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A351956
a(n) = 1 if either n = 1 or the primorial inflation of n is equal to k * p#, where p# is the primorial (A034386) of some prime p, and 1 <= k < p, otherwise 0.
4
1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0
OFFSET
1
COMMENTS
a(n) = 1 if A108951(n) is in A060735, 0 otherwise.
a(n) = 1 if A324886(n) is a power of prime (in A000961) and 0 otherwise.
When a(n) = 1 for any n > 1, then by necessity, prime p mentioned in the definition is A000040(1+A061395(n)), that is, the next prime larger than the greatest prime dividing n, A006530(n). Therefore, a(n) = 1 when A108951(n) / A002110(A061395(n)) < A000040(1+A061395(n)).
FORMULA
a(n) = A010055(A324886(n)).
a(n) = 1 if A329040(n) = 1, and 0 otherwise.
For n > 1, A010051(n) = a(n) * A351957(n).
PROG
(PARI)
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); };
A351956(n) = (!!isprimepower(A324886(n)));
(PARI)
\\ A faster program:
A002110(n) = prod(i=1, n, prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^f[i, 2]) };
A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A351956(n) = if(1==n, 1, my(gpfi=A061395(n)); (A108951(n)/A002110(gpfi)<prime(1+gpfi)));
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Apr 04 2022
STATUS
approved