A276154 -id:A276154 - cp's OEIS frontend

A373158 Fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

0, 2, 6, 4, 30, 8, 210, 6, 12, 32, 2310, 10, 30030, 212, 36, 8, 510510, 14, 9699690, 34, 216, 2312, 223092870, 12, 60, 30032, 18, 214, 6469693230, 38, 200560490130, 10, 2316, 510512, 240, 16, 7420738134810, 9699692, 30036, 36, 304250263527210, 218, 13082761331670030, 2314, 42, 223092872, 614889782588491410, 14

Offset: 1

Antti Karttunen, May 27 2024

nonn

Completely additive with a(p^e) = e * A002110(A000720(p)).

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences related to primorial numbers

Cf. also A000040, A000720, A002110, A003961, A034386, A060389, A108951, A276085, A276154, A373984 [= A108951(n)-a(n)], A373985 [= gcd(a(n), A108951(n))], A373986, A373987.

A373158(n) = { my(f=factor(n)); sum(i=1, #f~, f[i, 2]*prod(i=1,primepi(f[i, 1]),prime(i))); }; \\ corrected Jun 25 2024

From Antti Karttunen, Jun 25 2024, Oct 28 2024: (Start)
a(n) = A276085(A003961(n)).
For n >= 1, a(A000040(n)) = A002110(n), a(A002110(n)) = A060389(n).
(End)

Data [first incorrect term was at a(8)] and the faulty PARI-program corrected by Antti Karttunen, Jun 25 2024

A276152 a(n) = {smallest prime not dividing n} times {greatest primorial number which divides n} = A053669(n) * A053589(n).

Original entry on oeis.org

2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 210, 2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 210, 2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 210, 2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 30, 2, 6, 2, 6, 2, 30, 2, 6

Offset: 1

Antti Karttunen, Aug 24 2016

nonn

a(n) with n odd must = 2 because 1 is the only odd primorial, thereby the only primorial dividing odd n, and 2 is the smallest prime not dividing odd n. - Michael De Vlieger, Aug 25 2016

			a(6) = 30 because the smallest nondivisor prime 6 = 5 and the smallest primorial dividing 6 is 6 itself. 5 * 6 = 30.

Antti Karttunen, Table of n, a(n) for n = 1..2310

Cf. A002110, A053589, A053669, A257993.

Cf. also A276154.

Table[If[n == 1, 2, Prime@If[! MemberQ[#, 0], Length@ # + 1, Position[#, 0][[1, 1]]] (Times @@ Prime@ Flatten@ Position[TakeWhile[#, # > 0 &], 1]) &@ Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@# -> 1 &, f]]@ FactorInteger@ n], {n, 120}] (* or *)
Table[If[OddQ@ n, 2, Function[p, Prime[p + 1] Product[Prime@ k, {k, #[[p]]}]][LengthWhile[Differences@ #, # == 1 &] + 1] &@ PrimePi[FactorInteger[n][[All, 1]]]], {n, 120}] (* Michael De Vlieger, Aug 25 2016 *)

a(n) = A053589(n) * A053669(n).

a(n) = A002110(A257993(n)).

A328770 Numbers in whose primorial base expansion any digit is at most half of the maximal allowed digit for that position.

Original entry on oeis.org

0, 2, 6, 8, 12, 14, 30, 32, 36, 38, 42, 44, 60, 62, 66, 68, 72, 74, 90, 92, 96, 98, 102, 104, 210, 212, 216, 218, 222, 224, 240, 242, 246, 248, 252, 254, 270, 272, 276, 278, 282, 284, 300, 302, 306, 308, 312, 314, 420, 422, 426, 428, 432, 434, 450, 452, 456, 458, 462, 464, 480, 482, 486, 488, 492, 494, 510, 512, 516, 518, 522

Offset: 1

Antti Karttunen, Oct 31 2019

Equally, numbers in whose primorial base expansion there are no digits more than ((prime(k)-1)/2), where prime(k) is the modulus for the digit position k = 1 + maximal allowed digit for that position.

Differs from A276154, for example, this sequence does not contain term 120.

