A006094 -id:A006094 - cp's OEIS frontend

A209329 Decimal expansion of the sum over the inverse products of adjacent odd primes.

1, 3, 4, 4, 2, 6, 5, 0, 9, 6, 9, 1, 7, 3, 3, 2, 2, 8

Offset: 0

R. J. Mathar, Jan 19 2013

Contains the contribution from twin primes (A209328) plus other contributions from cousin primes (A143206) not already part of twin primes, sexy primes (A210477) not already accounted for, etc.

Summing up to (and including) 12-digit primes yields 0.134426509691698261. - Hans Havermann, Mar 17 2013

			0.134426509... = 1/(3*5) + 1/(5*7) + 1/(7*11) + 1/(11*13)+ ... = Sum_{n>=2} 1/A006094(n).

R. J. Mathar, Estimate of the Sum over Inverse Products of Adjacent Prime Pairs

Cf. A210473 (includes 1/(2*3)). Cf. also A085548.

{default(realprecision,19);s=0;q=1/3;forprime(p=1/q+1,10^9,s+=q*q=1./p);s} /* M. F. Hasler, Jan 22 2013 */

sum_{3 < p < 10^4} 1/(prevprime(p)*p) = 0.134416688[9]...
sum_{3 < p < 10^5} 1/(prevprime(p)*p) = 0.134425707...
sum_{3 < p < 10^6} 1/(prevprime(p)*p) = 0.1344264419...
sum_{3 < p < 10^7} 1/(prevprime(p)*p) = 0.13442650383...
sum_{3 < p < 10^8} 1/(prevprime(p)*p) = 0.13442650917[5]...
sum_{3 < p < 10^9} 1/(prevprime(p)*p) = 0.13442650964545...
Extrapolation of this data (using Aitken's method) indeed suggests a value of 0.134426509692, rounded to the last decimal place. Extrapolation of the ratios of the first differences (9.02e-6, 7.35e-7, 6.19e-8, 5.34e-9, 4.699e-10) yields subsequent terms (4.26e-11, 4.0e-12). - M. F. Hasler, Jan 22 2013

More terms from R. J. Mathar, Feb 08 2013

A250477 Number of times prime(n) (the n-th prime) occurs as the least prime factor among numbers 1 .. (prime(n)^2 * prime(n+1)): a(n) = A078898(A251720(n)).

Original entry on oeis.org

6, 8, 12, 21, 33, 45, 63, 80, 116, 148, 182, 232, 265, 296, 356, 433, 490, 548, 625, 674, 740, 829, 919, 1055, 1187, 1252, 1313, 1376, 1446, 1657, 1897, 2029, 2134, 2301, 2484, 2605, 2785, 2946, 3110, 3301, 3439, 3654, 3869, 3978, 4086, 4349, 4811, 5147, 5273, 5395, 5604, 5787, 6049, 6403, 6684, 6954, 7153

Offset: 1

Antti Karttunen, Dec 14 2014

nonn

a(n) = Position of 6 on row n of array A249821. This is always larger than A250474(n), the position of 4 on row n, as 4 is guaranteed to be the first composite term on each row of A249821.
From Antti Karttunen, Mar 29 2015: (Start)
a(n) = 1 + number of positive integers <= (prime(n)*prime(n+1)) whose smallest prime factor is at least prime(n).
That a(n) >  A250474(n) can also be seen by realizing that prime(n) must occur at least as many times as the smallest prime factor for the numbers in range 1 .. (prime(n)^2 * prime(n+1)) than for numbers in (smaller) range 1 .. (prime(n)^3), and also by realizing that a(n) cannot be equal to A250474(n) because each row of A249822 is a permutation of natural numbers.
Or more simply, by considering the comment given in A256447 which follows from the new interpretation given above.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..564
Antti Karttunen, Ratio a(n)/A250474(n+1), plotted up to n=65 with OEIS Plot2-utility

Column 6 of A249822. Cf. also A250474 (column 4), A250478 (column 8).
First differences: A256446. Cf. also A256447, A256448.
Cf. A002110, A006094, A020639, A078898, A249821, A251720, A281890.

