A120271 -id:A120271 - cp's OEIS frontend

A135212 a(n) = A078456(n)/A120271(n).

1, 1, 2, 4, 8, 576, 1152, 2304, 4608, 18432, 552960, 59719680, 2388787200, 100329062400, 200658124800, 802632499200, 1605264998400, 288947699712000, 6356849393664000, 444979457556480000, 10679506981355520000

Offset: 1

Alexander Adamchuk, Nov 23 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A078456, A120271.

Table[ Det[ DiagonalMatrix[ Table[ Prime[i+1]-1, {i, 1, n-1} ] ] + 1 ], {n, 1, 50} ] / Numerator[ Table[ Sum[ 1/(Prime[i]-1), {i, 1, n} ], {n, 1, 50}] ]

a(n) = matdet(matrix(n-1, n-1, j, k, if (j==k, prime(j+1), 1)))/numerator(sum(k=1, n, 1/(prime(k)-1))); \\ Michel Marcus, Oct 02 2016

a(n) = A078456(n)/A120271(n).

A128646 a(n) = denominator(Sum_{k=1..n} 1/(prime(k)-1)).

Original entry on oeis.org

1, 2, 4, 12, 60, 10, 80, 720, 7920, 55440, 55440, 18480, 18480, 18480, 425040, 5525520, 160240080, 53413360, 160240080, 160240080, 480720240, 480720240, 19709529840, 19709529840, 39419059680, 197095298400, 3350620072800

Offset: 1

Alexander Adamchuk, Mar 18 2007

A120271(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)); A128648(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)); numbers m such that a(m) = A128648(m) are listed in A128649.

Eric Weisstein's World of Mathematics, Prime Sums

Cf. A120271 (numerator(Sum_{k=1..n} 1/(prime(k)-1))).
Cf. A128649, A128647, A128648 (denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1))).
Cf. A119686, A006093, A000040.

Table[Denominator[Sum[1/(Prime[k]-1),{k,1,n}]],{n,1,36}]

a(n) = denominator(Sum_{k=1..n} 1/(prime(k)-1)).

A078456 Number of numbers less than prime(1)...prime(n) having exactly one prime factor among (prime(1),...,prime(n)) where prime(n) is the n-th prime.

Original entry on oeis.org

1, 3, 14, 92, 968, 12096, 199296, 3679488, 82607616, 2349508608, 71507128320, 2604912721920, 105300128563200, 4466750187110400, 207324589680230400, 10866166392736972800, 634672612705724006400, 38337584554108256256000

Offset: 1

Benoit Cloitre, Dec 31 2002

nonn

For n>1 a(n) is the determinant of the (n-1) X (n-1) matrix with elements M[i,j] = Prime[i+1] if i=j and 1 otherwise. (See example lines.) - Alexander Adamchuk, Jun 02 2006

Second column of A096294. - Eric Desbiaux, Jun 20 2013

			a(2)=3 since 2*3=6 and 2,3,4 have 1 prime factor among (2,3)
3 1 1 1 1 ...
1 5 1 1 1 ...
1 1 7 1 1 ...
1 1 1 11 1 ...
1 1 1 1 13 ...
and so a(2) = 3, a(3) = 3*5 - 1*1 = 14, a(4) = 3*5*7 + 1*1*1 + 1*1*1 - 7*1*1 - 5*1*1 - 3*1*1 = 92, etc.

Charles R Greathouse IV, Table of n, a(n) for n = 1..350

Cf. A135212, A120271.

Cf. A117494, A002110.

Table[ Det[ DiagonalMatrix[ Table[ Prime[i+1]-1, {i, 1, n-1} ] ] + 1 ], {n, 1, 20} ] (* Alexander Adamchuk, Jun 02 2006 *)

a(n)=sum(k=1,prod(i=1,n, prime(i)),if(isprime(gcd(k,prod(i=1,n, prime(i)))),1,0))

a(n) = matdet(matrix(n-1, n-1, j, k, if (j==k, prime(j+1), 1))); \\ after Mathematica; Michel Marcus, Oct 02 2016

a(n) = (prime(n)-1)*a(n-1) + A005867(n). - Matthew Vandermast, Jun 06 2004
a(n) = A120071(n) * A135212(n). - Alexander Adamchuk, Nov 23 2007
a(n) = A117494(A002110(n)). - Ridouane Oudra, Sep 18 2022

a(7) from Ralf Stephan, Mar 25 2003
a(8)-a(12) from Matthew Vandermast, Jun 06 2004
More terms from Alexander Adamchuk, Jun 02 2006

A092063 Numbers k such that numerator of Sum_{i=1..k} 1/(prime(i)-1) is prime.

Original entry on oeis.org

2, 3, 4, 7, 8, 15, 19, 21, 22, 25, 26, 31, 34, 45, 46, 52, 65, 69, 79, 85, 89, 98, 102, 122, 137, 149, 181, 195, 210, 220, 316, 325, 340, 385, 436, 466, 497, 934, 972, 1180, 1211, 1212, 1639, 1807, 2075, 2104, 3100, 3258, 3563, 3688, 4528, 4760, 4934, 6151, 6185, 7579, 8625, 8694, 9205

Offset: 1

Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Feb 20 2004

Note that the definition here is subtly different from that of A092065.
Terms a(k) < 1000 correspond to primes. Beyond, numerators are probable primes. Note that A120271(3100) has 2187 digits. - M. F. Hasler, Feb 06 2008
Intersection of A000040 (the primes) and A120271 (numerators of partial sums of 1/(prime(i)-1)). - M. F. Hasler, Feb 06 2008
a(60) > 10000. - Jason Yuen, Aug 26 2024

			1/(2-1) + 1/(3-1) = 3/2 and 3 is prime so a(1)=2.

