A036689 -id:A036689 - cp's OEIS frontend

A139116 a(n) = p*(p-1)/2, where p is A000043(n).

1, 3, 10, 21, 78, 136, 171, 465, 1830, 3916, 5671, 8001, 135460, 183921, 817281, 2425503, 2600340, 5172936, 9041878, 9779253, 46933516, 49406770, 62860078, 198732016, 235455850, 269317236, 989969256, 3718884403, 6105401253, 8718403176, 23347552095, 286402257541

Offset: 1

Omar E. Pol, May 10 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Harvey P. Dale)

Cf. A000043, A000217, A036689, A139306, A139307, A139116, A139223, A139224, A139225, A139226.

(#(#-1))/2&/@MersennePrimeExponent[Range[47]] (* Harvey P. Dale, Aug 13 2021 *)

a(n) = A000043(n)*(A000043(n)-1)/2.

a(24)-a(32) from Harvey P. Dale, Aug 13 2021

A117495 Product of a prime number p and the number of primes smaller than p.

Original entry on oeis.org

0, 3, 10, 21, 44, 65, 102, 133, 184, 261, 310, 407, 492, 559, 658, 795, 944, 1037, 1206, 1349, 1460, 1659, 1826, 2047, 2328, 2525, 2678, 2889, 3052, 3277, 3810, 4061, 4384, 4587, 5066, 5285, 5652, 6031, 6346, 6747, 7160, 7421, 8022, 8299, 8668, 8955, 9706

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 25 2006

			a(9) = 184 because (1) the 9th prime number is 23, (2) there are 8 primes smaller than 23 and (3) 23*8 = 184.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000720, A033286, A036689.

with(numtheory):a:=n->sum(ithprime(n), j=2..n):seq(a(n), n=1..47); # Zerinvary Lajos, Aug 24 2008

Table[(n - 1)Prime[n], {n, 60}] (* Zak Seidov, Aug 15 2010 *)

a(n)=prime(n)*(n-1) \\ Charles R Greathouse IV, Sep 15 2014

a(n) = (n-1)*prime(n). - Zak Seidov, Aug 15 2010

a(31) corrected by Jens Kruse Andersen, Sep 15 2014

A127922 1/24 of product of three numbers: n-th prime, previous and following number.

Original entry on oeis.org

1, 5, 14, 55, 91, 204, 285, 506, 1015, 1240, 2109, 2870, 3311, 4324, 6201, 8555, 9455, 12529, 14910, 16206, 20540, 23821, 29370, 38024, 42925, 45526, 51039, 53955, 60116, 85344, 93665, 107134, 111895, 137825, 143450, 161239, 180441, 194054

Offset: 2

Artur Jasinski, Feb 06 2007

nonn

The product of (n-1), n, and (n+1) = n^3 - n. - Harvey P. Dale, Jan 17 2011
For n > 2, a(n) = A001318(n-2) * A007310(n-1), if A007310(n-1) is prime. Also a(n) is a subsequence of A000330. - Richard R. Forberg, Dec 25 2013
If p is an odd prime it can always be the side length of a leg of a primitive Pythagorean triangle. However it constrains the other leg to have a side length of (p^2-1)/2 and the hypotenuse to have a side length of (p^2+1)/2. The resulting triangle has an area equal to (p-1)*p*(p+1)/4. a(n) is 1/6 the area of such triangles. - Frank M Jackson, Dec 06 2017

G. C. Greubel, Table of n, a(n) for n = 2..1000
César Aguilera, Two Prime Number Objects and The Velucchi Numbers, hal-02909691 [math.NT], 2020.

Cf. A000040, A000330, A011842.

Cf. A036689, A034953, A127917, A127918, A127919, A127920, A127921, A011842.

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/24, {n, 1, 100}] (#^3-#)/ 24&/@ Prime[Range[2,40]] (* Harvey P. Dale, Jan 17 2011 *)
((#-1)#(#+1))/24&/@Prime[Range[2,40]] (* Harvey P. Dale, Jan 20 2023 *)

for(n=2,25, print1((prime(n)+1)*prime(n)*(prime(n)-1)/24, ", ")) \\ G. C. Greubel, Jun 19 2017

a(n) = A011842(A000040(n) + 1) = A000330((A000040(n) - 1) / 2).