f[n_] := Count[Range[Prime[n]^2*Prime[n + 1]], x_ /; Min[First /@ FactorInteger[x]] == Prime@ n]; Array[f, 20] (* Michael De Vlieger, Mar 30 2015 *)

allocatemem(234567890);
A002110(n) = prod(i=1, n, prime(i));
A250477(n) = { my(m); m = (prime(n) * prime(n+1)); sumdiv(A002110(n-1), d, (moebius(d)*(m\d))); };
for(n=1, 23, print1(A250477(n),", "));
\\ A more practical program:

allocatemem(234567890);
vecsize = (2^24)-4;
v020639 = vector(vecsize);
v020639[1] = 1; for(n=2,vecsize, v020639[n] = vecmin(factor(n)[, 1]));
A020639(n) = v020639[n];
A250477(n) = { my(p=prime(n),q=prime(n+1),u=p*q,k=1,s=1); while(k <= u, if(A020639(k) >= p, s++); k++); s; };
for(n=1, 564, write("b250477.txt", n, " ", A250477(n)));
\\ Antti Karttunen, Mar 29 2015

a(n) = A078898(A251720(n)).
a(1) = 1, a(n) = Sum_{d | A002110(n-1)} moebius(d) * floor(A006094(n) / d). [Follows when A251720, (p_n)^2 * p_{n+1} is substituted to the similar formula given for A078898. Here p_n is the n-th prime (A000040(n)), A006094(n) gives the product p_n * p{n+1} and A002110(n) gives the product of the first n primes. Because the latter is always squarefree, one could use here also Liouville's lambda (A008836) instead of Moebius mu (A008683)].
a(n) = A250474(n) + A256447(n).

A023515 a(n) = prime(n)*prime(n-1) - 1.

Original entry on oeis.org

1, 5, 14, 34, 76, 142, 220, 322, 436, 666, 898, 1146, 1516, 1762, 2020, 2490, 3126, 3598, 4086, 4756, 5182, 5766, 6556, 7386, 8632, 9796, 10402, 11020, 11662, 12316, 14350, 16636, 17946, 19042, 20710, 22498, 23706, 25590, 27220, 28890

Offset: 1

Clark Kimberling

nonn

a(1) = 1 assumes the not generally accepted convention prime(0) = 1. - Klaus Brockhaus, Dec 23 2010

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Cf. A120875 (a subsequence).

[ NthPrime(n-1)*NthPrime(n)-1: n in [1..50] ]; // Vincenzo Librandi, Dec 23 2010; simplified by Klaus Brockhaus, Dec 23 2010

1,seq(ithprime(n)*ithprime(n-1)-1,n=2..40); # Muniru A Asiru, Apr 27 2019

Prepend[Table[Prime@ n Prime[n - 1] - 1, {n, 2, 12}], 1] (* Michael De Vlieger, Nov 10 2015 *)
Join[{1},Times@@#-1&/@Partition[Prime[Range[40]],2,1]] (* Harvey P. Dale, Jul 06 2024 *)

a(n) = if(n==1, 1, prime(n)*prime(n-1)-1) \\ Altug Alkan, Nov 10 2015

From Jason Kimberley, Oct 23 2015: (Start)
a(n) = A006094(n-1) - 1 = A000040(n-1)*A000040(n)-1, for n>1.
a(n) = 2*A102770(n-2), for n>2.
(End)

A127351 Prime numbers n such that A127350(k) = 2*n for some k.

Original entry on oeis.org

2003, 7883, 31151, 35363, 394739, 434939, 541007, 564983, 837929, 865979, 2453999, 2680493, 3479303, 3536219, 4145717, 4367267, 4706311, 5414159, 6541103, 6856019, 8804231, 9109223, 10227323, 10296059, 10701683, 10795507

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

Primes of the form (Sum_{i=k..k+3}Sum_{j=i+1..k+4}prime(i)*prime(j))/2.

Primes of the form a/2 where a is the coefficient of x^3 of the polynomial Prod_{j=0,4}(x-prime(k+j)) for some k.

Cf. A127350, A127345, A127346, A127347, A127348, A127349, A070934, A006094.

a = {}; Do[If[PrimeQ[(Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x] Prime[x + 3] + Prime[x] Prime[x + 4] + Prime[x + 1] Prime[x + 2] + Prime[x + 1] Prime[x + 3] + Prime[x + 1] Prime[x + 4] + Prime[x + 2] Prime[x + 3] + Prime[x + 2] Prime[x + 4] + Prime[x + 3] Prime[x + 4])/2], AppendTo[a, (Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x] Prime[x + 3] + Prime[x] Prime[x + 4] + Prime[x + 1] Prime[x + 2] + Prime[x + 1] Prime[x + 3] + Prime[x + 1] Prime[x + 4] + Prime[x + 2] Prime[x + 3] + Prime[x + 2] Prime[x + 4] + Prime[x + 3] Prime[x + 4])/2]], {x, 1, 1000}]; a

