A008864 -id:A008864 - cp's OEIS frontend

A138404 a(n) = prime(n)^5 - prime(n).

30, 240, 3120, 16800, 161040, 371280, 1419840, 2476080, 6436320, 20511120, 28629120, 69343920, 115856160, 147008400, 229344960, 418195440, 714924240, 844596240, 1350125040, 1804229280, 2073071520, 3077056320, 3939040560

Offset: 1

Artur Jasinski, Mar 19 2008

Subsequence of A061167. - Bernard Schott, Feb 06 2023

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index to sequences related to prime powers

Cf. A000040, A006093, A008864, A050997, A061167, A066872, A138430.

[NthPrime((n))^5 - NthPrime((n)): n in [1..30] ]; // Vincenzo Librandi, Jun 17 2011

a = {}; Do[p = Prime[n]; AppendTo[a, p^5 - p], {n, 1, 50}]; a
#^5-#&/@Prime[Range[30]] (* Harvey P. Dale, Dec 25 2022 *)

forprime(p=2,1e3,print1(p^5-p", ")) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = A050997(n) - A000040(n). - Elmo R. Oliveira, Jan 27 2023
From Bernard Schott, Feb 09 2023: (Start)
a(n) = A061167(A000040(n)).
a(n) = 30 * A138430(n).
a(n) = A000040(n) * A006093(n) * A008864(n) * A066872(n). (End)

A175221 a(n) = prime(n) + 4.

Original entry on oeis.org

6, 7, 9, 11, 15, 17, 21, 23, 27, 33, 35, 41, 45, 47, 51, 57, 63, 65, 71, 75, 77, 83, 87, 93, 101, 105, 107, 111, 113, 117, 131, 135, 141, 143, 153, 155, 161, 167, 171, 177, 183, 185, 195, 197, 201, 203, 215, 227, 231, 233, 237, 243, 245, 255, 261, 267, 273, 275

Offset: 1

Jaroslav Krizek, Mar 06 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Filtered([1..300], k-> IsPrime(k) ) + 4; # G. C. Greubel, May 20 2019

[(p+4):p in PrimesUpTo(500)]; // Vincenzo Librandi, Dec 04 2010

Prime[Range[60]]+4 (* Harvey P. Dale, Jan 01 2013 *)

{a(n) = prime(n) + 4}; \\ G. C. Greubel, May 20 2019

[nth_prime(n) + 4 for n in (1..60)] # G. C. Greubel, May 20 2019

a(n) = A000040(n) + 4 = A008864(n) + 3 = A052147(n) + 2 = A113395(n) + 1.
a(n) = A175222(n) - 1 = A139049(n) - 2 = A175223(n) - 3.
a(n) = A175224(n) - 4 = A140353(n) - 5 = A175225(n) - 6.

More terms from Vincenzo Librandi, Mar 14 2010

A216626 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} lcm(c,d) for n>=1, k>=1.

Original entry on oeis.org

1, 3, 3, 4, 7, 4, 7, 12, 12, 7, 6, 15, 10, 15, 6, 12, 18, 28, 28, 18, 12, 8, 28, 24, 27, 24, 28, 8, 15, 24, 30, 42, 42, 30, 24, 15, 13, 31, 32, 60, 16, 60, 32, 31, 13, 18, 39, 60, 56, 72, 72, 56, 60, 39, 18, 12, 42, 28, 51, 48, 70, 48, 51, 28, 42, 12, 28, 36

Offset: 1

Peter Luschny, Sep 12 2012

T(n,n) = A064950(n) = sum_{d|n} d*tau(d^2).
T(n,1) = T(1,n) = A000203(n) = sigma(n).
T(n,2) = T(2,n) = A062731(n) = sigma(2*n).
T(n+1,n) = A083539(n) = sigma(n+1)*sigma(n).
T(prime(n),1) = A008864(n) = prime(n)+1.