			2 is included, as in the primorial base (A049345) it is written as "10", thus 2 is included in the sequence as the maximal value that can occur in the second rightmost digit (in the primorial base representation) is 2 (as in "20" = 4 or "21" = 5 for example).

Antti Karttunen, Table of n, a(n) for n = 1..9072
Index entries for sequences related to primorial base.

Subsequence of A276154 (because of Bertrand's postulate).

Cf. A002110, A049345, A276086, A328834, A328849.

q[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; AllTrue[s/(Prime[Range[1, Length[s]]] - 1), # <= 1/2 &]]; Select[Range[0, 600], q] (* Amiram Eldar, Mar 13 2024 *)

isA328770(n) = { my(p=2); while(n, if((n%p)>((p-1)/2), return(0)); n = n\p; p = nextprime(1+p)); (1); };

a(n) = A328849(n)/2.
Because doubling these numbers in primorial base does not generate any carries, it follows that:
A276086(a(n)+a(n)) = A276086(a(n)) * A276086(a(n)) = A328834(n)^2.

A353572 Shifted variant of A342002: a(n) = A353571(A276086(n)), where A353571(x) = A003415(A003961(x)) / A003557(A003961(x)) and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 8, 2, 11, 1, 10, 12, 71, 19, 92, 2, 13, 17, 86, 24, 107, 3, 16, 22, 101, 29, 122, 4, 19, 27, 116, 34, 137, 1, 14, 16, 103, 27, 136, 18, 131, 167, 886, 244, 1117, 29, 164, 222, 1051, 299, 1282, 40, 197, 277, 1216, 354, 1447, 51, 230, 332, 1381, 409, 1612, 2, 17, 21, 118, 32, 151, 25, 152, 202, 991, 279

Offset: 0

Antti Karttunen, Apr 27 2022

Cf. A003415, A003557, A003961, A276154, A342002, A349905, A353571, A353573 [= gcd(a(n), A342002(n))], A353574.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A353571(n) = { my(s=A003961(n)); (A003415(s)/A003557(s)); };
A353572(n) = A353571(A276086(n));

a(n) = A353571(A276086(n)).
a(n) = A342002(A276154(n)).
For all n >= 0, a(n) >= A342002(n).

A286629 a(n) = (A000040(n)-1) * A002110(n).

Original entry on oeis.org

2, 12, 120, 1260, 23100, 360360, 8168160, 174594420, 4908043140, 181151410440, 6016814703900, 267146572853160, 12170010541088400, 549475975930141260, 28284929999070604860, 1694636240813882325960, 111520100308944333066060, 7037302881564418258996200, 518649222371297625688019940, 39055858108868927267719077300

Offset: 1

Antti Karttunen, Jul 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..120
Index entries for sequences related to primorial base

Cf. A000040, A002110, A006093, A061720, A276154, A286630.

Table[(Prime[n] - 1) Product[Prime[k], {k, n}], {n, 100}] (* Indranil Ghosh, Jul 07 2017 *)

a(n) = (prime(n)-1)*prod(k=1, n, prime(k)); \\ Michel Marcus, Jul 07 2017

from sympy import prime, primorial
def a002110(n): return 1 if n<1 else primorial(n)
def a(n): return (prime(n) - 1)*a002110(n)
print([a(n) for n in range(1, 21)]) # Indranil Ghosh, Jul 07 2017

(define (A286629 n) (* (- (A000040 n) 1) (A002110 n)))

a(n) = A006093(n) * A002110(n) = (A000040(n)-1) * A002110(n).
a(n) = A286630(n) - A002110(n).
a(n) = A276154(A061720(n-1)).

cp's OEIS Frontend

A373158 Fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A276152 a(n) = {smallest prime not dividing n} times {greatest primorial number which divides n} = A053669(n) * A053589(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A328770 Numbers in whose primorial base expansion any digit is at most half of the maximal allowed digit for that position.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A353572 Shifted variant of A342002: a(n) = A353571(A276086(n)), where A353571(x) = A003415(A003961(x)) / A003557(A003961(x)) and A276086 is the primorial base exp-function.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A286629 a(n) = (A000040(n)-1) * A002110(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

Formula