A070826 -id:A070826 - cp's OEIS frontend

A136354 a(n) is the smallest odd composite number m such that m+2 is prime and the set of distinct prime factors of m consists of the first n odd primes.

9, 15, 105, 3465, 15015, 765765, 33948915, 334639305, 3234846615, 100280245065, 3710369067405, 1369126185872445, 32706903329175075, 307444891294245705, 211829530101735290745, 961380175077106319535, 762374478836145311391255

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A070826 and A118750.

			a(1)=9 because k=1 with prime factor 3 and 9+2=11, prime

Cf. A136349-A136353, A136355-A136358, A070826, A118750.

a[n_]:=(c=Product[Prime[k],{k,2,n+1}]; For[m=1,!(!PrimeQ[c(2m-1)]&&PrimeQ[c(2m-1)+2]&&Length[FactorInteger[c(2m-1)]]==n),m++ ]; c(2m-1)); Table[a[n],{n,17}] (* Farideh Firoozbakht, Aug 12 2009 *)

Edited, corrected and extended by Farideh Firoozbakht, Aug 12 2009

A136355 Numbers of the form P = product of the first k odd primes where P+2 is composite.

Original entry on oeis.org

1155, 255255, 4849845, 111546435, 152125131763605, 6541380665835015, 16294579238595022365, 58644190679703485491635, 3929160775540133527939545, 278970415063349480483707695, 20364840299624512075310661735, 1608822383670336453949542277065

Offset: 1

Enoch Haga, Dec 25 2007

This sequence is different from A070826 and A118750.

			a(1)=1155 because 1157 is not prime.

Harvey P. Dale, Table of n, a(n) for n = 1..332

Cf. A136349-A136354, A136356-A136358, A070826, A118750.

v=Select[Range[21],!PrimeQ[Product[Prime[k+1],{k,#}]+2]&]; Table[Product[Prime[k+1],{k,v[[t]]}],{t,Length[v]}] (* Farideh Firoozbakht, Aug 12 2009 *)
Select[FoldList[Times,Prime[Range[2,22]]],CompositeQ[#+2]&] (* Harvey P. Dale, Jun 08 2022 *)

Edited with more terms by Farideh Firoozbakht, Aug 12 2009

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

A243054 a(0)=1, and for n >= 1, a(n) = p_n * A002110(n) / 2, where p_n is the n-th prime.

Original entry on oeis.org

1, 2, 9, 75, 735, 12705, 195195, 4339335, 92147055, 2565568005, 93810551835, 3108687597015, 137283655493985, 6237130402307805, 281279368630905645, 14449909890829548135, 863612699645536185345, 56721430329549272852565, 3577295631461912614989735, 263253771961188946371949515

Offset: 0

Antti Karttunen, May 29 2014

Fixed points of A243051.

From n >= 2 onward: a(n) = the product of the first n-1 odd primes multiplied by the n-th prime.

Cf. A000040, A002110, A006530, A070826, A243051.

(define (A243054 n) (if (zero? n) 1 (* (/ 1 2) (A000040 n) (A002110 n))))

a(0)=1, and for n >= 1, a(n) = A000040(n)*A002110(n)/2.

a(0)=1, a(1)=2 and for n >= 2, a(n) = A006530(A070826(n)) * A070826(n).

A355538 Partial sum of A001221 (number of distinct prime factors) minus 1, ranging from 2 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 14, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 23, 23, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 37, 37, 38, 39, 39, 40, 42, 42, 43, 44, 46, 46

Offset: 1

Gus Wiseman, Jul 23 2022

nonn

For initial terms up to 30 we have a(n) = Log_2 A355537(n).

The sum of the same range is A000096.
The product of the same range is A000142, Heinz number A070826.
For divisors (not just prime factors) we get A002541, also A006218, A077597.
A shifted variation is A013939.
The unshifted version is A022559, product A327486, w/o multiplicity A355537.
The ranges themselves are the rows of A131818 (shifted).
Partial sums of A297155 (shifted).
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A066843 gives partial sums of A000005.
Cf. A000720, A002110, A076610, A305054, A355538, A355731, A355733, A355741, A355744, A355745, A355746, A355747.

Table[Total[(PrimeNu[#]-1)&/@Range[2,n]],{n,1,100}]

a(n) = A013939(n) - n + 1.