			[-----1---2---3----4----5----6----7----8----9---10---11---12]
[ 1]  1,  3,  4,   7,   6,  12,   8,  15,  13,  18,  12,  28
[ 2]  3,  7, 12,  15,  18,  28,  24,  31,  39,  42,  36,  60
[ 3]  4, 12, 10,  28,  24,  30,  32,  60,  28,  72,  48,  70
[ 4]  7, 15, 28,  27,  42,  60,  56,  51,  91,  90,  84, 108
[ 5]  6, 18, 24,  42,  16,  72,  48,  90,  78,  48,  72, 168
[ 6] 12, 28, 30,  60,  72,  70,  96, 124,  84, 168, 144, 150
[ 7]  8, 24, 32,  56,  48,  96,  22, 120, 104, 144,  96, 224
[ 8] 15, 31, 60,  51,  90, 124, 120,  83, 195, 186, 180, 204
[ 9] 13, 39, 28,  91,  78,  84, 104, 195,  55, 234, 156, 196
[10] 18, 42, 72,  90,  48, 168, 144, 186, 234, 112, 216, 360
[11] 12, 36, 48,  84,  72, 144,  96, 180, 156, 216,  34, 336
[12] 28, 60, 70, 108, 168, 150, 224, 204, 196, 360, 336, 270
.
Displayed as a triangular array:
    1;
    3,  3;
    4,  7,  4;
    7, 12, 12,  7;
    6, 15, 10, 15,  6;
   12, 18, 28, 28, 18, 12;
    8, 28, 24, 27, 24, 28,  8;
   15, 24, 30, 42, 42, 30, 24, 15;
   13, 31, 32, 60, 16, 60, 32, 31, 13;

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Cf. A216620, A216621, A216622, A216623, A216624, A216625, A216627.

with(numtheory):
T:= (n, k) -> add(add(ilcm(c, d), c=divisors(n)), d=divisors(k)):
seq (seq (T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 12 2012

T[n_, k_] := Sum[LCM[c, d], {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2014 *)

def A216626(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(lcm, cp))
for n in (1..12): [A216626(n,k) for k in (1..12)]

A239703 Number of bases b > 1 for which the base-b digital sum of n is b.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 2, 0, 2, 2, 2, 1, 4, 0, 2, 1, 3, 1, 4, 1, 4, 2, 1, 1, 4, 1, 1, 2, 4, 0, 5, 0, 5, 3, 1, 2, 7, 0, 2, 3, 5, 0, 4, 0, 4, 3, 1, 1, 5, 1, 3, 2, 3, 0, 5, 2, 6, 1, 1, 0, 8, 0, 2, 2, 5, 3, 5, 1, 2, 2, 4, 1, 8, 0, 1, 4, 3, 2, 4, 1, 6, 3, 2, 0, 10, 2

Offset: 0

Hieronymus Fischer, Mar 31 2014

nonn

For the definition of the digital sum, see A007953.
For reference, we write digitSum_b(x) for the base-b digital sum of x according to A007953 (with general base b).
The bases counted exclude the special base 1. The base-1 expansion of a natural number is defined as 1=1_1, 2=11_1, 3=111_1 and so on. As a result, the base-1 digital sum of n is n. The inclusion of base b = 1 would lead to a(1) = 1 instead of a(1) = 0. All other terms remain unchanged.
For odd n > 1 and b := (n + 1)/2 we have digitSum_b(n) = b, and thus a(n) >= 1.
The digitSum_b(n) is < b for bases b which satisfy b > floor((n+1)/2), and thus a(n) <= floor((n+1)/2).
If b is a base such that the base-b digital sum of n is b, then b < n and b - 1 is a divisor of n - 1, thus the number of such bases is limited by the number of divisors of n - 1 (see formula section).
If p < n - 1 is a divisor of n - 1 which satisfy p >= sqrt(n - 1), then digitSum_b(n) = b for b := p + 1. This leads to a lower bound for a(n) (see formula section).
If b - 1 is a divisor of n - 1, then b is not necessarily a base such that base-b digital sum of n is b. Example: 1, 2, 3, 4, 6, 8, 12, 16, and 24 are the divisors < 48 of 48, but digitSum_2(49) = 3, digitSum_3(49) = 5, digitSum_5(49) = 9, digitSum_7(49) = 1.
a(b*n) > 0 for all b > 1 which satisfy digitSum_b(n) = b.
  Example 1: digitSum_2(3) = 2, hence a(2*3) > 0.
  Example 2: digitSum_3(5) = 3, hence a(3*5) > 0.
The first n with a(n) = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... are n = 3, 5, 17, 13, 31, 57, 37, 61, 81, 85, ... .