{m=235;k=4;for(n=1,m,a=sum(i=n,n+k-1,sum(j=i+1,n+k,prime(i)*prime(j)));if(isprime(b=a/2),print1(b,",")))} \\ Klaus Brockhaus, Jan 21 2007

{m=235;k=4;for(n=1,m,a=polcoeff(prod(j=0,k,(x-prime(n+j))),3);if(isprime(b=a/2),print1(b,",")))} \\ Klaus Brockhaus, Jan 21 2007

Edited by Klaus Brockhaus, Jan 21 2007

A205146 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j

Original entry on oeis.org

2, 3, 2, 3, 3, 4, 4, 5, 2, 3, 5, 5, 6, 4, 7, 5, 7, 5, 8, 3, 4, 5, 9, 6, 12, 6, 5, 7, 3, 7, 4, 5, 5, 7, 15, 5, 12, 8, 6, 8, 7, 4, 6, 7, 7, 9, 10, 6, 8, 12, 7, 10, 16, 5, 16, 13, 8, 10, 9, 7, 16, 4, 10, 5, 14, 5, 8, 10, 20, 16, 4, 6, 18, 12, 14, 13, 7, 6, 9, 11

Offset: 1

Clark Kimberling, Jan 25 2012

nonn

See A204892 for a discussion and guide to related sequences.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A006094, A204892, A205143.

s[n_] := s[n] = Prime[n] Prime[n + 1]; z1 = 400; z2 = 60;
Table[s[n], {n, 1, 30}]           (* A006094 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]           (* A205144 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
Table[d[n], {n, 1, z2}]           (* A205145 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}]           (* A205146 *)
Table[j[n], {n, 1, z2}]           (* A205147 *)
Table[s[k[n]], {n, 1, z2}]        (* A205148 *)
Table[s[j[n]], {n, 1, z2}]        (* A205149 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]        (* A205150 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}]    (* A205151 *)

s(m) = prime(m)*prime(m+1);
isok(k, n) = my(sk=s(k)); for (j=1, k-1, if (!Mod(sk-s(j), n), return (k)));
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Jul 23 2021

More terms from Michel Marcus, Jul 23 2021

A251720 a(n) = (p_n)^2 * p_{n+1}, where p_n is the n-th prime, A000040(n).

Original entry on oeis.org

12, 45, 175, 539, 1573, 2873, 5491, 8303, 15341, 26071, 35557, 56129, 72283, 86903, 117077, 165731, 212341, 249307, 318719, 367993, 420991, 518003, 613121, 768337, 950309, 1050703, 1135163, 1247941, 1342553, 1621663, 2112899, 2351057, 2608891, 2878829, 3352351

Offset: 1

Antti Karttunen, Dec 14 2014

nonn

Subsequence of A014612: a(1)=12=A014612(2), a(2)=45=A014612(10) - Zak Seidov, Apr 26 2016

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A001248, A006094, A250477, A014612, A268733.

a251720[n_Integer] := Prime[#]^2*Prime[# + 1] & /@ Range[n]; a251720[35] (* Michael De Vlieger, Dec 14 2014 *)
#[[1]]^2 #[[2]]&/@Partition[Prime[Range[40]],2,1] (* Harvey P. Dale, Mar 12 2015 *)

a(n) = A000040(n) * A000040(n) * A000040(n+1).
a(n) = A000040(n) * A006094(n).
a(n) = A001248(n) * A000040(n+1).

A286623 Square array A(n,k) = A276943(n,k)/A002110(n-1), read by descending antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

1, 3, 1, 4, 4, 1, 5, 6, 6, 1, 7, 7, 10, 8, 1, 9, 16, 11, 14, 12, 1, 10, 19, 36, 15, 22, 14, 1, 11, 21, 41, 78, 23, 26, 18, 1, 13, 22, 45, 85, 144, 27, 34, 20, 1, 15, 31, 46, 91, 155, 222, 35, 38, 24, 1, 16, 34, 71, 92, 165, 235, 324, 39, 46, 30, 1, 17, 36, 76, 155, 166, 247, 341, 438, 47, 58, 32, 1, 18, 37, 80, 162, 287, 248, 357, 457, 668, 59, 62, 38, 1

