A024451 -id:A024451 - cp's OEIS frontend

A241189 Numerator of Sum_{i=1..n} 1/(prime(i)*prime(i+1)).

1, 7, 11, 127, 1693, 29243, 561623, 13019431, 379503437, 11809225121, 438235268123, 18007758091069, 775817745542929, 36524284093223105, 1938403609207158571, 2160165866032831207, 131893095784520401909, 8844093116997411126541, 628373208972323386101329, 45900898298568589325230523

Offset: 1

N. J. A. Sloane, Apr 25 2014, based on a suggestion from Timothy Varghese

a(371) has 1002 decimal digits. - Michael De Vlieger, Jan 27 2016

			1/6, 7/30, 11/42, 127/462, 1693/6006, 29243/102102, 561623/1939938, 13019431/44618574, 379503437/1293938646, 11809225121/40112098026, 438235268123/1484147626962, ...

Michael De Vlieger, Table of n, a(n) for n = 1..370

Cf. A024451/A002110, A241189/A241190, A241191/A241192.

See also A061015/A061742, A115963/A115964.

g:= n-> add(1/(ithprime(i)*ithprime(i+1)),i=1..n);
t1:=[seq(g(n),n=1..20)];
t1a:=map(numer,t1); # A241189
t1b:=map(denom,t1); # A241190

Table[Numerator@ Sum[1/(Prime[i + 1] Prime@ i), {i, n}], {n, 20}] (* Michael De Vlieger, Jan 27 2016 *)
Accumulate[1/#&/@(Times@@@Partition[Prime[Range[25]],2,1])]//Numerator (* Harvey P. Dale, Mar 14 2023 *)

A241190 Denominator of Sum_{i=1..n} 1/(prime(i)*prime(i+1)).

Original entry on oeis.org

6, 30, 42, 462, 6006, 102102, 1939938, 44618574, 1293938646, 40112098026, 1484147626962, 60850052705442, 2616552266334006, 122977956517698282, 6517831695438008946, 7255699434544198638, 442597665507196116918, 29654043588982139833506, 2105437094817731928178926, 153696907921694430757061598

Offset: 1

N. J. A. Sloane, Apr 25 2014, based on a suggestion from Timothy Varghese

a(371) has 1003 decimal digits. - Michael De Vlieger, Jan 27 2016

			1/6, 7/30, 11/42, 127/462, 1693/6006, 29243/102102, 561623/1939938, 13019431/44618574, 379503437/1293938646, 11809225121/40112098026, 438235268123/1484147626962, ...

Michael De Vlieger, Table of n, a(n) for n = 1..370

Cf. A024451/A002110, A241189/A241190, A241191/A241192.

g:= n-> add(1/(ithprime(i)*ithprime(i+1)),i=1..n);
t1:=[seq(g(n),n=1..20)];
t1a:=map(numer,t1); # A241189
t1b:=map(denom,t1); # A241190

Table[Denominator@ Sum[1/(Prime[i + 1] Prime@ i), {i, n}], {n, 20}] (* Michael De Vlieger, Jan 27 2016 *)

a(n) = denominator(sum(k=1, n, 1/(prime(k)*prime(k+1)))); \\ Michel Marcus, Jan 27 2016

A241191 Numerator of Sum_{i=1..n} 1/(prime(i)prime(i+1)prime(i+2)).

Original entry on oeis.org

1, 3, 1, 93, 145, 213, 289, 365, 260511, 9645025, 395623447, 17017308303, 800016993275, 902324346127, 2502504187659113, 152669617912106167, 10229758110750827711, 726365554791051924279, 53027901964339123037045, 57389311757677767147397

Offset: 1

N. J. A. Sloane, Apr 25 2014, based on a suggestion from Timothy Varghese

			1/30, 3/70, 1/22, 93/2002, 145/3094, 213/4522, 289/6118, 365/7714, 260511/5500082, 9645025/203503034, 395623447/8343624394, 17017308303/358775848942, 800016993275/16862464900274, 902324346127/19015119993926, ...

Robert Israel, Table of n, a(n) for n = 1..361

Cf. A024451/A002110, A241189/A241190, A241191/A241192.

g:= n-> add(1/(ithprime(i)*ithprime(i+1)*ithprime(i+2)),i=1..n);
t1:=[seq(g(n),n=1..20)];
t1a:=map(numer,t1); # A241191
t1b:=map(denom,t1); # A241192

A241192 Denominator of Sum_{i=1..n} 1/(prime(i)prime(i+1)prime(i+2)).

Original entry on oeis.org

30, 70, 22, 2002, 3094, 4522, 6118, 7714, 5500082, 203503034, 8343624394, 358775848942, 16862464900274, 19015119993926, 52728927743156798, 3216464592332564678, 215503127686281833426, 15300722065726010173246, 1116952710797998742646958, 1208757043192354803686434

Offset: 1

N. J. A. Sloane, Apr 25 2014, based on a suggestion from Timothy Varghese

			1/30, 3/70, 1/22, 93/2002, 145/3094, 213/4522, 289/6118, 365/7714, 260511/5500082, 9645025/203503034, 395623447/8343624394, 17017308303/358775848942, 800016993275/16862464900274, 902324346127/19015119993926, ...

