A024528 -id:A024528 - cp's OEIS frontend

A125707 Numbers k such that A024528(k) is prime.

1, 2, 3, 4, 5, 7, 14, 15, 17, 23, 32, 36, 42, 309, 1384, 1588, 5631, 11226

Offset: 1

Alexander Adamchuk, Feb 01 2007

f=1;Do[p=Prime[n];f=f+1/p;g=Numerator[f];If[PrimeQ[g],Print[{n,p,g}]],{n,1,50}]

a(14)-a(16) from Ryan Propper, Feb 11 2008

a(17)-a(18) from Michael S. Branicky, Apr 27 2025

A368880 Primes in A024528.

Original entry on oeis.org

3, 11, 61, 457, 5237, 1226677, 34543329507310391, 1636619248175258407, 5186576044693944076609, 742779051038516950393163206833793, 1506853388294906471801157206440769816406928024502711, 651879075122842895567706351814676957742356330143458665568047

Offset: 1

Torlach Rush, Jan 08 2024

nonn

Primes which are the sum of the numerator and the denominator of partial sums of the reciprocals of primes.
Each term of this sequence can be expressed as the sum of an expression with exactly one odd term and n even terms, where the odd term is  A002110(n)/A002110(1), and n > 0 (see Alexander Adamchuk comment in A024528).
a(2) = 11 = 3 + 2 + 6 contains the only prime odd term 3.

			3 is a term because 1/2 = 1/2 and 1 + 2 = 3 which is prime.
11 is a term because 1/2 + 1/3 = 5/6 and 5 + 6 = 11 which is prime.
61 is a term because 5/6 + 1/5 = 31/30 and 31 + 30 = 61 which is prime.
457 is a term because 31/30 + 1/7 = 247/210 and 247 + 210 = 457 which is prime.

Intersection of A000040 and A024528.

Cf. A002110.

A075986 Numerator of 1+1/prime(1)^2+ ... + 1/prime(n)^2 where prime(k) is the k-th prime.

Original entry on oeis.org

1, 5, 49, 1261, 62689, 7629469, 1294716361, 375074829229, 135662633811769, 71859617272521901, 60483708554835755641, 58166700851687469003901, 79670437976161330893757369, 133981073592392620630139873389

Offset: 0

Zak Seidov, Sep 28 2002

The sum is similar to that in A061015 with an additional 1. The sum in the definition has limit about 1.45224742. The case of reciprocal cubes is in A075987.

For n>0 a(n) is the determinant of the n X n matrix with elements M[i,j] = 1+Prime[i]^2 if i=j and 1 otherwise. - Alexander Adamchuk, Jul 08 2006

			a(2) = 49 so a(3) = 49*p(3)^2 + (2*3)^2 = 49*25 + 36 = 1261.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 94-98.

Steven R. Finch, Meissel-Mertens Constants [Broken link]
Steven R. Finch, Meissel-Mertens Constants [From the Wayback machine]

Cf. A061015, A075987, A024528.

Table[Det[DiagonalMatrix[Table[Prime[i]^2,{i,1,n}]]+1],{n,1,15}] (* Alexander Adamchuk, Jul 08 2006 *)
Accumulate[Join[{1},1/Prime[Range[20]]^2]]//Numerator (* Harvey P. Dale, May 08 2023 *)

a(0)=1; a(n)=a(n-1)*prime(n)^2+(prime(1)*...*prime(n-1))^2.

Edited by Dean Hickerson, Sep 30 2002

A136365 Numbers k such that A075986(k) is prime.

Original entry on oeis.org

1, 171, 210, 550, 1445, 1809, 2176, 2719

Offset: 1

Alexander Adamchuk, Dec 27 2007

Cf. A075986 (numerator of 1 + 1/p(1)^2 + ... + 1/p(n)^2, where p(k) = prime(k)).
Cf. A024528, A125707.
Cf. A136366, A136367, A136368, A136369, A136370, A136371.

f=1; Do[ p=Prime[n]; f=f + 1/p^2; g=Numerator[ f ]; If[ PrimeQ[ g ], Print[ {n, g} ] ], {n, 1, 210} ]

a(4)-a(8) from Robert Price, Aug 26 2019

A120292 Absolute value of numerator of determinant of n X n matrix with elements M[i,j] = prime(i)/(1+prime(i)) if i=j and 1 otherwise.