Cf. A092064, A120271.

Position[Accumulate[1/(Prime[Range[3100]]-1)],?(PrimeQ[ Numerator[ #]]&)]//Flatten (* _Harvey P. Dale, Oct 16 2016 *)

A120271(n) = numerator(sum(k=1, n, 1/(prime(k)-1)));
for (i=1,500,if(isprime(A120271(i)),print1(i,",")));

print_A092063( i=0 /* start testing at i+1 */)={local(s=sum(j=1,i,1/(prime(j)-1))); while(1, while(!ispseudoprime(numerator(s+=1/(prime(i++)-1))),); print1(i", "))} \\  M. F. Hasler, Feb 06 2008

More terms from M. F. Hasler, Feb 06 2008
Edited by T. D. Noe, Oct 30 2008
Corrected by Harvey P. Dale, Oct 16 2016
a(48)-a(59) from Jason Yuen, Aug 26 2024

A128647 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).

Original entry on oeis.org

1, 1, 3, 7, 41, 3, 53, 437, 5167, 34189, 36037, 3833, 3987, 11521, 274223, 3458639, 103063291, 100392623, 34273501, 33510453, 308270747, 302107667, 12626774467, 12402802537, 25216220279, 124110148411, 2142721739387, 111888942151111

Offset: 1

Alexander Adamchuk, Mar 18 2007

Numbers m such that A128648(m) = A128646(n) are listed in A128649.

Eric Weisstein's World of Mathematics, Prime sums

Cf. A128648 (denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1))).
Cf. A120271 (numerator(Sum_{k=1..n} 1/(prime(k)-1))).
Cf. A128646, A128649, A119686, A006093, A000040.

Table[Numerator[Sum[(-1)^(k+1)*1/(Prime[k]-1),{k,1,n}]],{n,1,36}]

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).

A128648 a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).

Original entry on oeis.org

1, 2, 4, 12, 60, 5, 80, 720, 7920, 55440, 55440, 6160, 6160, 18480, 425040, 5525520, 160240080, 160240080, 53413360, 53413360, 480720240, 480720240, 19709529840, 19709529840, 39419059680, 197095298400, 3350620072800, 177582863858400

Offset: 1

Alexander Adamchuk, Mar 18 2007

Numbers m such that a(m) equals A128646(m) are listed in A128649.

Eric Weisstein's World of Mathematics, Prime Sums

Cf. A128647 (numerator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1))).
Cf. A128646 (denominator(Sum_{k=1..n} 1/(prime(k)-1))).
Cf. A128649, A120271, A119686, A006093, A000040.

Table[Denominator[Sum[(-1)^(k+1)*1/(Prime[k]-1),{k,1,n}]],{n,1,36}]

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).

A128649 Numbers m such that A128646(m) = A128648(m).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 14, 15, 16, 17, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 65, 66, 71, 72, 73, 74, 75, 76, 77, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 539, 540, 541, 542, 543, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610

Offset: 1

Alexander Adamchuk, Mar 18 2007

nonn

Terms of this sequence are 1..5, 7..11, 14..17, 21..35, 65..66, 71..77, 81..93, 539..543, 600..639, 644..650, 707..818, 1152..1185, 4502..4577, 4601..4823, 4893..5003, 7483..7633, ...

Eric Weisstein's World of Mathematics, Prime Sums

Cf. A128648 (denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1))).
Cf. A128646 (denominator(Sum_{k=1..n} 1/(prime(k)-1))).
Cf. A128647, A120271, A119686, A006093, A000040.

f=0;g=0;Do[p=Prime[n];f=f+1/(p-1);g=g+(-1)^(n+1)*1/(p-1);kf=Denominator[f];kg=Denominator[g];If[Equal[kf,kg],Print[n]],{n,1,10000}]

A092064 Prime numbers in A092063.

Original entry on oeis.org

2, 3, 7, 19, 31, 79, 89, 137, 149, 181, 6151

Offset: 1

Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Feb 20 2004

6151 corresponds to a probable prime. - Jason Yuen, Aug 25 2024

Cf. A120271.

A120271(n) = numerator(sum(k=1, n, 1/(prime(k)-1)));
for (i=1,500,if(isprime(i) && isprime(A120271(i)),print1(i,",")));

a(11) from Jason Yuen, Aug 25 2024

cp's OEIS Frontend

A135212 a(n) = A078456(n)/A120271(n).

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A128646 a(n) = denominator(Sum_{k=1..n} 1/(prime(k)-1)).

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A078456 Number of numbers less than prime(1)*...*prime(n) having exactly one prime factor among (prime(1),...,prime(n)) where prime(n) is the n-th prime.

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A092063 Numbers k such that numerator of Sum_{i=1..k} 1/(prime(i)-1) is prime.

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A128647 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).

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A128648 a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).

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A128649 Numbers m such that A128646(m) = A128648(m).

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A092064 Prime numbers in A092063.

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A078456 Number of numbers less than prime(1)...prime(n) having exactly one prime factor among (prime(1),...,prime(n)) where prime(n) is the n-th prime.