A373848 -id:A373848 - 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

A373842 a(n) = A003415(A276085(n)), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

Original entry on oeis.org

0, 0, 1, 1, 5, 1, 31, 1, 4, 1, 247, 4, 2927, 1, 12, 4, 40361, 1, 716167, 12, 80, 1, 14117683, 1, 16, 1, 5, 80, 334406399, 6, 9920878441, 1, 216, 568, 60, 5, 314016924901, 33975, 3740, 6, 11819186711467, 14, 492007393304957, 216, 7, 28300, 21460568175640361, 5, 92, 1, 60080, 3740, 1021729465586766997, 1, 540, 14

Offset: 1

Antti Karttunen, Jun 20 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..481

Cf. A003415, A024451, A276085, A373843 [= gcd(n, a(n))], A373846 (positions of 1's), A373847 [k such that a(n)<=k], A373848.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
A373842(n) = A003415(A276085(n));

For n >= 1, a(A000040(n)) = A024451(n-1).

A373603 The second smallest k such that A003415(k) == A276086(k) mod A002110(n), or -1 if no such k exists, where A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and A002110 gives the n-th primorial.

Original entry on oeis.org

2, 9, 26, 122, 1382, 21446, 204566, 9699686, 90387605

Offset: 1

Antti Karttunen, Jun 22 2024

For n > 1, the index of the next term in A373849, after its sixth term 0, that is a multiple of A002110(n), as for n >= 1, the smallest k such that A003415(k) == A276086(k) mod A002110(n) gives the sequence 1, 6, 6, 6, 6, 6, 6, 6, ..., because A003415(6) = A276086(6).
Provided that such k exists for every n (and the escape clause is not needed), then the sequence is by necessity monotonic. If it is strictly monotonic, then it implies that k=6 is the only k such that A003415(k) = A276086(k). See also comments in A351228.
Note that if we instead search for the smallest k such that A276086(k) is a multiple of A002110(n) we obtain A143293, partial sums of the primorial numbers. See also A368703.

Cf. A002110, A003415, A143293, A276086, A373849.

Cf. also A351228, A368703, A371104, A373848.

A002110(n) = prod(i=1,n,prime(i));
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); };
A373603(n) = { my(m=A002110(n), c=2); for(i=1,oo,if(0==((A276086(i)-A003415(i))%m), c--; if(0==c, return(i)))); };

A373844 Triangle read by rows: T(n,k) = A276086(1 + A002110(n) + A002110(k)), 1 <= k <= n, where A276086 is the primorial base exp-function.

Original entry on oeis.org

18, 30, 50, 42, 70, 98, 66, 110, 154, 242, 78, 130, 182, 286, 338, 102, 170, 238, 374, 442, 578, 114, 190, 266, 418, 494, 646, 722, 138, 230, 322, 506, 598, 782, 874, 1058, 174, 290, 406, 638, 754, 986, 1102, 1334, 1682, 186, 310, 434, 682, 806, 1054, 1178, 1426, 1798, 1922, 222, 370, 518, 814, 962, 1258, 1406, 1702, 2146, 2294, 2738

Offset: 1

Antti Karttunen, Jun 21 2024

Triangle giving all products of three primes, of which one is even (2) and two are odd (not necessarily distinct), so that the product is of the form 4m+2.

The only terms such that T(n, k) > A373845(n, k) > 1 are 30, 42, 110 at positions T(2,1), T(3,1), T(4,2), and the corresponding terms in A373845 are 6, 14, 38.

			Triangle begins as:
   18,
   30,  50,
   42,  70,  98,
   66, 110, 154, 242,
   78, 130, 182, 286, 338,
  102, 170, 238, 374, 442,  578,
  114, 190, 266, 418, 494,  646,  722,
  138, 230, 322, 506, 598,  782,  874, 1058,
  174, 290, 406, 638, 754,  986, 1102, 1334, 1682,
  186, 310, 434, 682, 806, 1054, 1178, 1426, 1798, 1922,
  222, 370, 518, 814, 962, 1258, 1406, 1702, 2146, 2294, 2738,
etc.

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 rows of the triangle

Cf. A002110, A276086, A370121, A373845.

Cf. also A087112, A369979, A373848.

A002110(n) = prod(i=1,n,prime(i));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A373844(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2, x=A002110(1+n - binomial(c + 1, 2))); A276086(1+(A002110(1+c)+x)); };

For n, k >= 1, T(n, k) = A276086(1+A370121(n, k)).

For n, k >= 1, T(n, k) = 2*A087112(n+1, k+1).

A373845 Triangle read by rows: T(n,k) = arithmetic derivative of (1 + A002110(n) + A002110(k)), 1 <= k <= n, where A002110(n) is the n-th primorial number.

Original entry on oeis.org

1, 6, 1, 14, 1, 1, 74, 38, 1, 1, 1551, 338, 1, 1, 1, 21084, 8631, 1330, 1, 1, 3550, 172655, 72938, 1970, 3410, 1, 1, 5822, 3233234, 4157356, 421750, 228491, 10190, 13610, 537398, 289610, 297753138, 32805527, 5188250, 8698439, 761710, 1, 18344100, 1, 6954431, 2156564414, 929540471, 68769335, 335525472, 4283242, 21900155, 348965439, 109820278, 185002, 32593310

Offset: 1

Antti Karttunen, Jun 21 2024

Arithmetic derivatives of the sums of three primorials, of which one is 1 [= A002110(0)], and two are > 1.
Ones occur in positions where 1 + A002110(n) + A002110(k) is a prime.
See also comments in A373844, and in A373848.

			Triangle begins as:
        1,
        6,        1,
       14,        1,       1,
       74,       38,       1,       1,
     1551,      338,       1,       1,      1,
    21084,     8631,    1330,       1,      1,  3550,
   172655,    72938,    1970,    3410,      1,     1,     5822,
  3233234,  4157356,  421750,  228491,  10190, 13610,   537398, 289610,
297753138, 32805527, 5188250, 8698439, 761710,     1, 18344100,      1, 6954431,
etc.

Cf. A002110, A003415, A370121, A373844, A373848.

Cf. also A024451, A370129, A370138 (arithmetic derivative applied to the sums of a constant number of primorials).

A002110(n) = prod(i=1,n,prime(i));
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A373845(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2, x=A002110(1+n - binomial(c + 1, 2))); A003415(1+(A002110(1+c)+x)); };

For n, k >= 1, T(n, k) = A003415(1+A370121(n, k)).

A373847 Numbers k for which A373842(k) <= k, where A373842 is the arithmetic derivative of the primorial base log-function.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 27, 30, 32, 36, 40, 42, 45, 48, 50, 54, 56, 60, 63, 64, 70, 72, 74, 75, 80, 81, 84, 88, 90, 96, 98, 100, 105, 108, 110, 112, 120, 125, 126, 128, 132, 135, 140, 144, 147, 150, 154, 160, 162, 165, 168, 175, 176, 180, 182, 189, 192, 196, 198, 200, 210

Offset: 1

Antti Karttunen, Jun 20 2024

nonn

Cf. A373842.

Subsequences: A373846, A373848.

PARI
```
isA373847(n) = (A373842(n)<=n);
```

cp's OEIS Frontend

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

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A373842 a(n) = A003415(A276085(n)), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

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A373603 The second smallest k such that A003415(k) == A276086(k) mod A002110(n), or -1 if no such k exists, where A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and A002110 gives the n-th primorial.

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A373844 Triangle read by rows: T(n,k) = A276086(1 + A002110(n) + A002110(k)), 1 <= k <= n, where A276086 is the primorial base exp-function.

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A373845 Triangle read by rows: T(n,k) = arithmetic derivative of (1 + A002110(n) + A002110(k)), 1 <= k <= n, where A002110(n) is the n-th primorial number.

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A373847 Numbers k for which A373842(k) <= k, where A373842 is the arithmetic derivative of the primorial base log-function.

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