A139115 a(n) = p*(p - 1), where p is A000043(n).

Original entry on oeis.org

2, 6, 20, 42, 156, 272, 342, 930, 3660, 7832, 11342, 16002, 270920, 367842, 1634562, 4851006, 5200680, 10345872, 18083756, 19558506, 93867032, 98813540, 125720156, 397464032, 470911700, 538634472, 1979938512, 7437768806, 12210802506

Offset: 1

Omar E. Pol, May 10 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Harvey P. Dale)

Cf. A000043, A036689, A139306, A139307, A139116, A139223, A139224, A139225, A139226.

#(#-1)&/@MersennePrimeExponent[Range[30]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 15 2020 *)

a(n) = A000043(n)*(A000043(n) - 1).

More terms from Vincenzo Librandi, May 11 2010

A139223 M*(M-1), where M is Mersenne prime A000668(n).

Original entry on oeis.org

6, 42, 930, 16002, 67084290, 17179475970, 274876334082, 4611686011984936962, 5316911983139663484697699213480296450, 383123885216472214589586754930667236976614368197214210

Offset: 1

Omar E. Pol, May 10 2008

nonn

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000668, A036689, A139115, A139116, A139224, A139225, A139226, A139256.

a(n) = A000668(n)*(A000668(n)-1).

More terms from R. J. Mathar, Jun 24 2009

A139225 M(M-1)/3, where M is Mersenne prime A000668(n).

Original entry on oeis.org

2, 14, 310, 5334, 22361430, 5726491990, 91625444694, 1537228670661645654, 1772303994379887828232566404493432150, 127707961738824071529862251643555745658871456065738070, 8776024305713098891493168973639040433964428736682367693182293334, 9649340769776349618630915417390658987602357538676244438223111363610210030934

Offset: 1

Omar E. Pol, May 10 2008

nonn

Terms from a(13) on have 314 or more digits and are not listed for that reason. - R. J. Mathar, May 11 2008

Cf. A000668, A036689, A139115, A139116, A139223, A139224, A139226.

a(n)=A000668(n)*(A000668(n)-1)/3.

More terms from R. J. Mathar, May 11 2008

A139226 M(M-1)/6, where M is Mersenne prime A000668(n).

Original entry on oeis.org

1, 7, 155, 2667, 11180715, 2863245995, 45812722347, 768614335330822827, 886151997189943914116283202246716075, 63853980869412035764931125821777872829435728032869035, 4388012152856549445746584486819520216982214368341183846591146667, 4824670384888174809315457708695329493801178769338122219111555681805105015467

Offset: 1

Omar E. Pol, May 10 2008

nonn

Perfect number A000396(n) minus Mersenne prime A000668(n), divided by 3.

Terms from a(13) on have 313 or more digits and are not listed for that reason. - R. J. Mathar, May 11 2008

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000217, A000396, A000668, A036689, A139096, A139115, A139116, A139223, A139224, A139225, A139256, A139306.

a(n) = A000668(n)*(A000668(n)-1)/6 = A139223(n)/6 = A139224(n)/3.

a(n) = (A000396(n)-A000668(n))/3.

More terms from R. J. Mathar, May 11 2008

A257252 Transpose of square array A257251.

Original entry on oeis.org

2, 6, 2, 20, 6, 2, 42, 10, 6, 2, 110, 28, 20, 6, 2, 156, 22, 14, 10, 6, 2, 272, 52, 44, 28, 20, 6, 2, 342, 34, 26, 22, 14, 10, 6, 2, 506, 76, 68, 52, 44, 28, 20, 6, 2, 812, 138, 114, 102, 78, 66, 42, 10, 6, 2, 930, 58, 46, 38, 34, 26, 22, 14, 20, 6, 2, 1332, 186, 174, 138, 114, 102, 78, 66, 42, 10, 6, 2

Offset: 1

Antti Karttunen, Apr 29 2015

See A257251.

