A328836 -id:A328836 - cp's OEIS frontend

A276156 Numbers obtained by reinterpreting base-2 representation of n in primorial base: a(0) = 0, a(2n) = A276154(a(n)), a(2n+1) = 1 + A276154(a(n)).

0, 1, 2, 3, 6, 7, 8, 9, 30, 31, 32, 33, 36, 37, 38, 39, 210, 211, 212, 213, 216, 217, 218, 219, 240, 241, 242, 243, 246, 247, 248, 249, 2310, 2311, 2312, 2313, 2316, 2317, 2318, 2319, 2340, 2341, 2342, 2343, 2346, 2347, 2348, 2349, 2520, 2521, 2522, 2523, 2526, 2527, 2528, 2529, 2550, 2551, 2552, 2553, 2556, 2557, 2558, 2559, 30030, 30031

Offset: 0

Antti Karttunen, Aug 24 2016

Numbers that are sums of distinct primorial numbers, A002110.

Numbers with no digits larger than one in primorial base, A049345.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to primorial base

Cf. A000040, A001511, A002110, A007088, A007814, A019565, A049345, A257993, A276084, A276085, A276154, A351073, A328461, A328473, A328474, A328571, A328831, A328836, A341518, A344591.
Complement of A177711.
Subsequences: A328233, A328832, A328462 (odd bisection).
Conjectured subsequences: A328110, A380527.
Fixed points of A328841, positions of zeros in A328828, A328842, and A329032, positions of ones in A328581, A328582, and A381032.
Positions of terms < 2 in A328114.
Indices where A327860 and A329029 coincide.
Cf. also table A328464 (and its rows).
Cf. also A059590, A283985, A290249, A342921.

nn = 65; b = MixedRadix[Reverse@ Prime@ Range[IntegerLength[nn, 2] - 1]]; Table[FromDigits[IntegerDigits[n, 2], b], {n, 0, 65}] (* Version 10.2, or *)
Table[Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ # - 1]], Reverse@ #}] &@ IntegerDigits[n, 2], {n, 0, 65}] (* Michael De Vlieger, Aug 26 2016 *)

A276156(n) = { my(s=0, p=1, r=1); while(n, if(n%2, s += r); n>>=1; p = nextprime(1+p); r *= p); (s); }; \\ Antti Karttunen, Feb 03 2022

from sympy import prime, primorial, primepi, factorint
from operator import mul
def a002110(n): return 1 if n<1 else primorial(n)
def a276085(n):
    f=factorint(n)
    return sum([f[i]*a002110(primepi(i) - 1) for i in f])
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) # after Chai Wah Wu
def a(n): return 0 if n==0 else a276085(a019565(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 23 2017

a(0) = 0, a(2n) = A276154(a(n)), a(2n+1) = 1+A276154(a(n)).
Other identities. For all n >= 0:
a(n) = A276085(A019565(n)).
A049345(a(n)) = A007088(n).
A257993(a(n)) = A001511(n).
A276084(a(n)) = A007814(n).
A051903(a(n)) = A351073(n).

A328832 Numbers that are sums of distinct primorial numbers, A002110, and do not have a factor of the form p^p.

Original entry on oeis.org

1, 2, 3, 6, 7, 9, 30, 31, 33, 37, 38, 39, 210, 211, 213, 217, 218, 219, 241, 242, 246, 247, 249, 2310, 2311, 2313, 2317, 2318, 2319, 2341, 2342, 2343, 2346, 2347, 2521, 2522, 2523, 2526, 2527, 2529, 2550, 2551, 2553, 2557, 2558, 2559, 30030, 30031, 30033, 30037, 30038, 30039, 30061, 30062, 30063, 30066, 30067, 30069, 30241

Offset: 1

Antti Karttunen, Oct 30 2019

nonn

Numbers n such that A129251(n) = 0 and A328828(n) = 0 (or equally, A328114(n) = 1).

Terms k in A276156 for which A276086(A276085(k)) = k, i.e., those terms of A276156 which are in the range of A276086.

Antti Karttunen, Table of n, a(n) for n = 1..23330
Index entries for sequences related to primorial base

Intersection of A048103 and A276156.

Cf. A002110, A129251, A276085, A276086, A328114, A328828, A328831, A328833, A328836.

A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
A328828(n) = { my(i=1, p=2); while(n, if((n%p)>1, return(i)); i++; n = n\p; p = nextprime(1+p)); (0); };
isA328832(n) = ((0==A129251(n)) && (0==A328828(n)));

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); };
k=0; for(n=1,(2^15)-1, if(!A129251(u=A276156(n)), k++; write("b328832.txt", k, " ", u, " ")));

a(n) = A276086(A328833(n)).

A328844 a(n) = A328842(A276086(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 12, 2, 2, 6, 0, 6, 60, 18, 12, 36, 120, 6, 210, 92, 2, 126, 510, 876, 2010, 378, 1002, 1626, 1200, 3066, 8700, 0, 6, 12, 6, 30, 90, 2, 32, 66, 0, 66, 420, 138, 102, 276, 840, 1326, 630, 632, 1502, 66, 2730, 5316, 13230, 1818, 6192, 10566, 23730, 6996, 46200, 12, 60, 108, 48, 222, 636, 2, 242, 486, 1260, 1956, 1890, 978, 102

Offset: 0

Antti Karttunen, Oct 30 2019

nonn

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

Cf. A276086, A328842, A328843.