Offset: 1

Antti Karttunen, Jun 28 2017

			The top left 12 X 12 corner of the array:
  1,  3,  4,  5,    7,    9,   10,   11,   13,   15,   16,   17
  1,  4,  6,  7,   16,   19,   21,   22,   31,   34,   36,   37
  1,  6, 10, 11,   36,   41,   45,   46,   71,   76,   80,   81
  1,  8, 14, 15,   78,   85,   91,   92,  155,  162,  168,  169
  1, 12, 22, 23,  144,  155,  165,  166,  287,  298,  308,  309
  1, 14, 26, 27,  222,  235,  247,  248,  443,  456,  468,  469
  1, 18, 34, 35,  324,  341,  357,  358,  647,  664,  680,  681
  1, 20, 38, 39,  438,  457,  475,  476,  875,  894,  912,  913
  1, 24, 46, 47,  668,  691,  713,  714, 1335, 1358, 1380, 1381
  1, 30, 58, 59,  900,  929,  957,  958, 1799, 1828, 1856, 1857
  1, 32, 62, 63, 1148, 1179, 1209, 1210, 2295, 2326, 2356, 2357
  1, 38, 74, 75, 1518, 1555, 1591, 1592, 3035, 3072, 3108, 3109

Transpose: A286625.
Cf. A002110, A276943.
Row 1: A276155.
Column 1: A000012, Column 2: A008864, Column 3: A100484, Column 4: A072055, Column 5: A023523 (from its second term onward), Column 6: A286624 (= 1 + A123134), Column 11: 2*A123134, Column 13: 3*A006094.
Cf. A276616 (analogous array).

(define (A286623 n) (A286623bi (A002260 n) (A004736 n)))
(define (A286623bi row col) (/ (A276943bi row col) (A002110 (- row 1))))

A(n,k) = A276943(n, k) / A002110(n-1).

A073484 Number of gaps in factors of the n-th squarefree number.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 2, 0

Offset: 1

Reinhard Zumkeller, Aug 03 2002

nonn

			The 69th squarefree number is 110=2*5*11, therefore a(69)=2, as there are two gaps: between 2 and 5 and between 5 and 11.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A073483.

gaps[n_] := SequenceCount[FactorInteger[n][[;; , 1]], {p1_, p2_} /; p2 != NextPrime[p1], Overlaps -> True]; gaps /@ Select[Range[200], SquareFreeQ] (* Amiram Eldar, Apr 10 2021 *)

a(A000040(n))=0; a(A006094(n))=0; a(A002110(n))=0; a(A073485(n))=0.
a(A073486(n))>0; a(A073487(n))=1; a(A073488(n))=2; a(A073489(n))=3.
a(n)=0 iff A073483(n)=1.

A127349 a(n) = Sum_{i=n..n+1} Sum_{j=i+1..n+2} Sum_{k=j+1..n+3} prime(i)prime(j)prime(k).

Original entry on oeis.org

247, 886, 2556, 6288, 12900, 22392, 40808, 63978, 105000, 161142, 216232, 294168, 385544, 507782, 658820, 858000, 1067502, 1251952, 1518910, 1783854, 2114748, 2618148, 3147710, 3696090, 4239528, 4626300, 5033232, 5898936, 6871200

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) = absolute value of the coefficient of x^1 of the polynomial Product_{j=0..3} (x - prime(n+j)) of degree 4; the roots of this polynomial are prime(n), ..., prime(n+3); cf. Vieta's formulas.
All terms with exception of the first one are even.
Arithmetic derivative (see A003415) of prime(n)*prime(n+1)*prime(n+2)*prime(n+3). - Giorgio Balzarotti, May 26 2011

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Vieta's Formulas

Cf. A046302, A127345, A127346, A127347, A127348, A127350, A127351, A070934, A006094.

[NthPrime(n)*NthPrime(n+1)*NthPrime(n+2) + NthPrime(n)*NthPrime(n+2)*NthPrime(n+3) + NthPrime(n)*NthPrime(n+1)* NthPrime(n+3) + NthPrime(n+1)*NthPrime(n+2)*NthPrime(n+3): n in [1..30]]; // Vincenzo Librandi, Feb 12 2018

P := select(isprime, [2, seq(i, i = 1 .. 1000, 2)]):
f := L) -> convert(L, `*`)*add(1/t, t = L):
seq(f(P[i..i+3]),i=1..nops(P)-3); # Robert Israel, Feb 11 2018

Table[Prime[n] Prime[n+1] Prime[n+2] + Prime[n] Prime[n+2] Prime[n+3] + Prime[n] Prime[n+1] Prime[n+3] + Prime[n+1] Prime[n+2] Prime[n+3], {n, 100}]

{m=29;h=3;for(n=1,m,print1(sum(i=n,n+h-2,sum(j=i+1,n+h-1,sum(k=j+1,n+h,prime(i)*prime(j)*prime(k)))),","))} \\ Klaus Brockhaus, Jan 21 2007

{m=29;k=3;for(n=1,m,print1(abs(polcoeff(prod(j=0,k,(x-prime(n+j))),1)),","))} \\ Klaus Brockhaus, Jan 21 2007

a(n) = A046302(n)*Sum_{i=n..n+3} 1/prime(i). - Robert Israel, Feb 11 2018

Edited by Klaus Brockhaus, Jan 21 2007

A138109 Positive integers k whose smallest prime factor is greater than the cube root of k and strictly less than the square root of k.