A046967 a(1) = 2, a(n) = a(n-1)^prime(n) for n > 1.

Original entry on oeis.org

2, 8, 32768, 40564819207303340847894502572032

Offset: 1

Felice Russo

			a(3) = 32768 because (2^3)^5 = 32768.

Amiram Eldar, Table of n, a(n) for n = 1..5

Cf. A070826.

RecurrenceTable[{a[1]==2,a[n]==a[n-1]^Prime[n]},a[n],{n,4}] (* Harvey P. Dale, Jul 18 2011 *)
2^FoldList[Times, 1, Prime[Range[2, 4]]] (* Amiram Eldar, Oct 12 2024 *)

a(n)=if(n<2,1,a(n-1)^prime(n)) \\ Charles R Greathouse IV, Jun 20 2013

a(n) = 2^A070826(n). - Amiram Eldar, Oct 12 2024

A074107 a(n) = Product of (prime + 1) for first n primes - primorial (n); Sum of proper divisors of the n-th primorial.

Original entry on oeis.org

0, 1, 6, 42, 366, 4602, 66738, 1231314, 25136790, 612982650, 18612572370, 602072009070, 23079296834790, 976751205195990, 43281303292150770, 2090585319354906990, 113506497027753468870, 6842978980142398176930, 426187457118982899608730, 29098035465450244144376910, 2102916875063497845451016610, 156173789584825539524342644530

Offset: 0

Amarnath Murthy, Aug 22 2002

nonn

			a(3) = (2+1)*(3+1)*(5+1) - 2*3*5 = 72 - 30 = 42.

Antti Karttunen, Table of n, a(n) for n = 0..349 (terms 1..349 from Harvey P. Dale)
Index entries for sequences related to primorial numbers

Cf. A001065, A002110, A003415, A024451, A051674, A054640, A070826, A348507.

for n from 1 to 25 do a[n] := product(ithprime(i)+1,i=1..n)-product(ithprime(i),i=1..n): od:seq(a[j],j=1..25);

Module[{nn=20,p,pr,pr1},p=Prime[Range[nn]];pr=FoldList[Times,1,p];pr1= FoldList[Times,1,p+1];#[[2]]-#[[1]]&/@Rest[Thread[{pr,pr1}]]](* Harvey P. Dale, Feb 07 2015 *)

A074107(n) = (prod(i=1,n,1+prime(i))-prod(i=1,n,prime(i))); \\ Antti Karttunen, Nov 19 2024

From Antti Karttunen, Nov 19 2024: (Start)
a(n) = A348507(A002110(n)) = A054640(n) - A002110(n) = A001065(A002110(n)).
a(n) >= A024451(n), because A348507(n) >= A003415(n).
For n >= 1, a(n) <= A070826(1+n) [= A002110(1+n)/2] < A051674(n).
(End)

More terms from Sascha Kurz, Feb 01 2003

Term a(0)=0 prepended, data section further extended, and secondary definition added by Antti Karttunen, Nov 19 2024

A091852 Numerator of Sum_{i=1..n} prime(i)/prime(i+1).

Original entry on oeis.org

2, 19, 208, 3023, 52004, 1079263, 24845332, 663589691, 21809669044, 769910292199, 31595368408378, 1432693760237483, 67842962092519574, 3469898586979325623, 198354535000733806154, 12566530264688830748431

Offset: 1

N. J. A. Sloane, Mar 13 2004

Proposed by Mohammed BOUAYOUN, Feb 16 2004; computed by R. K. Guy, Feb 17 2004

			The sums are f(1)=2/3, f(2)=2/3 + 3/5, f(3)=2/3+3/5+5/7, f(4)=2/3+3/5+5/7+7/11, ...

Denominators are in A070826.

A091852[n_] := Numerator[ Sum[ Prime[i] / Prime[i + 1], {i, n}]]; Table[ A091852[n], {n, 16}] (* Robert G. Wilson v, Mar 16 2004 *)

More terms from Robert G. Wilson v, Mar 16 2004

A098206 A first order iteration: n-th term is obtained from (n-1)-th by adding n-th prime and then multiplying by the n-th prime; initial value is 1.

Original entry on oeis.org

1, 12, 85, 644, 7205, 93834, 1595467, 30314234, 697227911, 20219610260, 626807919021, 23191893005146, 950867613212667, 40887307368146530, 1921703446302889119, 101850282654053126116, 6009166676589134444325

Offset: 1

Labos Elemer, Oct 19 2004

nonn

			n=4: a(4)=(a(3)+7)*7=(85+7)*7=644.

Harvey P. Dale, Table of n, a(n) for n = 1..350

Cf. A070826, A019461.

a:= n -> mul(ithprime(j),j=2..n) + add(ithprime(k)*mul(ithprime(j),j=k..n),k=2..n):
seq(a(n), n=1..30); # Robert Israel, Feb 12 2015

f[x_]:=(f[x-1]+Prime[x])*Prime[x];f[1]=0;Table[f[w], {w, 1, 25}]
nxt[{n_,a_}]:=Module[{p=Prime[n+1]},{n+1,p(a+p)}]; NestList[nxt,{1,1},20][[All,2]] (* Harvey P. Dale, Jun 18 2021 *)

a(n) = (a(n-1)+prime(n))*prime(n), a(1)=1.

a(n) = product(j=2..n, prime(j)) + sum(k=2..n, prime(k)*product(j=k..n, prime(j))). - Robert Israel, Feb 12 2015

A136353 First odd composite N divisible by precisely the first n odd primes with N-2 prime.

Original entry on oeis.org

9, 15, 105, 1155, 15015, 255255, 4849845, 111546435, 9704539845, 100280245065, 18551845337025, 152125131763605, 98120709987525225, 7071232499767651215, 16294579238595022365, 33648306127698721183725, 527797716117331369424715

Offset: 1

Enoch Haga, Dec 25 2007

nonn

This sequence is different from A070826 and A118750.

A clearer definition of the sequence: a(n) is the smallest odd composite number m such that m - 2 is prime and the set of the distinct prime factors of m equals the set of the first n odd primes. - Farideh Firoozbakht, Jun 30 2009

			The first odd prime is 3, 3*3-2 = 7 is prime so a(1) = 9.
The product of the first two odd primes is 15, and 15-2 is prime, so a(2) = 15.

Charles R Greathouse IV, Table of n, a(n) for n = 1..200

Cf. A136349, A136350, A136351, A136352, A136354, A136355, A136356, A136357, A136358.

Cf. A070826, A118750.

a[n_]:=(c=Product[Prime[k],{k,2,n+1}];For[m=1,!(!PrimeQ[c (2m-1)]&&PrimeQ[c(2m-1)-2]&&Length[FactorInteger[c(2m-1)]]==n),m++ ];c(2m-1));Table[a[n],{n,20}] (* Enoch Haga, Jul 02 2009 *)

sm(n,x)=forprime(p=2,x, n/=p^valuation(n,p)); n==1
a(n)=my(m=factorback(primes(n+1)[2..n+1]),k,p=prime(n+1)); while(!isprime(k++*m-2) && sm(k,p),); k*m \\ Charles R Greathouse IV, Sep 14 2015

Compute N = product of the first n odd primes. If N-2 is prime, add N to the sequence. Otherwise test 3N, 5N, 7N, 9N, ... until kN - 2 is prime, subject to A006530(k) <= n+1.

More terms, better title, and Mathematica program from Farideh Firoozbakht received Jun 30 2009. - Enoch Haga, Jul 02 2009

Further editing by Charles R Greathouse IV, Oct 05 2009

cp's OEIS Frontend

A136354 a(n) is the smallest odd composite number m such that m+2 is prime and the set of distinct prime factors of m consists of the first n odd primes.

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A136355 Numbers of the form P = product of the first k odd primes where P+2 is composite.

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

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A243054 a(0)=1, and for n >= 1, a(n) = p_n * A002110(n) / 2, where p_n is the n-th prime.

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A355538 Partial sum of A001221 (number of distinct prime factors) minus 1, ranging from 2 to n.

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A046967 a(1) = 2, a(n) = a(n-1)^prime(n) for n > 1.

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A074107 a(n) = Product of (prime + 1) for first n primes - primorial (n); Sum of proper divisors of the n-th primorial.

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A091852 Numerator of Sum_{i=1..n} prime(i)/prime(i+1).

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