A276151 -id:A276151 - cp's OEIS frontend

A328473 a(n) = A276156(n) - A002110(A007814(n)).

0, 0, 2, 0, 6, 6, 8, 0, 30, 30, 32, 30, 36, 36, 38, 0, 210, 210, 212, 210, 216, 216, 218, 210, 240, 240, 242, 240, 246, 246, 248, 0, 2310, 2310, 2312, 2310, 2316, 2316, 2318, 2310, 2340, 2340, 2342, 2340, 2346, 2346, 2348, 2310, 2520, 2520, 2522, 2520, 2526, 2526, 2528, 2520, 2550, 2550, 2552, 2550, 2556, 2556, 2558, 0, 30030

Offset: 1

Antti Karttunen, Oct 18 2019

A276156(n) converts the binary expansion of n to a number whose primorial base representation has the same digits of 0's and 1's, thus each one of its terms is a unique sum of distinct primorial numbers. This sequence is otherwise similar, but the primorial number corresponding to the least significant 1-bit of n is dropped from the sum, so the sum is not unique anymore.

Cf. A002110, A053589, A129760, A276151, A276156, A328461, A328474.

A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328473(n) = (A276156(n)-A002110(valuation(n,2)));

a(n) = A276156(A129760(n)).

a(n) = A276151(A276156(n)) = A276156(n) - A002110(A007814(n)).

A324384 a(n) = gcd(n, A276154(n)), where A276154 is the primorial base left shift.

Original entry on oeis.org

0, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 11, 12, 1, 2, 1, 8, 1, 18, 1, 4, 7, 2, 1, 24, 1, 2, 1, 4, 1, 30, 1, 8, 1, 2, 7, 12, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 12, 1, 2, 1, 52, 1, 6, 1, 56, 1, 2, 1, 60, 1, 2, 1, 16, 1, 6, 1, 4, 1, 14, 1, 24, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 12, 1, 2, 1, 8, 1, 90, 1, 4, 1, 2, 1, 12, 1, 2, 1, 4, 1, 6, 1, 8, 1

Offset: 0

Antti Karttunen, Feb 26 2019

nonn

For a very few primes, a(p) > 1 (then by necessity a(p) = p). In range 2 .. 2^25 there are three: 2, 11, 119039.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A002110, A049345, A276085, A276086, A276151, A276152, A276154, A323879, A324198, A324350, A324351.

A276151(n) = { my(s=1); forprime(p=2, , if(n%p, return(n-s), s *= p)); };
A276152(n) = { my(s=1); forprime(p=2, , if(n%p, return(s*p), s *= p)); };
A276154(n) = if(!n,n,(A276152(n) + A276154(A276151(n))));
\\ Alternatively, A276154 can be defined with A276085, A276086 and A003961:
A002110(n) = prod(i=1,n,prime(i));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A276154(n) = A276085(A003961(A276086(n)));
A324384(n) = gcd(n,A276154(n));

a(n) = gcd(n, A276154(n)).

A328575 a(n) = A003557(A032742(A276086(n))).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 5, 5, 5, 5, 15, 5, 25, 25, 25, 25, 75, 25, 125, 125, 125, 125, 375, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 5, 5, 5, 5, 15, 5, 25, 25, 25, 25, 75, 25, 125, 125, 125, 125, 375, 1, 7, 7, 7, 7, 21, 7, 7, 7, 7, 7, 21, 7, 35, 35, 35, 35, 105, 35, 175, 175, 175, 175, 525, 175, 875, 875, 875, 875, 2625, 7

Offset: 0

Antti Karttunen, Oct 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..32768
Index entries for sequences related to primorial base

Cf. A003557, A032742, A051903, A276086, A276088, A328114, A328572, A328576.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328575(n) = A003557(A032742(A276086(n)));

a(n) = A003557(A032742(A276086(n))).
For n >= 1, a(n) = A003557(A276086(A276151(n))).
For n >= 1, max(A276088(n),1+A051903(a(n))) = A328114(n).

cp's OEIS Frontend

A328473 a(n) = A276156(n) - A002110(A007814(n)).

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A324384 a(n) = gcd(n, A276154(n)), where A276154 is the primorial base left shift.

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A328575 a(n) = A003557(A032742(A276086(n))).

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