A378885 -id:A378885 - cp's OEIS frontend

A378884 Numbers that are not powers of primes and whose two smallest prime divisors are consecutive primes.

6, 12, 15, 18, 24, 30, 35, 36, 42, 45, 48, 54, 60, 66, 72, 75, 77, 78, 84, 90, 96, 102, 105, 108, 114, 120, 126, 132, 135, 138, 143, 144, 150, 156, 162, 165, 168, 174, 175, 180, 186, 192, 195, 198, 204, 210, 216, 221, 222, 225, 228, 234, 240, 245, 246, 252, 255, 258

Offset: 1

Amiram Eldar, Dec 09 2024

Subsequence of A104210 and first differs from at an n = 15: A104210(15) = 70 = 2 * 5 * 7 is not a term of this sequence.
All the positive multiples of 6 (A008588 \ {0}) are terms.
Numbers k such that nextprime(lpf(k)) = A151800(A020639(k)) | k.
The asymptotic density of this sequence is Sum_{k>=1} (Product_{j=1..k-1} (1-1/prime(j)))/(prime(k)*prime(k+1)) = 0.2178590011934... .

			12 = 2^2 * 3 is a term since 2 and 3 are consecutive primes.
70 = 2 * 5 * 7 is not a term since 2 and 5 are not consecutive primes.
165 = 3 * 5 * 11 is a term since 3 and 5 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A024619, A104210 and A378885.
Subsequences: A006094, A256617.
Cf. A008588, A020639, A151800.

q[k_] := Module[{p = FactorInteger[k][[;; , 1]]}, Length[p] > 1 && p[[2]] == NextPrime[p[[1]]]]; Select[Range[300], q]

is(k) = if(k == 1, 0, my(p = factor(k)[,1]); #p > 1 && p[2] == nextprime(p[1]+1));

A378886 The number of consecutive primes in the prime factorization of n starting from the smallest prime dividing n; a(1) = 0.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3

Offset: 1

Amiram Eldar, Dec 10 2024

First differs from A300820 at n = 70 = 2 * 5 * 7: A300820(70) = 2 while a(70) = 1.

			a(42) = 2 since 42 = 2 * 3 * 7 and 2 and 3 are 2 consecutive primes.
a(28) = 1 since 28 = 2^2 * 7 and 3 is not a divisor of 28.
a(4095) = 3 since 4095 = 3^2 * 5 * 7 * 13 and 3, 5 and 7 are 3 consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A020639, A073491, A276084, A300820, A378884, A378885.

a[n_] := Module[{p = FactorInteger[n][[;; , 1]], c = 1, q}, q = p[[1]]; Do[q = NextPrime[q]; If[q == p[[i]], c++, Break[]], {i, 2, Length[p]}]; c]; a[1] = 0; Array[a, 100]

a(n) = if(n == 1, 0, my(p = factor(n)[,1], c = 1, q); q = p[1]; for(i = 2, #p, q = nextprime(q+1); if(q == p[i], c++, break)); c);

a(n) >= A276084(n).
a(n) <= A300820(n).
a(n) = A001221(n) if and only if n is in A073491.
a(n) >= 1 for n >= 2.
a(n) >= 2 if and only if n is in A378884.
a(n) >= 3 if and only if n is in A378885.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * (d(k) - d(k+1)) = 1.2630925015039..., where d(1) = 1 and d(k) = Sum_{i>=1} (Product_{j=1..i-1} (1-1/prime(j)))/(Product_{j=0..k-1} prime(i+j)), for k >= 2. d(k) is the asymptotic density of numbers m for which a(m) >= k.

cp's OEIS Frontend

A378884 Numbers that are not powers of primes and whose two smallest prime divisors are consecutive primes.

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