Original entry on oeis.org

2, 1, 1, 5, 1, 23, 1, 1, 1, 23, 17, 13, 5, 1, 1, 1, 1, 37, 293, 47, 61, 29, 1, 29, 271, 593, 43, 233, 29, 811, 1, 941, 101, 1, 1, 1231, 131, 29, 1, 521, 1, 109, 1, 149, 509, 89, 59, 107, 617, 1, 1, 47, 173, 3067, 47, 1, 3463, 3599, 89, 431, 4021, 521, 2161, 2239, 103, 1, 1

Offset: 1

Alexander Adamchuk, Jul 08 2006, Jul 04 2008

Some a(n) are equal to 1 (n = 2, 3, 5, 7, 8, 9, 14, 15, 16, 17, 23, 31, 34, 35, 39, 41, 43, 50, 51, 56, ...).
a(58) = 3599 = 59*61 is not prime. - T. D. Noe, Nov 15 2006
Most terms are prime or 1.
Numbers n such that a(n)>1 and is not prime are listed in A141779(n) = {58, 282, 367, 743, 808, 1015, 1141, 1299, 1962, 2109, 2179, 2397, 2501, ...}.
Composite terms are listed in A141781 = {3599, 118477, 210589, 971573, 1164103, 1901959, 2446681, 3230069, ...}.
Note that all listed terms of A141781 are semiprime, for example: 3599 = 59*61, 118477 = 257*461, 210589 = 251*839, 971573 = 643*1511.
Conjecture: All composite terms are semiprime.

Alexander Adamchuk, Jul 04 2008, Table of n, a(n) for n = 1..282

Cf. A024528, A118679, A125716, A141780, A141779, A141781.

Abs[Numerator[Table[Det[DiagonalMatrix[Table[Prime[i]/(Prime[i]+1)-1,{i,1,n}]]+1],{n,1,60}]]]
Table[Numerator[Abs[(1 - Sum[Prime[k] + 1,{k, 1, n}])/Product[Prime[k] + 1, {k, 1, n}] ]],{n,1,282}]

a(n)=abs(numerator(matdet(matrix(n,n,i,j,if(i==j,prime(i)/(1+prime(i)),1))))) \\ Charles R Greathouse IV, Feb 07 2013

A120291 Numerator of determinant of n X n matrix with elements M[i,j] = (1+Prime[i])/Prime[i] if i=j and 1 otherwise.

Original entry on oeis.org

3, 1, 11, 3, 29, 1, 59, 1, 101, 1, 1, 3, 239, 47, 1, 191, 21, 251, 569, 64, 1, 12, 25, 482, 1061, 1, 1, 98, 1481, 797, 1721, 926, 3, 8, 3, 1214, 1, 458, 1, 1544, 99, 1724, 1213, 1916, 1, 2, 1, 3, 4889, 853, 5351, 1, 49, 3041, 2113, 3301, 6871, 3571, 2473, 10, 2661

Offset: 1

Alexander Adamchuk, Jul 08 2006, Aug 19 2006

Many a(n), such as 3,11,29,59,101,239,569,1061,1481,1721,4889.., are primes of form p(1)+...+p(k)+1 where p(i) =i-th prime A053845. It appeares that all primes of this form are presented in a(n) in their natural order.

Indices n such that a(n) = 1 are {2,6,8,10,11,15,21,26,27,37,39,45,47,52,75,84,87,88,91,94,...} = A121744[n] Numbers n such that (1 + Sum[Prime[k],{k,1,n}]) = (1 + A007504[n]) divides primorial number p(n)# = Product[Prime[k],{k,1,n}] = A002110[n].

Cf. A024528, A053845.

Cf. A121744, A007504, A002110.

Numerator[Table[Det[DiagonalMatrix[Table[1/Prime[i],{i,1,n}]]+1],{n,1,70}]]
Table[Numerator[(1+Sum[Prime[k],{k,1,n}])/Product[Prime[k],{k,1,n}]],{n,1,100}]

a(n) = numerator[Det[DiagonalMatrix[Table[1/Prime[i],{i,1,n}]]+1]].

a(n) = Numerator[ (1 + Sum[ Prime[k], {k,1,n} ]) / Product[ Prime[k], {k,1,n} ] ]. a(n) = Numerator[ (1 + A007504[n]) / A002110[n] ].