			a(1) = 1, since digitSum_1(1) = 1 and digitSum_b(1) <> b for all b > 1.
a(2) = 0, since digitSum_1(2) = 2 (because of 2 = 11_1), and digitSum_2(2) = 1 (because of 2 = 10_2), and digitSum_b(2) = 2 for all b > 2.
a(3) = 1, since digitSum_1(3) = 3 (because of 3 = 111_1), and digitSum_2(3) = 2 (because of 3 = 11_2), and digitSum_3(3) = 1 (because of 3 = 10_3), and digitSum_b(3) = 3 for all b > 3.
a(5) = 2, since digitSum_1(5) = 5 (because of 5 = 11111_1), and digitSum_2(5) = 2 (because of 5 = 101_2), and digitSum_3(5) = 3 (because of 5 = 12_3), and digitSum_4(5) = 2 (because of 5 = 11_4), and digitSum_5(5) = 1 (because of 5 = 10_5), and digitSum_b(5) = 5 for all b > 5.

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Cf. A007953, A079696, A008864, A187813, A239707, A239708.

Cf. A000040; A000005 (definition of sigma_0(n)).

"> Version 1: simple calculation for small numbers.
  Answer the number of bases b such that the digital sum of n in base b is b.
  Valid for bases b >= 1, thus returning a(1) = 1.
  Using digitalSum from A007953.
  Usage: n numOfBasesWithAltDigitalSumEQBase
  Answer: a(n)"
numOfBasesWithDigitalSumEQBase
  | numBases b bmax |
  numBases := 0.
  bmax := self + 1 // 2.
  b := 0.
  [b < bmax] whileTrue: [
     b := b + 1
     (self digitalSum: b) = b
     ifTrue: [numBases := numBases + 1]].
  ^numBases
-----------
"> Version 2: accelerated calculation for large numbers.
  Answer the number of bases b such that the digital sum of n in base b is b.
  Valid for bases b > 1, thus returning a(1) = 0.
  Using digitalSum from A007953.
  Usage: n numOfBasesWithAltDigitalSumEQBase
  Answer: a(n)"
numOfBasesWithDigitalSumEQBase
  | numBases div b bsize |
  self < 3 ifTrue: [^0].
  div := (self - 1) divisors.
  numBases := 0.
  bsize := div size - 1.
  1 to: bsize do: [ :i | b := (div at: i) + 1.
   (self digitalSum: b) = b
       ifTrue: [numBases := numBases + 1] ].
  ^numBases

a(n) = 0, if and only if n is a term of A187813.
a(A187813(n)) = 0.
a(A239708(n)) = 1, for n > 0.
a(A018900(n)) > 0, for n > 0.
a(A079696(n)) > 0, for n > 0.
a(A008864(n)) <= 1, for n > 0.
a(n) <= 1, if n - 1 is a prime.
a(n) <= sigma_0(n - 1) - 1, for n > 1.
a(n) >= floor((sigma_0(n-1)-1)/2), for n > 1.

A377781 First differences of A065514(n) = greatest number < prime(n) that is 1 or a prime-power.

Original entry on oeis.org

1, 2, 1, 4, 2, 5, 1, 2, 8, 2, 3, 5, 4, 2, 6, 4, 6, 5, 3, 4, 2, 8, 2, 6, 8, 4, 2, 4, 2, 16, 3, 3, 6, 2, 10, 2, 6, 6, 6, 4, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 4, 13, 1, 6, 6, 2, 6, 4, 8, 4, 14, 4, 2, 4, 14, 12, 4, 2, 4, 8, 6, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10

Offset: 1

Gus Wiseman, Nov 14 2024

nonn

Note 1 is a power of a prime but not a prime-power.

Differences of A065514, which is the restriction of A031218 (differences A377782).
The opposite is A377703 (restriction of A000015), differences of A345531.
The opposite for nonsquarefree is A377784, differences of A377783.
For nonsquarefree we have A378034, differences of A378032 (restriction of A378033).
The opposite for squarefree is A378037, differences of A112926 (restriction of A067535).
For squarefree we have A378038, differences of A112925 (restriction of A070321).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime-powers, differences A057820.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A361102 lists the non-powers of primes, differences A375708.
Prime-powers between primes:
- A053607 primes
- A080101 count (exclusive)
- A304521 by bits
- A366833 count
- A377057 positive
- A377286 zero
- A377287 one
- A377288 two
Cf. A001597, A007916, A008864, A053289, A120327, A343249, A375706, A377281, A377282, A377289.

