 67, 3, 73, 3, 19, 79, 83, 43, 89, 3, 47, 3, 97, 3, 103, 109, 113, 3, 29, 59, 3, 61, 127, 131, 67, 3, 137, 139, 3, 73, 151, 7, 3, 157, 163, 167, 173, 3, 11, 179, 181, 23, 47, 191

Offset: 3

Reinhard Zumkeller, Jan 28 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 3..10000

Cf. A078701, A052180.

a[n_] := Max[FactorInteger[#/2^IntegerExponent[#, 2]][[1, 1]] & /@ Range[Prime[n] + 1, Prime[n + 1] - 1]]; Array[a, 100, 3] (* Amiram Eldar, Mar 28 2025 *)

A338101 Smallest odd prime dividing n is a(n)-th prime, or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 2, 0, 3, 2, 4, 0, 2, 3, 5, 2, 6, 4, 2, 0, 7, 2, 8, 3, 2, 5, 9, 2, 3, 6, 2, 4, 10, 2, 11, 0, 2, 7, 3, 2, 12, 8, 2, 3, 13, 2, 14, 5, 2, 9, 15, 2, 4, 3, 2, 6, 16, 2, 3, 4, 2, 10, 17, 2, 18, 11, 2, 0, 3, 2, 19, 7, 2, 3, 20, 2, 21, 12, 2, 8, 4, 2, 22, 3, 2, 13, 23, 2, 3, 14, 2, 5, 24, 2

Offset: 1

Ilya Gutkovskiy, Nov 10 2020

nonn

			70 = 2 * 5 * 7 = prime(1) * prime(3) * prime(4), 3 < 4, so a(70) = 3.

Cf. A000079 (positions of 0's), A000265, A000720, A020639, A055396, A078701.

f:= proc(n) local w;
   w:= numtheory:-factorset(n) minus {2};
   if w = {} then 0 else numtheory:-pi(min(w)) fi
end proc:
map(f, [$1..100]); # Robert Israel, Nov 13 2020

Array[If[Or[# == 1, ! IntegerQ@ #], 0, PrimePi@ #] &@ SelectFirst[FactorInteger[#][[All, 1]], OddQ] &, 90] (* Michael De Vlieger, Nov 13 2020 *)

a(n) = my(v = select(x->((x%2)==1), factor(n)[, 1])); if (#v, primepi(vecmin(v)), 0); \\ Michel Marcus, Nov 13 2020

a(n) = A000720(A020639(A000265(n))).

a(n) = A000720(A078701(n)).

A372263 Least odd prime factor of the n-th sum of two consecutive primes, A001043(n) = prime(n) + prime(n+1), or 2 if there is no odd prime factor.

Original entry on oeis.org

5, 2, 3, 3, 3, 3, 3, 3, 13, 3, 17, 3, 3, 3, 5, 7, 3, 2, 3, 3, 19, 3, 43, 3, 3, 3, 3, 3, 3, 3, 3, 67, 3, 3, 3, 7, 5, 3, 5, 11, 3, 3, 3, 3, 3, 5, 7, 3, 3, 3, 59, 3, 3, 127, 5, 7, 3, 137, 3, 3, 3, 3, 3, 3, 3, 3, 167, 3, 3, 3, 89, 3, 5, 47, 3, 193, 3, 3, 3, 3, 3, 3, 3, 109, 3, 223

Offset: 1

M. F. Hasler, Apr 24 2024

nonn

Since the sum of any two primes > 2 is even, we rather consider odd prime factors.
Can it be proved or disproved that there are primes that occur only finitely many times (or never) in this sequence? If so, which is the smallest such prime?
From Robert Israel, Dec 29 2024: (Start)
Dickson's conjecture implies that every odd prime occurs infinitely many times in the sequence.
a(n) = 2 if and only if n = A000720(2^k) where k is in A226178. (End)

			Sums of two consecutive primes are given as s(n) = A001043(n). The least odd prime factor (or 2 if there's no odd prime factor) of these terms is a(n):
n = 1, 2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14,  15,  16,  17,  18, ...
s = 5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, ...
a = 5, 2,  3,  3,  3,  3,  3,  3, 13,  3, 17,  3,  3,  3,   5,   7,   3,   2, ...
Also, a(21) = spf(152) = 19; a(23) = spf(172) = 43; a(32) = spf(268) = 67, ...