Cf. A328836 (positions of zeros).

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328842(n) = { my(p=2, r=1, s=0); while(n, if((n%p)>0, s += ((n%p)-1)*r); r *= p; n = n\p; p = nextprime(1+p)); (s); };
A328844(n) = A328842(A276086(n));

a(n) = A328842(A276086(n)).

a(n) = A276086(n) - A328843(n).

A328829 Index of the least significant digit > 1 in the primorial base expansion of A276086(n), 0 if no such digit exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 2, 2, 3, 0, 3, 4, 3, 3, 3, 4, 3, 5, 2, 2, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 0, 3, 3, 3, 4, 4, 2, 2, 3, 0, 3, 5, 3, 3, 3, 5, 3, 5, 2, 2, 3, 5, 3, 5, 3, 3, 3, 5, 3, 6, 3, 4, 3, 3, 3, 3, 2, 2, 3, 5, 3, 5, 3, 3, 3, 5, 3, 5, 2, 2, 3, 5, 3, 5, 3, 3, 3, 5, 3, 5, 3, 3, 4, 3, 3, 3, 2, 2, 3, 5, 3, 5, 3, 3, 3, 5

Offset: 0

Antti Karttunen, Oct 29 2019

nonn

a(n) = index of the least non-unitary prime divisor of A276087(n) or 0 if no such prime-divisor exists.

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

Cf. A276086, A276087, A277885, A328828, A328836 (positions of zeros).

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328828(n) = { my(i=1, p=2); while(n, if((n%p)>1, return(i)); i++; n = n\p; p = nextprime(1+p)); (0); };
A328829(n) = A328828(A276086(n));

A277885(n) = if(1==n,0,my(f=factor(n)); for(i=1,#f~,if(f[i,2]>1,return(primepi(f[i,1])))); (0));
A328829(n) = A277885(A276086(A276086(n)));

a(n) = A328828(A276086(n)) = A277885(A276087(n)).

A328833 A276085 applied to the intersection of A048103 (p^p-free numbers) and A276156 (sums of distinct primorials).

Original entry on oeis.org

0, 1, 2, 3, 30, 4, 9, 6469693230, 212, 200560490130, 510511, 2312, 39, 7799922041683461553249199106329813876687996789903550945093032474868511536164700810, 7858321551080267055879092, 6469693260, 2566376117594999414479597815340071648394471, 557940830126698960967415392, 1062411448280052319722448549835623701226301211611796930357321893850294264731624591303255041960530, 421, 7420738134813, 512820, 3217644767340672907899084554132, 249

Offset: 1

Antti Karttunen, Oct 30 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..34
Index entries for sequences related to primorial base

Cf. A002110, A129251, A048103, A276085, A276156, A328828, A328832.
Cf. A328836 (same terms, sorted into ascending order).
Cf. A328313 (a subsequence).

A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
A328828(n) = { my(i=1, p=2); while(n, if((n%p)>1, return(i)); i++; n = n\p; p = nextprime(1+p)); (0); };
isA328832(n) = ((0==A129251(n)) && (0==A328828(n)));
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
for(n=1,32768,if(isA328832(n),print1(A276085(n),", ")));

a(n) = A276085(A328832(n)).

A328837 Numbers k for which A328403(k) = A276086(A276086(A276086(k))) is squarefree.

Original entry on oeis.org

0, 1, 2, 4, 9, 2312

Offset: 1

Antti Karttunen, Oct 30 2019

Numbers k such that A276086(k) is in A328836, or equally, that A276087(k) is in A276156, i.e., numbers k for which A328828(A276087(k)) is zero, that is, numbers k such that in the primorial base expansion of A276087(k) there are no digits larger than 1.

All the terms of A328313 are also included in this sequence. Questions: Is that sequence finite? Even if it is, is this one also? Are there any terms here between 2312 and 3217644767340672907899084554132? Are there only finitely many numbers k for which A328828(A328403(k)) is zero? (See comments in A328398.)

Index entries for sequences related to primorial base

Cf. A008966, A276086, A276087, A276156, A328313, A328403, A328828, A328836.

Positions of ones in A328394. See also comments in A328398.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328828(n) = { my(i=1, p=2); while(n, if((n%p)>1, return(i)); i++; n = n\p; p = nextprime(1+p)); (0); };
isA328837(n) = !A328828(A276086(A276086(n)));

cp's OEIS Frontend

A276156 Numbers obtained by reinterpreting base-2 representation of n in primorial base: a(0) = 0, a(2n) = A276154(a(n)), a(2n+1) = 1 + A276154(a(n)).

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A328832 Numbers that are sums of distinct primorial numbers, A002110, and do not have a factor of the form p^p.

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A328844 a(n) = A328842(A276086(n)).

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A328829 Index of the least significant digit > 1 in the primorial base expansion of A276086(n), 0 if no such digit exists.

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A328833 A276085 applied to the intersection of A048103 (p^p-free numbers) and A276156 (sums of distinct primorials).

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A328837 Numbers k for which A328403(k) = A276086(A276086(A276086(k))) is squarefree.

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