A121281 Triangle T(n,k) read by rows: T(n,0) = A002110(n) and T(n,k) = A002110(n)/prime(k) for 1<=k<=n.

Original entry on oeis.org

1, 2, 1, 6, 3, 2, 30, 15, 10, 6, 210, 105, 70, 42, 30, 2310, 1155, 770, 462, 330, 210, 30030, 15015, 10010, 6006, 4290, 2730, 2310, 510510, 255255, 170170, 102102, 72930, 46410, 39270, 30030, 9699690, 4849845, 3233230, 1939938, 1385670, 881790, 746130, 570570, 510510

Offset: 0

Philippe Deléham, Aug 24 2006

			Triangle begins
1;
2, 1;
6, 3, 2;
30, 15, 10, 6;
210, 105, 70, 42, 30;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Cf. A000040, A002110, A024528, A070826.

a121281 n k = a121281_tabl !! n !! k
a121281_row n = a121281_tabl !! n
a121281_tabl = [1] : f [1] a000040_list where
   f xs@(x:_) (p:ps) = ys : f ys ps where ys = (map (* p) xs) ++ [x]
-- Reinhard Zumkeller, Nov 20 2015

tabl(nn) = {for (n=0, nn, pr = prod(i=1, n, prime(i)); for (k=0, n, if (k==0, v = pr, v = pr/prime(k)); print1(v, ", ");); print(););} \\ Michel Marcus, Apr 06 2015

Sum_{0<=k<=n} T(n,k) = A024528(n).

T(n+1,k) = prime(n+1) * T(n,k) for k=0..n and T(n+1,n+1) = T(n,0). - Reinhard Zumkeller, Nov 20 2015

Corrected and extended by Michel Marcus, Apr 06 2015

A191645 Numerators of the n-th partial "harmonic" sum of 1 + inverse semiprimes.

Original entry on oeis.org

5, 17, 55, 293, 2141, 445, 457, 5153, 131597, 1745411, 1772711, 30586537, 31024117, 597115873, 604577173, 14050770329, 99311504603, 100230122303, 101081931443, 101903852543

Offset: 1

Jonathan Vos Post, Jun 09 2011

Denominators appear to be the same as A140123. The fractions begin: 5/4, 17/12, 55/36, 293/180, 2141/1260, 445/252, 457/252, 5153/2772, 131597/69300, ...

This is the semiprime analog of A024528.

			a(1) =  5 because 1 + 1/4 = 5/4.
a(2) = 17 because 1 + 1/4 + 1/6 = 17/12.
a(3) = 55 because 1 + 1/4 + 1/6 + 1/9 = 55/36.

Cf. A001358, A112141, A001008, A140123, A024528.

A191645 := proc(n) 1+add(1/A001358(i),i=1..n) ; numer(%) ; end proc:
seq(A191645(n),n=1..20) ; # R. J. Mathar, Jun 16 2011

With[{sp=Join[{1},Select[Range[100],PrimeOmega[#]==2&]]},Rest[ Numerator[ Accumulate[1/sp]]]] (* Harvey P. Dale, May 01 2015 *)

s=1; for(k=1,99, bigomega(k)==2 & print1(numerator(s+=1/k)", "))  \\ M. F. Hasler, Jun 17 2011

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A125707 Numbers k such that A024528(k) is prime.

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A368880 Primes in A024528.

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A075986 Numerator of 1+1/prime(1)^2+ ... + 1/prime(n)^2 where prime(k) is the k-th prime.

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A136365 Numbers k such that A075986(k) is prime.

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A120292 Absolute value of numerator of determinant of n X n matrix with elements M[i,j] = prime(i)/(1+prime(i)) if i=j and 1 otherwise.

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A120291 Numerator of determinant of n X n matrix with elements M[i,j] = (1+Prime[i])/Prime[i] if i=j and 1 otherwise.

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A121281 Triangle T(n,k) read by rows: T(n,0) = A002110(n) and T(n,k) = A002110(n)/prime(k) for 1<=k<=n.

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A191645 Numerators of the n-th partial "harmonic" sum of 1 + inverse semiprimes.

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