Differences[Table[NestWhile[#-1&,Prime[n]-1,#>1&&!PrimePowerQ[#]&],{n,100}]]

A077068 Semiprimes of the form prime + 1.

Original entry on oeis.org

4, 6, 14, 38, 62, 74, 158, 194, 278, 314, 398, 422, 458, 542, 614, 662, 674, 734, 758, 878, 998, 1094, 1154, 1202, 1214, 1238, 1322, 1382, 1454, 1622, 1658, 1754, 1874, 1934, 1994, 2018, 2138, 2342, 2474, 2558, 2594, 2798, 2858, 2918, 3062, 3218, 3254

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

a(n) = A005383(n)+1 = 2*A005382(n).
There are 672 semiprimes of form prime+1 below 100000.
a(n) = A232342(n) + A077065(n). - Reinhard Zumkeller, Dec 16 2013

			A001358(25)=74=2*37 is a term as 74=A000040(21)+1=73+1.

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

Cf. A008864, A001358, A000040, A077065, A077067.

a077068 n = a077068_list !! (n-1)
a077068_list = filter ((== 1) . a010051 . (`div` 2)) a008864_list
-- Reinhard Zumkeller, Nov 22 2013

Select[Range[6000],Plus@@Last/@FactorInteger[#]==2&&PrimeQ[#-1]&] (* Vladimir Joseph Stephan Orlovsky, May 08 2011 *)
Select[ Prime@ Range@ 460 +1,PrimeOmega@# == 2 &] (* Robert G. Wilson v, Feb 18 2014 *)

[x+1|x<-primes(10^5),bigomega(x+1)==2] \\ Charles R Greathouse IV, Nov 22 2013

is(n)=isprime(n-1) && isprime(n\2) \\ Charles R Greathouse IV, Mar 20 2016

A010051(A008864(n)/2) = A064911(A008864(n)) = 1. - Reinhard Zumkeller, Nov 22 2013

A131991 a(n) = 1 + prime(n) + prime(n)^2 + prime(n)^3.

Original entry on oeis.org

15, 40, 156, 400, 1464, 2380, 5220, 7240, 12720, 25260, 30784, 52060, 70644, 81400, 106080, 151740, 208920, 230764, 305320, 363024, 394420, 499360, 578760, 712980, 922180, 1040604, 1103440, 1236600, 1307020, 1455780, 2064640, 2265384

Offset: 1

Reinhard Zumkeller, Aug 06 2007

Number of points and lines for the prime(n)-Cremona-Richmond configuration. - Carlos Segovia Gonzalez, Jul 30 2020

			a(4)=400 because the 4th prime is 7, 7^3=343, 7^2=49, and 343+49+7+1=400.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. Segovia and M. Winklmeier, Calculating the dimension of the universal embedding of the symplectic dual polar space using languages, arXiv:1312.4315 [math.CO], 2013-2019.
C. Segovia and M. Winklmeier, Calculating the dimension of the universal embedding of the symplectic dual polar space using languages, The Elec. Jour. of Comb. 27(4) (2020), P4.39.

Cf. A030078, A008864, A131992, A131993.

Cf. A000040, A000203. - Zak Seidov, Feb 13 2016

[1+NthPrime(n)+NthPrime(n)^2+NthPrime(n)^3: n in [1..40]]; // Vincenzo Librandi, Dec 27 2010

A131991:= n -> map (p -> p^(3)+p^(2)+p+1, ithprime(n)):
seq (A131991(n), n=1..32); # Jani Melik, Jan 25 2011

#^3 + #^2 + # + 1 &/@Prime[Range[100]] (* Vincenzo Librandi, Mar 20 2014 *)

a(n) = 1 + A060800(n)*A000040(n).
a(n) = (A030514(n) - 1)/A006093(n).
a(n) = A000203(A000040(n)^3). - Zak Seidov, Feb 13 2016

A131993 a(n) = 1 + prime(n) + prime(n)^2 + prime(n)^3 + prime(n)^4 + prime(n)^5.

