A370114 -id:A370114 - cp's OEIS frontend

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

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

A327859 a(n) = A276086(A003415(n)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 1, 2, 2, 9, 2, 18, 2, 25, 5, 10, 2, 225, 2, 30, 15, 21, 2, 750, 2, 625, 45, 50, 2, 525, 45, 150, 3750, 21, 2, 14, 2, 18375, 75, 250, 25, 49, 2, 750, 225, 735, 2, 630, 2, 875, 210, 1250, 2, 385875, 75, 1050, 375, 13125, 2, 36750, 225, 1029, 1125, 14, 2, 1029, 2, 42, 5250, 2941225, 125, 98, 2, 1225, 1875, 78750

Offset: 0

Antti Karttunen, Sep 30 2019

Sequence contains only terms of A048103.
Are there fixed points other than 1, 2, 10, 15, 5005? (There are none in the range 5006 .. 402653184.) See A369650.
Records occur at n = 0, 2, 4, 6, 8, 12, 18, 27, 32, 48, 64, 80, 144, 224, 256, 336, 448, 480, 512, 1728, ... (see also A131117).
a(n) and n are never multiples of 9 at the same time, thus the fixed points certainly exclude any terms of A008591. For a proof, consider my comment in A047257 and that A003415(9*n) is always a multiple of 3. - Antti Karttunen, Feb 08 2024

Antti Karttunen, Table of n, a(n) for n = 0..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003415, A008591, A048103, A131117, A276086, A327858, A327860, A341517 [= mu(a(n))], A341518 (k where a(k) is squarefree), A369641 (composite k where a(k) is squarefree), A369642.

Cf. A370114 (where a(k) is a multiple of k), A370115 (where k is a multiple of a(k)), A369650.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327859(n) = A276086(A003415(n));

a(n) = A276086(A003415(n)).

a(p) = 2 for all primes p.

A328387 Numbers k such that A276086(k) is a multiple of k.

Original entry on oeis.org

1, 3, 15, 25, 75, 105, 147, 175, 343, 385, 525, 539, 625, 735, 825, 1029, 1155, 1225, 1331, 1375, 1617, 1815, 2695, 3003, 3025, 3675, 3773, 3993, 4375, 5005, 5145, 5577, 5775, 6655, 6825, 8085, 8125, 8281, 8575, 9075, 9555, 9625, 10725, 11011, 11319, 12675, 12705, 13013, 13377, 15015, 15379, 15925, 17303, 17745, 17787, 17875

Offset: 1

Antti Karttunen, Oct 15 2019

nonn

All terms are odd. Of the first 3003 terms, 1709 are multiples of five.

Antti Karttunen, Table of n, a(n) for n = 1..3003

Cf. A276086, A324198.
Indices of 0's in A328386. Indices of 1's in A351250.
Subsequence of A048103 and of A358226.
Cf. also A370114, A358231.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA328387(n) = (0==(A276086(n)%n));

A369964 a(n) = gcd(n, A276086(A003415(n))), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 10, 1, 3, 1, 2, 15, 1, 1, 6, 1, 5, 3, 2, 1, 3, 5, 2, 3, 7, 1, 2, 1, 1, 3, 2, 5, 1, 1, 2, 3, 5, 1, 42, 1, 1, 15, 2, 1, 3, 1, 50, 3, 1, 1, 6, 5, 7, 3, 2, 1, 3, 1, 2, 21, 1, 5, 2, 1, 1, 3, 70, 1, 1, 1, 2, 25, 1, 1, 6, 1, 5, 1, 2, 1, 21, 5, 2, 3, 1, 1, 6, 1, 1, 3, 2, 5, 3, 1, 98, 3, 25

Offset: 0

Antti Karttunen, Feb 11 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A003415, A276086, A327859, A370114 (fixed points, see also A369650), A370116, A370117.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A369964(n) = gcd(n, A276086(A003415(n)));

a(n) = gcd(n, A327859(n)) = gcd(n, A276086(A003415(n))).
For n >= 1, a(n) = n / A370116(n).
For n >= 0, a(n) = A327859(n) / A370117(n).

A370115 Numbers k for which k is a multiple of A276086(A003415(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 2, 10, 15, 161, 2189, 5005, 27030, 29861, 510221, 223092341

Offset: 1

Antti Karttunen, Feb 11 2024

nonn

Question: Is the squarefreeness a necessary condition for the nonzero terms of this sequence?
Many of the terms occur also in A368703, because the arithmetic derivative of those terms is one of the primorial numbers, A002110.
If it exists, a(13) > 1241513984.

Positions of 1's in A370117, positions of 0's in A370120.
Intersection of A048103 and A369650 is a subsequence of this sequence. See the comments in latter.
Cf. A002110, A003415, A276086, A327859, A368703.
Cf. also A369970, A370114.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA370115(n) = !(n%A276086(A003415(n)));

A370116 Numerator of n/A276086(A003415(n)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 1, 7, 8, 9, 1, 11, 4, 13, 7, 1, 16, 17, 3, 19, 4, 7, 11, 23, 8, 5, 13, 9, 4, 29, 15, 31, 32, 11, 17, 7, 36, 37, 19, 13, 8, 41, 1, 43, 44, 3, 23, 47, 16, 49, 1, 17, 52, 53, 9, 11, 8, 19, 29, 59, 20, 61, 31, 3, 64, 13, 33, 67, 68, 23, 1, 71, 72, 73, 37, 3, 76, 77, 13, 79, 16, 81, 41, 83, 4, 17, 43

Offset: 0

Antti Karttunen, Feb 11 2024

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A003415, A276086, A327859, A369964, A370114 (positions of 1's), A370117 (denominators).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A370116(n) = (n/gcd(n, A276086(A003415(n))));

a(n) = n / A369964(n) = n / gcd(n, A276086(A003415(n))).

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

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A327859 a(n) = A276086(A003415(n)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A328387 Numbers k such that A276086(k) is a multiple of k.

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A369964 a(n) = gcd(n, A276086(A003415(n))), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A370115 Numbers k for which k is a multiple of A276086(A003415(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A370116 Numerator of n/A276086(A003415(n)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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