A328832 -id:A328832 - cp's OEIS frontend

A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

0, 1, 2, 2, 6, 3, 30, 3, 4, 7, 210, 4, 2310, 31, 8, 4, 30030, 5, 510510, 8, 32, 211, 9699690, 5, 12, 2311, 6, 32, 223092870, 9, 6469693230, 5, 212, 30031, 36, 6, 200560490130, 510511, 2312, 9, 7420738134810, 33, 304250263527210, 212, 10, 9699691, 13082761331670030, 6, 60, 13, 30032, 2312, 614889782588491410, 7, 216, 33, 510512, 223092871, 32589158477190044730, 10

Offset: 1

Antti Karttunen, Aug 21 2016

nonn

Completely additive with a(p^e) = e * A002110(A000720(p)-1).
This is a left inverse of A276086 ("primorial base exp-function"), hence the name "primorial base log-function". When the domain is restricted to the terms of A048103, this works also as a right inverse, as A276086(a(A048103(n))) = A048103(n) for all n >= 1. - Antti Karttunen, Apr 24 2022
On average, every third term is a multiple of 4. See A369001. - Antti Karttunen, May 26 2024

A left inverse of A276086.
Cf. A000040, A000720, A002110, A028234, A034386, A048103, A049345, A055396, A067029, A108951, A143293, A276154, A328316, A328624, A328625, A328768, A328832, A346105, A351576, A376398 (partial sums).
Positions of multiples of k in this sequence, for k=2, 3, 4, 5, 8, 27, 3125: A003159, A339746, A369002, A373140, A373138, A377872, A377878.
Cf. A036554 (positions of odd terms), A035263, A096268 (parity of terms).
Cf. A372575 (rgs-transform), A372576 [a(n) mod 360], A373842 [= A003415(a(n))].
Cf. A373145 [= gcd(A003415(n), a(n))], A373361 [= gcd(n, a(n))], A373362 [= gcd(A001414(n), a(n))], A373485 [= gcd(A083345(n), a(n))], A373835 [= gcd(bigomega(n), a(n))], and also A373367 and A373147 [= A003415(n) mod a(n)], A373148 [= a(n) mod A003415(n)].
Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A001414 (with a(p)=p), A059975 (with a(p)=p-1), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).
Cf. also A276075 for factorial base and A048675, A054841 for base-2 and base-10 analogs.

nn = 60; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[FromDigits[#, b] &@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, nn}] (* Version 10.2, or *)
f[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ w - 1]], Reverse@ w}]; Table[f@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, 60}] (* Michael De Vlieger, Aug 30 2016 *)

A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); }; \\ Antti Karttunen, Nov 11 2024

from sympy import primorial, primepi, factorint
def a002110(n):
    return 1 if n<1 else primorial(n)
def a(n):
    f=factorint(n)
    return sum(f[i]*a002110(primepi(i) - 1) for i in f)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 22 2017

a(1) = 0; for n > 1, a(n) = a(A028234(n)) + (A067029(n) * A002110(A055396(n)-1)).
a(1) = 0, a(n) = (e1*A002110(i1-1) + ... + ez*A002110(iz-1)) when n = prime(i1)^e1 * ... * prime(iz)^ez.
Other identities.
For all n >= 0:
a(A276086(n)) = n.
a(A000040(1+n)) = A002110(n).
a(A002110(1+n)) = A143293(n).
From Antti Karttunen, Apr 24 & Apr 29 2022: (Start)
a(A283477(n)) = A283985(n).
a(A108951(n)) = A346105(n). [The latter has a similar additive formula as this sequence, but instead of primorials, uses their partial sums]
When applied to sequences where a certain subset of the divisors of n has been multiplicatively encoded with the help of A276086, this yields a corresponding number-theoretical sequence, i.e. completes their computation:
a(A319708(n)) = A001065(n) and a(A353564(n)) = A051953(n).
a(A329350(n)) = A069359(n) and a(A329380(n)) = A323599(n).
In the following group, the sum of the rhs-sequences is n [on each row, as say, A328841(n)+A328842(n)=n], because the pointwise product of the corresponding lhs-sequences is A276086:
a(A053669(n)) = A053589(n) and a(A324895(n)) = A276151(n).
a(A328571(n)) = A328841(n) and a(A328572(n)) = A328842(n).
a(A351231(n)) = A351233(n) and a(A327858(n)) = A351234(n).
a(A351251(n)) = A351253(n) and a(A324198(n)) = A351254(n).
The sum or difference of the rhs-sequences is A108951:
a(A344592(n)) = A346092(n) and a(A346091(n)) = A346093(n).
a(A346106(n)) = A346108(n) and a(A346107(n)) = A346109(n).
Here the two sequences are inverse permutations of each other:
a(A328624(n)) = A328625(n) and a(A328627(n)) = A328626(n).
a(A346102(n)) = A328622(n) and a(A346233(n)) = A328623(n).
a(A346101(n)) = A289234(n). [Self-inverse]
Other correspondences:
a(A324350(x,y)) = A324351(x,y).
a(A003961(A276086(n))) = A276154(n). [The primorial base left shift]
a(A276076(n)) = A351576(n). [Sequence reinterpreting factorial base representation as a primorial base representation]
(End)

Name amended by Antti Karttunen, Apr 24 2022

Name simplified, the old name moved to the comments - Antti Karttunen, Jun 23 2024

A048103 Numbers not divisible by p^p for any prime p.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98