Cf. A024451/A002110, A241189/A241190, A241191/A241192.

g:= n-> add(1/(ithprime(i)*ithprime(i+1)*ithprime(i+2)),i=1..n);
t1:=[seq(g(n),n=1..20)];
t1a:=map(numer,t1); # A241191
t1b:=map(denom,t1); # A241192

A244622 Primes in the sequence of first arithmetic derivative of primorials.

Original entry on oeis.org

5, 31, 2927, 40361, 201015517717077830328949, 13585328068403621603022853, 5692733621468679832887230172131, 3215488142498485484492183158345029261034221047849345857469577412562094716564064084247

Offset: 1

Freimut Marschner, Jul 02 2014

nonn

A002110 is the sequence of primorial numbers (product of consecutive prime numbers, written prime(n)#). A024451 = numerator of Sum_{i = 1..n} 1/prime(i) is the first arithmetic derivative of prime(n)#, written (prime(n)#)'. The second arithmetic derivative of prime(n)#, written (prime(n)#)'' [= A369651(n)] is 1 if (prime(n)#)' is prime. This case leads to a selection of 13 primorials out of the first 100 primorials. The table shows the counting number n of this selection, the primorial notation, the index i used in A002110 and A024451 and the 2nd arithmetic derivative of the 13 prime numbers of A024451. Remark: i [= A109628(n)] is the prime number index of A000040.
------------------------------------------------------
n              a(n) = (prime(i)#)’   i         (a(n))'
------------------------------------------------------
            (3#)’                 2            1
            (5#)’                 3            1
           (11#)’                 5            1
           (13#)’                 6            1
           (61#)’                18            1
           (67#)’                19            1
           (79#)’                22            1
          (211#)’                47            1
          (269#)’                57            1
         (271#)’                58            1
         (307#)’                63            1
         (349#)’                70            1
         (367#)’                73            1
A-number links for A109628 and A369651 added by Antti Karttunen, Feb 08 2024

			a(1) = (3#)' = (2*3 = 6)' = 2+3 = 5.

Freimut Marschner, Table of n, a(n) for n = 1..13

Cf. A000040, A002110, A024451, A003415, A109628, A244621, A369651.

Cf. also A351088.

a(1) = (prime(2)#)' = (3#)' = (6)' = 5, (5)' = 1 ; a(4) = (prime(6)#)' = (13#)' =(30030)' = 40361, (40361)' = 1.

f[n_] := Numerator[Accumulate[Table[1/Prime[i], {i, 1, n}]]];
Select[f[50], PrimeQ] (* Ivan N. Ianakiev, Jul 08 2019 *)

lista() = {vadp = readvec("/gp/bfiles/b024451.txt"); for (i=1, #vadp, if (isprime(vadp[i]), print1(vadp[i], ", ");););} \\ Michel Marcus, Jul 05 2014

a(n) = (prime(i)#)' if (prime(i)#)'' = 1.
a(n) = (prime(i)#)' if A003415(A002110(i)) is prime or A003415(A024451(i)) = 1.
a(n) = A024451(A109628(n)). - Antti Karttunen, Feb 08 2024

A328232 Numbers whose arithmetic derivative (A003415) is a primorial number, including cases where it is the first primorial, A002110(0) = 1.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 161, 163, 167, 173, 179, 181, 191, 193, 197, 199, 209, 211, 221, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317

Offset: 1

Antti Karttunen, Oct 09 2019

nonn

Numbers n such that A327859(n) = A276086(A003415(n)) is a prime.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial numbers

Cf. A002110, A003415, A024451 (arith. deriv. of primorials), A068346, A276086, A327859, A328233.
Union of A000040 and A327978 (gives the composite terms).
Differs from A189710 for the first time by having term a(39) = 161, which is not included in A189710, while A189710(44) = 185 is the first term in latter that is not included here.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A327859(n) = A276086(A003415(n));
isA328232(n) = isprime(A327859(n));

A369651 The second arithmetic derivative of primorial numbers.

Original entry on oeis.org

0, 0, 1, 1, 32, 1, 1, 37712, 381596, 9835475, 36880858, 5221912091, 149759057759, 186024798, 456607833524310, 655374897748042, 32188411533060, 3002590335363139484, 1, 1, 1489192548103758778136571, 905624730150525686386253700, 1, 110832638787270947169636975150, 331748330219014906068858, 349669183443106226663011852002384

Offset: 0

Antti Karttunen, Jan 31 2024

nonn

See comments in A024451.

Antti Karttunen, Table of n, a(n) for n = 0..49

Cf. A002110, A003415, A024451, A068346, A109628 (positions of 1's).

Cf. also A368702.

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]));
A369651(n) = A003415(A003415(A002110(n)));

a(n) = A003415(A024451(n)) = A068346(A002110(n)).