Original entry on oeis.org

6, 15, 21, 35, 55, 65, 77, 85, 91, 95, 115, 119, 133, 143, 161, 187, 203, 209, 217, 221, 247, 253, 259, 287, 299, 301, 319, 323, 329, 341, 377, 391, 403, 407, 437, 451, 473, 481, 493, 517, 527, 533, 551, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 703

Offset: 1

David S. Newman, May 04 2008

nonn

This sequence was suggested by Moshe Shmuel Newman.
A020639(n)^2 < a(n) < A020639(n)^3. - Reinhard Zumkeller, Dec 17 2014
In other words, k = p*q with primes p, q satisfying p < q < p^2. - Charles R Greathouse IV, Apr 03 2017
If "strictly less than" in the definition were changed to "less than or equal to" then this sequence would also include the squares of primes (A001248), resulting in A251728. - Jon E. Schoenfield, Dec 27 2022

			6 is a term because the smallest prime factor of 6 is 2 and 6^(1/3) = 1.817... < 2 < 2.449... = sqrt(6).
From _Michael De Vlieger_, Apr 27 2024: (Start):
Table of p*q where p = prime(n) and q = prime(n+k):
n\k   1     2     3     4     5     6     7     8     9    10    11
-------------------------------------------------------------------
1:    6;
2:   15,   21;
3:   35,   55,   65,   85,   95,  115;
4:   77,   91,  119,  133,  161,  203,  217,  259,  287,  301,  329;
     ... (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequence of A251728 and of A006881. A006094 is a proper subset.

Cf. A020639, A010846, A079047, A162306.

a138109 n = a138109_list !! (n-1)
a138109_list = filter f [1..] where
   f x = p ^ 2 < x && x < p ^ 3 where p = a020639 x
-- Reinhard Zumkeller, Dec 17 2014

s = {}; Do[f = FactorInteger[i]; test = f[[1]][[1]]; If [test < N[i^(1/2)] && test > N[i^(1/3)], s = Union[s, {i}]], {i, 2, 2000}]; Print[s]
Select[Range[1000],Surd[#,3]Harvey P. Dale, May 10 2015 *)

is(n)=my(f=factor(n)); f[,2]==[1,1]~ && f[1,1]^3 > n \\ Charles R Greathouse IV, Mar 28 2017

list(lim)=if(lim<6, return([])); my(v=List([6])); forprime(p=3,sqrtint(1+lim\=1)-1, forprime(q=p+2, min(p^2-2,lim\p), listput(v,p*q))); Set(v) \\ Charles R Greathouse IV, Mar 28 2017

from math import isqrt
from sympy import primepi, primerange
def A138109(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(min(x//p,p**2)) for p in primerange(s+1)))
    return bisection(f,n,n) # Chai Wah Wu, Mar 05 2025

From Michael De Vlieger, Apr 27 2024: (Start)
Let k = a(n); row k of A162306 = {1, p, q, p^2, p*q}, therefore A010846(k) = 5.
A079047(n) = card({ q : p < q < p^2 }), p and q primes. (End)

cp's OEIS Frontend

A209329 Decimal expansion of the sum over the inverse products of adjacent odd primes.

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A250477 Number of times prime(n) (the n-th prime) occurs as the least prime factor among numbers 1 .. (prime(n)^2 * prime(n+1)): a(n) = A078898(A251720(n)).

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A023515 a(n) = prime(n)*prime(n-1) - 1.

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A127351 Prime numbers n such that A127350(k) = 2*n for some k.

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A205146 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j

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A251720 a(n) = (p_n)^2 * p_{n+1}, where p_n is the n-th prime, A000040(n).

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A286623 Square array A(n,k) = A276943(n,k)/A002110(n-1), read by descending antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A127349 a(n) = Sum_{i=n..n+1} Sum_{j=i+1..n+2} Sum_{k=j+1..n+3} prime(i)prime(j)prime(k).