Offset: 1

N. J. A. Sloane

If a(n) = Product p_i^e_i then p_i > e_i for all i.
Complement of A100716; A129251(a(n)) = 0. - Reinhard Zumkeller, Apr 07 2007
Density is 0.72199023441955... = Product_{p>=2} (1 - p^-p) where p runs over the primes. - Charles R Greathouse IV, Jan 25 2012
A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). - Reinhard Zumkeller, Apr 28 2012
Range of A276086. Also numbers not divisible by m^m for any natural number m > 1. - Antti Karttunen, Nov 18 2024

			6 = 2^1 * 3^1 is OK but 12 = 2^2 * 3^1 is not.
625 = 5^4 is present because it is not divisible by 5^5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement: A100716.
Positions of 0's in A129251, A342023, A376418, positions of 1's in A327936, A342007, A359550 (characteristic function).
Cf. A048102, A048104, A051674 (p^p), A054743, A054744, A377982 (a left inverse, partial sums of char. fun, see also A328402).
Cf. A276086 (permutation of this sequence, see also A376411, A376413).
Subsequences: A002110, A005117, A006862, A024451 (after its initial 0), A057588, A099308 (after its initial 0), A276092, A328387, A328832, A359547, A370114, A371083, A373848, A377871, A377992.
Disjoint union of {1}, A327934 and A358215.
Also A276078 is a subsequence, from which this differs for the first time at n=451 where a(451)=625, while that value is missing from A276078.

a048103 n = a048103_list !! (n-1)
a048103_list = filter (\x -> and $
   zipWith (>) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012

{1}~Join~Select[Range@ 120, Times @@ Boole@ Map[First@ # > Last@ # &, FactorInteger@ #] > 0 &] (* Michael De Vlieger, Aug 19 2016 *)

isok(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,1] <= f[i,2], return(0))); return(1); \\ Michel Marcus, Nov 13 2020

A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); }; \\ (A359550 is the characteristic function for A048103) - Antti Karttunen, Nov 18 2024

from itertools import count, islice
from sympy import factorint
def A048103_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda d:d[1]A048103_list = list(islice(A048103_gen(),30)) # Chai Wah Wu, Jan 05 2023

;; With Antti Karttunen's IntSeq-library.
(define A048103 (ZERO-POS 1 1 A129251))
;; Antti Karttunen, Aug 18 2016

a(n) ~ kn with k = 1/Product_{p>=2}(1 - p^-p) = Product_{p>=2}(1 + 1/(p^p - 1)) = 1.3850602852..., where the product is over all primes p. - Charles R Greathouse IV, Jan 25 2012

For n >= 1, A377982(a(n)) = n. - Antti Karttunen, Nov 18 2024

More terms from James Sellers, Apr 22 2000

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)).

Original entry on oeis.org

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).

A328828 Index of the least significant digit larger than one in the primorial base expansion of n, 0 if no such digit exists.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 3, 3, 3, 3

Offset: 0

Antti Karttunen, Oct 29 2019

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

Cf. A000720, A049345, A055396, A276086, A276156 (positions of 0's), A277885, A328572, A328829 [= a(A276086(n))], A328832, A381032 [= A008578(1+a(n))].

a[n_] := Module[{k = n, p = 2, s = {}, r, i}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; i = FirstPosition[s, ?(# > 1 &)]; If[MissingQ[i], 0, i[[1]]]]; Array[a, 100] (* _Amiram Eldar, Mar 13 2024 *)

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

a(n) = A277885(A276086(n)) = A055396(A328572(n)).

a(n) = A000720(A381032(n)). - Antti Karttunen, Feb 23 2025

A328836 Numbers k such that A276086(k) is a sum of distinct primorial numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 30, 39, 212, 249, 421, 2312, 2559, 30045, 32589, 510511, 512820, 543099, 1021050, 9729723, 10242789, 233335659, 446185742

Offset: 1

Antti Karttunen, Oct 29 2019

Numbers k such that A276086(k) is in A276156, i.e., numbers k for which A328828(A276086(k)) is zero, i.e., numbers k such that in the primorial base expansion of A276086(k) there are no digits larger than 1.
Numbers k for which A276087(k) is squarefree.
No more terms below 2^31.

Index entries for sequences related to primorial base

Sequence A328833 sorted into ascending order.
Cf. A008966, A276086, A276087, A276156, A328828, A328832, A328833.
Positions of zeros in A328829 and in A328844, positions of ones in A328389.
Cf. A143293 (a subsequence).
All the terms of A328313 are included in this sequence, like also in A328837.

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); };
isA328836(n) = !A328828(A276086(n));

A328831 Number of distinct prime factors p such that p^p is a divisor of n-th number > 0 that is a sum of distinct primorial numbers, A276156(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0

Offset: 1

Antti Karttunen, Oct 30 2019

nonn

Cf. A129251, A276156.

Cf. A328832 (gives A276156(k) for those k for which a(k) = 0).

A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
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); };
A328831(n) = A129251(A276156(n));

a(n) = A129251(A276156(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)).

cp's OEIS Frontend

A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A048103 Numbers not divisible by p^p for any prime p.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Python

Scheme

Formula

Extensions

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)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A328828 Index of the least significant digit larger than one in the primorial base expansion of n, 0 if no such digit exists.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A328836 Numbers k such that A276086(k) is a sum of distinct primorial numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A328831 Number of distinct prime factors p such that p^p is a divisor of n-th number > 0 that is a sum of distinct primorial numbers, A276156(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula