A329348 -id:A329348 - cp's OEIS frontend

A345941 a(n) = gcd(n, A329044(n)).

1, 2, 3, 4, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 4, 17, 9, 19, 5, 7, 11, 23, 3, 25, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 9, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 25, 17, 13, 53, 9, 11, 7, 19, 29, 59, 5, 61, 31, 7, 4, 13, 11, 67, 17, 23, 7, 71, 9, 73, 37, 25, 19, 11, 13, 79, 5, 3, 41, 83, 7, 17, 43, 29, 11, 89

Offset: 1

Antti Karttunen, Jul 03 2021

nonn

Only powers of primes (A000961) occur as terms. A346087 gives the exponents. - Antti Karttunen, Jul 07 2021

Cf. A000961, A006530, A020639, A071178, A108951, A324886, A329348, A329044, A345942, A345943, A346087, A346097.

A345941(n) = gcd(n,A329044(n)); \\ Rest of program given in A329044.

a(n) = gcd(n, A329044(n)).
a(n) = n / A345942(n).
a(n) = A329044(n) / A345943(n).
a(p) = p for all primes p.
From Antti Karttunen, Jul 07 2021: (Start)
a(n) = A006530(n)^A346087(n) = A006530(n)^min(A071178(n), A329348(n)).
a(n) = gcd(n, A346097(n)).
A006530(a(n)) = A020639(A329044(n)) = A006530(n).
(End)

A329349 Number of occurrences of the largest primorial present in the greedy sum of primorials adding to A108951(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 6, 2, 1, 2, 1, 4, 6, 2, 1, 1, 4, 2, 1, 4, 1, 1, 1, 1, 6, 2, 2, 4, 1, 2, 6, 1, 1, 1, 1, 4, 5, 2, 1, 3, 1, 8, 6, 4, 1, 2, 2, 8, 6, 2, 1, 3, 1, 2, 3, 2, 1, 12, 1, 4, 6, 5, 1, 1, 1, 2, 2, 4, 16, 12, 1, 2, 6, 2, 1, 2, 1, 2, 6, 8, 1, 10, 12, 4, 6, 2, 1, 6, 1, 2, 2, 1, 1, 12, 1, 8, 1

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

The greedy sum is also the sum with the minimal number of primorials, used for example in the primorial base representation.

			For n = 21 = 3 * 7, A108951(21) = A034386(3) * A034386(7) = 6 * 210, so the factor of the largest primorial present (210) in the greedy sum is 6 (as 1260 = 210 + 210 + 210 + 210 + 210 + 210), thus a(21) = 6.
For n = 24 = 2^3 * 3, A108951(24) = A034386(2)^3 * A034386(3) = 2^3 * 6 = 48 = 1*30 + 3*6, and as the factor of the largest primorial in the sum is 1, we have a(24) = 1.

Cf. A002110, A034386, A071178, A108951, A276086, A276153, A324886, A324888, A329040, A329343, A329344, A329345, A329348.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276153(n) = { my(e=0, p=2); while(n, e=n%p; n = n\p; p = nextprime(1+p)); (e); };
A329349(n) = A276153(A108951(n));

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A071178(n) = if(1==n,0,my(es=factor(n)[,2]); es[#es]);
A329349(n) = A071178(A324886(n));

a(n) = A276153(A108951(n)) = A071178(A324886(n)).

a(n) <= A324888(n).

A329346 a(n) = A322356(A324886(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 7, 1, 1, 1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 7, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 7, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 19, 1, 1, 1, 1, 13, 1, 7, 1, 1, 13, 1, 1, 1, 1, 1, 7, 1, 1, 13, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 7, 1, 13, 1, 13, 1, 1, 1, 1, 13

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

			For n = 128 = 2^7, A108951(128) = A034386(2)^7 = 128. As 128 = 4 * 30 + 1*6 + 1* 2, A276086(128) = 36015 = 7^4 * 5^1 * 3^1, and there are two such primes that both p and p-2 divide n, and p-2 is also prime, namely, 7 and 5, thus a(128) = 7*5 = 35. This is also the first occurrence of composite number in this sequence.

Cf. A002110, A034386, A108951, A276086, A324886, A329040, A322356, A329343, , A329344, A329347, A329348.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A322356(n) = { my(f = factor(n), m=1); for(i=1, #f~, if(isprime(f[i,1]+2)&&!(n%(f[i,1]+2)), m *= (f[i,1]+2))); (m); };
A329346(n) = A322356(A324886(n));

a(n) = A322356(A324886(n)).

cp's OEIS Frontend

A345941 a(n) = gcd(n, A329044(n)).

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A329349 Number of occurrences of the largest primorial present in the greedy sum of primorials adding to A108951(n).

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PARI

PARI

Formula

A329346 a(n) = A322356(A324886(n)).

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PARI

Formula