			The top left corner of the array:
  2, 6, 20, 42, 110, 156, 272, 342, 506, 812, 930, 1332
  2, 6, 10, 28,  22,  52,  34,  76, 138,  58, 186,  148
  2, 6, 20, 14,  44,  26,  68, 114,  46, 174, 124,   74
  2, 6, 10, 28,  22,  52, 102,  38, 138, 116,  62,  148
  2, 6, 20, 14,  44,  78,  34, 114,  92,  58, 124,  222
  2, 6, 10, 28,  66,  26, 102,  76,  46, 116, 186,  222
  2, 6, 20, 42,  22,  78,  68,  38,  92, 174, 186,   74
  2, 6, 10, 14,  66,  52,  34,  76, 138, 174,  62,  222
  2, 6, 20, 42,  44,  26,  68, 114, 138,  58, 186,  148
  2, 6, 10, 28,  22,  52, 102, 114,  46, 174, 124,   74
  2, 6, 20, 14,  44,  78, 102,  38, 138, 116,  62,  222
  2, 6, 10, 28,  66,  78,  34, 114,  92,  58, 186,  148
  2, 6, 20, 14,  66,  26, 102,  76,  46, 174, 124,  222
  2, 6, 10, 28,  22,  78,  68,  38, 138, 116, 186,  296
  2, 6, 20, 42,  66,  52,  34, 114,  92, 174, 248,  148
  2, 6, 10, 14,  44,  26, 102,  76, 138, 232, 124,   74
  ...

Antti Karttunen, Table of n, a(n) for n = 1..3321; the first 81 antidiagonals of the array

Transpose: A257251.
Row 1: A036689.
Cf. also A083140, A257254 (same array but with terms divided by 2).

(define (A257252 n) (A257251bi (A004736 n) (A002260 n))) ;; Other code as in A257251.

A138416 a(n) = (p^3 - p^2)/2, where p = prime(n).

Original entry on oeis.org

2, 9, 50, 147, 605, 1014, 2312, 3249, 5819, 11774, 14415, 24642, 33620, 38829, 50807, 73034, 100949, 111630, 148137, 176435, 191844, 243399, 282449, 348524, 451632, 510050, 541059, 606797, 641574, 715064, 1016127, 1115465, 1276292, 1333149

Offset: 1

Artur Jasinski, Mar 19 2008

Differences (p^k - p^m)/q with k > m:
    expression     OEIS sequence
  --------------   -------------
   p^2 - p            A036689
  (p^2 - p)/2         A008837
   p^3 - p            A127917
  (p^3 - p)/2         A127918
  (p^3 - p)/3         A127919
  (p^3 - p)/6         A127920
   p^3 - p^2          A135177
  (p^3 - p^2)/2    this sequence
   p^4 - p            A138401
  (p^4 - p)/2         A138417
   p^4 - p^2          A138402
  (p^4 - p^2)/2       A138418
  (p^4 - p^2)/3       A138419
  (p^4 - p^2)/4       A138420
  (p^4 - p^2)/6       A138421
  (p^4 - p^2)/12      A138422
   p^4 - p^3          A138403
  (p^4 - p^3)/2       A138423
   p^5 - p            A138404
  (p^5 - p)/2         A138424
  (p^5 - p)/3         A138425
  (p^5 - p)/5         A138426
  (p^5 - p)/6         A138427
  (p^5 - p)/10        A138428
  (p^5 - p)/15        A138429
  (p^5 - p)/30        A138430
   p^5 - p^2          A138405
  (p^5 - p^2)/2       A138431
   p^5 - p^3          A138406
  (p^5 - p^3)/2       A138432
  (p^5 - p^3)/3       A138433
  (p^5 - p^3)/4       A138434
  (p^5 - p^3)/6       A138435
  (p^5 - p^3)/8       A138436
  (p^5 - p^3)/12      A138437
  (p^5 - p^3)/24      A138438
   p^5 - p^4          A138407
  (p^5 - p^4)/2       A138439
   p^6 - p            A138408
  (p^6 - p)/2         A138440
   p^6 - p^2          A138409
  (p^6 - p^2)/2       A138441
  (p^6 - p^2)/3       A138442
  (p^6 - p^2)/4       A138443
  (p^6 - p^2)/5       A138444
  (p^6 - p^2)/6       A138445
  (p^6 - p^2)/10      A138446
  (p^6 - p^2)/12      A138447
  (p^6 - p^2)/15      A138448
  (p^6 - p^2)/20      A122220
  (p^6 - p^2)/30      A138450
  (p^6 - p^2)/60      A138451
   p^6 - p^3          A138410
  (p^6 - p^3)/2       A138452
   p^6 - p^4          A138411
  (p^6 - p^4)/2       A138453
  (p^6 - p^4)/3       A138454
  (p^6 - p^4)/4       A138455
  (p^6 - p^4)/6       A138456
  (p^6 - p^4)/8       A138457
  (p^6 - p^4)/12      A138458
  (p^6 - p^4)/24      A138459
   p^6 - p^5          A138412
  (p^6 - p^5)/2       A138460
.
We can prove that for n>1, a(n) is the remainder of the Euclidean division of Sum_{k=0..p-1} k^p by p^3 where p = prime(n). - Pierre Vandaële, Nov 30 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..168

