A347101 -id:A347101 - cp's OEIS frontend

A347102 Totally additive with a(prime(k)) = A001223(k), where A001223 gives the distance from the k-th prime to the next larger prime.

0, 1, 2, 2, 2, 3, 4, 3, 4, 3, 2, 4, 4, 5, 4, 4, 2, 5, 4, 4, 6, 3, 6, 5, 4, 5, 6, 6, 2, 5, 6, 5, 4, 3, 6, 6, 4, 5, 6, 5, 2, 7, 4, 4, 6, 7, 6, 6, 8, 5, 4, 6, 6, 7, 4, 7, 6, 3, 2, 6, 6, 7, 8, 6, 6, 5, 4, 4, 8, 7, 2, 7, 6, 5, 6, 6, 6, 7, 4, 6, 8, 3, 6, 8, 4, 5, 4, 5, 8, 7, 8, 8, 8, 7, 6, 7, 4, 9, 6, 6, 2, 5, 4, 7, 8

Offset: 1

Antti Karttunen, Aug 19 2021

nonn

			For n = 12 = 2*2*3, the corresponding prime gaps are 1, 1 and 2, thus a(12) = 1+1+2 = 4.
For n = 42 = 2*3*7, the corresponding prime gaps are 1, 2 and 4, thus a(42) = 1+2+4 = 7.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for primes, gaps between

Cf. A001223, A001414, A003961, A007814, A056239, A064989, A347101, A347123.

A347102(n) = { my(f=factor(n), s=0); for(i=1, #f~, s += f[i, 2]*(nextprime(f[i, 1]+1)-f[i,1])); (s); };

a(n) = A001414(A003961(n)) - A001414(n).
a(n) = A007814(n) + 2*A056239(A064989(A347123(n))).
For all n >= 0, a(2^n) = n.

A347123 Fully multiplicative with a(prime(k)) = prime(1+floor(A001223(k)/2)).

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 5, 8, 9, 6, 3, 12, 5, 10, 9, 16, 3, 18, 5, 12, 15, 6, 7, 24, 9, 10, 27, 20, 3, 18, 7, 32, 9, 6, 15, 36, 5, 10, 15, 24, 3, 30, 5, 12, 27, 14, 7, 48, 25, 18, 9, 20, 7, 54, 9, 40, 15, 6, 3, 36, 7, 14, 45, 64, 15, 18, 5, 12, 21, 30, 3, 72, 7, 10, 27, 20, 15, 30, 5, 48, 81, 6, 7, 60, 9, 10, 9, 24, 11, 54

Offset: 1

Antti Karttunen, Aug 19 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for primes, gaps between

Cf. A001222, A001223, A003586, A007814, A347101, A347102.

Cf. also A336855.

A347123(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 1] = prime(1+((nextprime(f[i, 1]+1)-f[i,1])\2))); factorback(f); };

For all n >= 1, A001222(a(n)) = A001222(n), A007814(a(n)) = A007814(n).

For all n >= 1, a(A003586(n)) = A003586(n).

A345161 If n = Product (p_j^k_j) then a(n) = max (nextprime(p_j) - p_j), where nextprime = A151800.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 2, 2, 4, 4, 2, 1, 2, 2, 4, 2, 4, 2, 6, 2, 2, 4, 2, 4, 2, 2, 6, 1, 2, 2, 4, 2, 4, 4, 4, 2, 2, 4, 4, 2, 2, 6, 6, 2, 4, 2, 2, 4, 6, 2, 2, 4, 4, 2, 2, 2, 6, 6, 4, 1, 4, 2, 4, 2, 6, 4, 2, 2, 6, 4, 2, 4, 4, 4, 4, 2, 2, 2, 6, 4, 2, 4, 2, 2, 8, 2

Offset: 1

Ilya Gutkovskiy, Aug 26 2021

nonn

			a(39) = a(3 * 13) = a(prime(2) * prime(6)), prime(3) - prime(2) = 5 - 3 = 2, prime(7) - prime(6) = 17 - 13 = 4, so a(39) = max(2, 4) = 4.

Cf. A000079 (positions of 1's), A000720, A001223, A006530, A061395, A063095, A151800, A327441, A347101, A347102.

a[n_] := Max @@ (NextPrime[#[[1]]] - #[[1]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 90}]

If n = Product (p_j^k_j) then a(n) = max (prime(pi(p_j) + 1) - p_j), where pi = A000720.
a(2^j*n) = a(n).
a(n^j) = a(n), j > 0.
a(prime(n)^j) = A001223(n), j > 0.
a(n!) = A327441(n).
a(prime(n)#) = A063095(n).
2 + Sum_{k=1..n-1} a(prime(k)^j) = prime(n), j > 0.
Sum_{d|n} mu(n/d) * a(d) = 0 if n is an even number or an odd number divisible by a square > 1.

cp's OEIS Frontend

A347102 Totally additive with a(prime(k)) = A001223(k), where A001223 gives the distance from the k-th prime to the next larger prime.

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A347123 Fully multiplicative with a(prime(k)) = prime(1+floor(A001223(k)/2)).

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A345161 If n = Product (p_j^k_j) then a(n) = max (nextprime(p_j) - p_j), where nextprime = A151800.

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