A369970 Numbers k such that A003415(k) is a multiple of A276086(k), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 6, 2315, 510510

Offset: 1

Antti Karttunen, Feb 07 2024

For the general dynamics of this phenomenon, see the scatter plots of A351231 and A351233.
Question: Are the terms by necessity all squarefree?
As a subsequence this sequence includes all primorials with indices k such that A024451(k) is a multiple of A000040(1+k). See A369972 and A369973.
872415232 < a(6) <= 13082761331670030 [= A369973(4)].

			2315 is included as A003415(2315) = 5+463 = 468 = 2^2 * 3^2 * 13 (note that 2315 is a semiprime = 5*463, thus its arithmetic derivative is the sum of its two prime factors), and because that 468 is a multiple of A276086(2315) = 234 = 2 * 3^2 * 13 [the exponents of primes are here read from the primorial base expansion of 2315, A049345(2315) = 100021].
510510 is included because A003415(510510) = 19*37693, which is a multiple of A276086(510510) = 19.

Index entries for sequences related to primorial base

Cf. A000040, A003415, A024451, A276086, A369972, A369973 (subsequence).
Positions of 1's in A351231, positions of 0's in A351233 and in A369971.
After the two initial terms, a subsequence of A351228.
Cf. also A358221.

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); };
isA369970(n) = !(A003415(n)%A276086(n));

A370138 Arithmetic derivatives of the sums of three primorials > 1.

Original entry on oeis.org

5, 7, 9, 21, 19, 21, 41, 33, 61, 123, 109, 111, 191, 165, 211, 459, 213, 361, 705, 951, 1361, 1319, 3537, 1173, 2195, 2479, 1481, 2111, 3295, 3421, 2313, 5415, 5885, 5891, 11091, 15019, 16371, 35067, 15033, 25061, 33373, 15123, 26057, 31309, 42955, 16691, 48573, 36329, 45845, 62385, 31167, 72201, 62123, 80969, 141399, 151113

Offset: 1

Antti Karttunen, Mar 09 2024

nonn

For n > 20, a(n) > A369979(n).

Antti Karttunen, Table of n, a(n) for n = 1..10660
Index entries for sequences related to primorial numbers

Cf. A000292, A002110, A003415, A369979, A370137.

Cf. also A024451, A370129.

up_to = 15180;
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]));
A370137list(up_to) = { my(v = vector(up_to), i=0); for(x=1,oo, for(y=1,x, for(z=1,y, i++; if(i > up_to, return(v)); v[i] = A002110(x)+A002110(y)+A002110(z)))); (v); };
v370137 = A370137list(up_to);
A370137(n) = v370137[n];
A370138(n) = A003415(A370137(n));

a(n) = A003415(A370137(n)).

A203008 (n-1)-st elementary symmetric function of the first n odd primes; a(0) = 0.

Original entry on oeis.org

0, 1, 8, 71, 886, 12673, 230456, 4633919, 111429982, 3343015913, 106868339918, 4054408822031, 169941130770676, 7459593754902673, 357142287146260646, 19235986110046059943, 1151217759731312559002, 71185663518687172418657

Offset: 0

Clark Kimberling, Dec 29 2011

nonn

Arithmetic derivative of the product of first n odd primes. - Antti Karttunen, Jan 31 2024

Primes occur at indices: 3, 19, 23, 117, 119, 127, 161, 209, ..., and they are: 71, 346723099672193960193396979, 15360643606799479140185671512081451, ... - Antti Karttunen, Feb 06 2024

Antti Karttunen, Table of n, a(n) for n = 0..349

Cf. A000035, A003415, A024451, A060389, A070826 (n-th. symm. function), A071148 (1st symm. func), A327860.

f[k_] := Prime[k + 1]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A203008 *)

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]));
A203008(n) = if(!n,n,A003415(A002110(1+n)/2)); \\ Antti Karttunen, Jan 31 2024

From Antti Karttunen, Jan 31 2024 and Feb 06 2024: (Start)
a(n) = A003415(A070826(1+n)) = (1/2)*(A024451(1+n)-A070826(1+n)).
For n >= 1, a(n) = A327860(A060389(n)).
A000035(a(n)) = A000035(n).
(End)

Term a(0) = 0 prepended by Antti Karttunen, Jan 31 2024

cp's OEIS Frontend

A241189 Numerator of Sum_{i=1..n} 1/(prime(i)*prime(i+1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A241190 Denominator of Sum_{i=1..n} 1/(prime(i)*prime(i+1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A241191 Numerator of Sum_{i=1..n} 1/(prime(i)*prime(i+1)*prime(i+2)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A241192 Denominator of Sum_{i=1..n} 1/(prime(i)*prime(i+1)*prime(i+2)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

A244622 Primes in the sequence of first arithmetic derivative of primorials.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A328232 Numbers whose arithmetic derivative (A003415) is a primorial number, including cases where it is the first primorial, A002110(0) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A369651 The second arithmetic derivative of primorial numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A369970 Numbers k such that A003415(k) is a multiple of A276086(k), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Views

Author

Keywords

Comments

Examples

Links

A241191 Numerator of Sum_{i=1..n} 1/(prime(i)prime(i+1)prime(i+2)).

A241192 Denominator of Sum_{i=1..n} 1/(prime(i)prime(i+1)prime(i+2)).