[(p^3-p^2)/2: p in PrimesUpTo(1000)]; // Vincenzo Librandi, Jun 17 2011

a = {}; Do[p = Prime[n]; AppendTo[a, (p^3 - p^2)/2], {n, 1, 50}]; a
(#^3-#^2)/2&/@Prime[Range[50]] (* Harvey P. Dale, Nov 01 2020 *)

forprime(p=2,1e3,print1((p^3-p^2)/2", ")) \\ Charles R Greathouse IV, Jun 16 2011

Definition corrected by T. D. Noe, Aug 25 2008

A174857 The minimum distance k > 0 such that A020639(n+k) = A020639(n).

Original entry on oeis.org

2, 6, 2, 20, 2, 42, 2, 6, 2, 110, 2, 156, 2, 6, 2, 272, 2, 342, 2, 6, 2, 506, 2, 10, 2, 6, 2, 812, 2, 930, 2, 6, 2, 20, 2, 1332, 2, 6, 2, 1640, 2, 1806, 2, 6, 2, 2162, 2, 28, 2, 6, 2, 2756, 2, 10, 2, 6, 2, 3422, 2, 3660, 2, 6, 2, 20, 2, 4422, 2, 6, 2, 4970, 2, 5256, 2, 6, 2, 14, 2, 6162

Offset: 2

Vladimir Shevelev, Mar 31 2010

nonn

The sequence has the same records as A002618.

Antti Karttunen, Table of n, a(n) for n = 2..16385

Cf. A020639, A002618, A036689, A174869.

A174857 := proc(n) local k,aref ; aref := A020639(n) ; for k from 1 do if A020639(n+k) = aref then return k; end if; end do: end proc:
seq(A174857(n),n=2..80) ; # R. J. Mathar, Dec 07 2010

Block[{s = Array[FactorInteger[#][[1, 1]] &, 10^4]}, Array[If[EvenQ[#], 2, Block[{k = 1, n = s[[#]]}, While[n != s[[# + k]], k++; If[# + k > Length[s], AppendTo[s, FactorInteger[# + k][[1, 1]] ]] ]; k]] &, 78, 2]] (* Michael De Vlieger, Apr 06 2021 *)

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A174857(n) = if(isprime(n), (n-1)*n, my(spf=A020639(n)); for(k=1,oo,if(A020639(n+k)==spf,return(k)))); \\ Antti Karttunen, Apr 06 2021

If n is even, then a(n) = 2.
If n = 3k and A020639(k) >= 3, then a(n) = 6.
If n is prime, then a(n) = A036689(n).

cp's OEIS Frontend

A139116 a(n) = p*(p-1)/2, where p is A000043(n).

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A117495 Product of a prime number p and the number of primes smaller than p.

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A127922 1/24 of product of three numbers: n-th prime, previous and following number.

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A139115 a(n) = p*(p - 1), where p is A000043(n).

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A139223 M*(M-1), where M is Mersenne prime A000668(n).

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A139225 M(M-1)/3, where M is Mersenne prime A000668(n).

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A139226 M(M-1)/6, where M is Mersenne prime A000668(n).

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A257252 Transpose of square array A257251.

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