Original entry on oeis.org

63, 364, 3906, 19608, 177156, 402234, 1508598, 2613660, 6728904, 21243690, 29583456, 71270178, 118752606, 150508644, 234330768, 426237714, 727250580, 858672906, 1370581548, 1830004056, 2101864254, 3116505840, 3987077724, 5647514670, 8676791718, 10615201506

Offset: 1

Reinhard Zumkeller, Aug 06 2007

a(n) = 1 + A131992(n)*A000040(n).

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A050997, A008864, A060800, A131991.

[1+(&+[NthPrime(n)^(k): k in [1..5]]): n in [1..30]]; // Bruno Berselli and Vincenzo Librandi, Apr 20 2011

Total[#^Range[0,5]]&/@Prime[Range[30]]  (* Harvey P. Dale, Apr 20 2011 *)

a(n) = (A030516(n) - 1)/A006093(n).

A139049 a(n) = prime(n) + 6.

Original entry on oeis.org

8, 9, 11, 13, 17, 19, 23, 25, 29, 35, 37, 43, 47, 49, 53, 59, 65, 67, 73, 77, 79, 85, 89, 95, 103, 107, 109, 113, 115, 119, 133, 137, 143, 145, 155, 157, 163, 169, 173, 179, 185, 187, 197, 199, 203, 205, 217, 229, 233, 235, 239, 245, 247, 257, 263, 269, 275, 277

Offset: 1

Odimar Fabeny, Jun 02 2008

a(n) = A000040(n) + 6 = A008864(n) + 5 = A052147(n) + 4 = A113395(n) + 3 = A175221(n) + 2 = A175222(n) + 1 = A175223(n) - 1 = A175224(n) - 2 = A140353(n) - 3 = A175225(n) - 4. - Jaroslav Krizek, Mar 06 2010

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A140353.

Filtered([1..300], k-> IsPrime(k) ) + 6; # G. C. Greubel, May 20 2019

[NthPrime(n)+6: n in [1..70]]; // G. C. Greubel, May 20 2019

A139049:=n->ithprime(n)+6: seq(A139049(n), n=1..100); # Wesley Ivan Hurt, Apr 18 2017

6+Prime[Range[70]] (* G. C. Greubel, May 20 2019 *)

PARI
```
A139049(n) = prime(n)+6
```

[nth_prime(n) +6 for n in (1..70)] # G. C. Greubel, May 20 2019

Edited by Michael B. Porter, Jan 28 2010

A175223 a(n) = prime(n) + 7.

Original entry on oeis.org

9, 10, 12, 14, 18, 20, 24, 26, 30, 36, 38, 44, 48, 50, 54, 60, 66, 68, 74, 78, 80, 86, 90, 96, 104, 108, 110, 114, 116, 120, 134, 138, 144, 146, 156, 158, 164, 170, 174, 180, 186, 188, 198, 200, 204, 206, 218, 230, 234, 236, 240, 246, 248, 258, 264, 270, 276, 278

Offset: 1

Jaroslav Krizek, Mar 06 2010

a(n) = A000040(n) + 7 = A008864(n) + 6 = A052147(n) + 5 = A113395(n) + 4 = A175221(n) + 3 = A175222 (n) + 2 = A139049(n) + 1 = A175224(n) - 1 = A140353(n) - 2 = A175225(n) - 3.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Filtered([1..300], k-> IsPrime(k) ) + 7; # G. C. Greubel, May 20 2019

[p+7: p in PrimesUpTo(500)]; // Vincenzo Librandi, Dec 04 2010

Prime[Range[70]] + 7 (* Vincenzo Librandi, Nov 27 2013 *)

{a(n) = prime(n) + 7}; \\ G. C. Greubel, May 20 2019

[nth_prime(n) + 7 for n in (1..70)] # G. C. Greubel, May 20 2019

More terms from Vincenzo Librandi, Mar 14 2010

cp's OEIS Frontend

A138404 a(n) = prime(n)^5 - prime(n).

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A175221 a(n) = prime(n) + 4.

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A216626 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} lcm(c,d) for n>=1, k>=1.

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A239703 Number of bases b > 1 for which the base-b digital sum of n is b.

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Smalltalk

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A377781 First differences of A065514(n) = greatest number < prime(n) that is 1 or a prime-power.

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Mathematica

A077068 Semiprimes of the form prime + 1.

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Haskell

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PARI

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A131991 a(n) = 1 + prime(n) + prime(n)^2 + prime(n)^3.

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