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

Cf. A001043 (sums of two consecutive primes),  A078701 (least odd prime divisor), A020639 (spf: least prime factor), A000265 (odd part of n), A000079 (powers of 2).
Cf. A024677 (spf of A024675(n) = A001043(n)/2).
Cf. A226178.

f:= proc(n) subs(infinity=2,min(numtheory:-factorset(ithprime(n)+ithprime(n+1)) minus {2})) end proc:
map(f, [$1..100]); # Robert Israel, Dec 29 2024

apply( {a(n) = max(A078701(A001043(n)), 2)}, [1..99])
/* a "self-contained" but less efficient definition:
a(n) = factor(max((n=prime(n)+prime(n+1))>>valuation(n,2),2))[1,1] */

a(n) = max(A078701(A001043(n)), 2) = A020639(max(A000265(A001043(n)), 2)), where A000265(m) > 2 unless m is in A000079.

A023511 Least odd prime divisor of prime(n) + 1, or 1 if prime(n) + 1 is a power of 2.

Original entry on oeis.org

3, 1, 3, 1, 3, 7, 3, 5, 3, 3, 1, 19, 3, 11, 3, 3, 3, 31, 17, 3, 37, 5, 3, 3, 7, 3, 13, 3, 5, 3, 1, 3, 3, 5, 3, 19, 79, 41, 3, 3, 3, 7, 3, 97, 3, 5, 53, 7, 3, 5, 3, 3, 11, 3, 3, 3, 3, 17, 139, 3, 71, 3, 7, 3, 157, 3, 83, 13, 3, 5, 3, 3, 23, 11, 5, 3, 3, 199, 3, 5, 3, 211, 3, 7, 5, 3, 3

Offset: 1

Clark Kimberling

nonn

Note that a(n) = 1 for n= 2, 4, 11, 31, 1028, ... (A059305). - Michel Marcus, Oct 01 2013

Cf. A008864, A078701.

a(n) = my(p = prime(n) + 1, v = p/(2^valuation(p, 2))) ; if (v == 1, 1, factor(v)[1, 1]); \\ Michel Marcus, Oct 01 2013

a(n) = A078701(A008864(n)). - Michel Marcus, Jun 06 2019

A023519 Least odd prime divisor of prime(n)prime(n-1) - 1, or 1 if prime(n)prime(n-1) - 1 is a power of 2.

Original entry on oeis.org

1, 5, 7, 17, 19, 71, 5, 7, 109, 3, 449, 3, 379, 881, 5, 3, 3, 7, 3, 29, 2591, 3, 11, 3, 13, 31, 7, 5, 7, 3079, 5, 4159, 3, 9521, 5, 7, 3, 3, 5, 3, 3, 97, 5, 7, 5, 17, 3, 3, 5, 7, 13339, 3, 31, 5, 3, 3, 3, 7, 3, 11, 39761, 11, 5, 23869, 7, 5, 23, 3, 59, 151, 19, 3, 43, 3, 3, 11, 3, 19, 39799, 13

Offset: 1

Clark Kimberling

nonn

Assumes the not generally accepted convention prime(0) = 1. - Michel Marcus, Jun 06 2019

Cf. A023515, A078701.

p(n) = if (n==0, 1, prime(n));
f(n) = my(v = select(x->((x%2)==1), factor(n)[, 1])); if (#v, vecmin(v), 1);
a(n) = f(p(n)*p(n-1) - 1); \\ Michel Marcus, Jun 06 2019

a(n) = A078701(A023515(n)). - Michel Marcus, Jun 06 2019

Terms corrected by Sean A. Irvine, Jun 05 2019

cp's OEIS Frontend

A079602 Greatest of smallest odd prime factors of all composite numbers between n-th prime and next prime.

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A338101 Smallest odd prime dividing n is a(n)-th prime, or 0 if no such prime exists.

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A372263 Least odd prime factor of the n-th sum of two consecutive primes, A001043(n) = prime(n) + prime(n+1), or 2 if there is no odd prime factor.

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A023511 Least odd prime divisor of prime(n) + 1, or 1 if prime(n) + 1 is a power of 2.

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A023519 Least odd prime divisor of prime(n)*prime(n-1) - 1, or 1 if prime(n)*prime(n-1) - 1 is a power of 2.

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A023519 Least odd prime divisor of prime(n)prime(n-1) - 1, or 1 if prime(n)prime(n-1) - 1 is a power of 2.