A286624 -id:A286624 - cp's OEIS frontend

A328464 Square array A(n,k) = A276156((2^(n-1)) * (2k-1)) / A002110(n-1), read by descending antidiagonals.

1, 3, 1, 7, 4, 1, 9, 16, 6, 1, 31, 19, 36, 8, 1, 33, 106, 41, 78, 12, 1, 37, 109, 386, 85, 144, 14, 1, 39, 121, 391, 1002, 155, 222, 18, 1, 211, 124, 421, 1009, 2432, 235, 324, 20, 1, 213, 1156, 426, 1079, 2443, 4200, 341, 438, 24, 1, 217, 1159, 5006, 1086, 2575, 4213, 7430, 457, 668, 30, 1, 219, 1171, 5011, 17018, 2586, 4421, 7447, 12674, 691, 900, 32, 1

Offset: 1

Antti Karttunen, Oct 16 2019

Array is read by falling antidiagonals with n (row) and k (column) ranging as: (n,k) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

Row n contains all such sums of distinct primorials whose least significant summand is A002110(n-1), with each sum divided by that least significant primorial, which is also the largest primorial which divides that sum.

			Top left 9 X 11 corner of the array:
1: | 1,  3,   7,   9,    31,    33,    37,    39,    211,    213,    217
2: | 1,  4,  16,  19,   106,   109,   121,   124,   1156,   1159,   1171
3: | 1,  6,  36,  41,   386,   391,   421,   426,   5006,   5011,   5041
4: | 1,  8,  78,  85,  1002,  1009,  1079,  1086,  17018,  17025,  17095
5: | 1, 12, 144, 155,  2432,  2443,  2575,  2586,  46190,  46201,  46333
6: | 1, 14, 222, 235,  4200,  4213,  4421,  4434,  96578,  96591,  96799
7: | 1, 18, 324, 341,  7430,  7447,  7753,  7770, 215442, 215459, 215765
8: | 1, 20, 438, 457, 12674, 12693, 13111, 13130, 392864, 392883, 393301
9: | 1, 24, 668, 691, 20678, 20701, 21345, 21368, 765050, 765073, 765717

Cf. A328463 (transpose).
Cf. A000265, A002110, A007814, A135764, A276154, A276156,
Rows 1 - 5: A328462, A328465, A328466, A328467, A328468.
Column 2: A008864.
Column 3: A023523 (after its initial term).
Column 4: A286624.
Cf. also arrays A276945, A286625.

up_to = 105;
A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328464sq(n,k) = (A276156((2^(n-1)) * (k+k-1)) / A002110(n-1));
A328464list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A328464sq(col,(a-(col-1))))); (v); };
v328464 = A328464list(up_to);
A328464(n) = v328464[n];

A(n,k) = A276156((2^(n-1)) * (2k-1)) / A002110(n-1).

a(n) = A328461(A135764(n)). [When all sequences are considered as one-dimensional]

A060389 a(1)=p_1, a(2)=p_1 + p_1p_2, a(3)=p_1 + p_1p_2 + p_1p_2p_3, ... where p_i is the i-th prime.

Original entry on oeis.org

2, 8, 38, 248, 2558, 32588, 543098, 10242788, 233335658, 6703028888, 207263519018, 7628001653828, 311878265181038, 13394639596851068, 628284422185342478, 33217442899375387208, 1955977793053588026278

Offset: 1

Jason Earls, Apr 04 2001

nonn

Partial sums of the primorials A002110 starting from 2. - Charles R Greathouse IV, Feb 05 2014
All terms are even. From a(98) on, all terms are multiples of 523. - Charles R Greathouse IV, Feb 05 2014
The only values of n for which a(n)/2 is prime are: 3, 5, 7, 11, 15, 47, 49. The corresponding 7 primes are: 19, 1279, 271549, 103631759509, 314142211092671239, 826811434211869939736645732264127163964562391958563586838409421490271014424018927729, 41839936239750050346953677118447851613901200239299781782205558511980130628486398081201749. - Amiram Eldar, May 04 2017

			a(4) = 248 because p_1 + p_1*p_2 + p_1*p_2*p_3 + p_1*p_2*p_3*p_4 = 2 + 6 + 30 + 210 = 248.
a(5) = 2558: A002110(3) = 30, A286624(4) = 85, a(2) = 8; 30*85 + 8 = 2558. - _Bob Selcoe_, Oct 12 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for sequences related to primorial numbers

Cf. A002110, A070826, A084737, A143293, A203008 [= A327860(a(n))], A276085, A286624, A373158.

for n from 1 to 30 do printf(`%d,`,sum(product(ithprime(i), i=1..j), j=1..n)) od:

Accumulate[Denominator[Accumulate[1/Prime[Range[20]]]]] (* Alonso del Arte, Mar 21 2013 *)
Accumulate@ FoldList[Times, Prime@ Range@ 17] (* Michael De Vlieger, May 04 2017 *)

a(n)=my(s,P=1); forprime(p=2,prime(n),s+=P*=p); s \\ Charles R Greathouse IV, Feb 05 2014

a(n) = A002110(n-2)*A286624(n-1) + a(n-3), n >= 4. - Bob Selcoe, Oct 12 2017

a(n) = A276085(A070826(1+n)) = A084737(2+n)-2 = A373158(A002110(n)). - Antti Karttunen, Feb 06 2024, Oct 28 2024

More terms from James Sellers, Apr 05 2001

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).

A286625 Square array A(n,k) = A276945(n,k)/A002110(k-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, 1, 3, 1, 4, 4, 1, 6, 6, 5, 1, 8, 10, 7, 7, 1, 12, 14, 11, 16, 9, 1, 14, 22, 15, 36, 19, 10, 1, 18, 26, 23, 78, 41, 21, 11, 1, 20, 34, 27, 144, 85, 45, 22, 13, 1, 24, 38, 35, 222, 155, 91, 46, 31, 15, 1, 30, 46, 39, 324, 235, 165, 92, 71, 34, 16, 1, 32, 58, 47, 438, 341, 247, 166, 155, 76, 36, 17, 1, 38, 62, 59, 668, 457, 357, 248, 287, 162, 80, 37, 18

Offset: 1

Antti Karttunen, Jun 28 2017

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

Transpose: A286623.
Cf. A002110, A276945.
Column 1: A276155.
Row 1: A000012, Row 2: A008864, Row 3: A100484, Row 4: A072055, Row 5: A023523 (from its second term onward), Row 6: A286624.
Cf. A276617 (analogous array).

(define (A286625 n) (A286625bi (A002260 n) (A004736 n)))
(define (A286625bi row col) (/ (A276945bi row col) (A002110 (- col 1))))

A(n,k) = A276945(n, k) / A002110(k-1).

A123134 a(n) = prime(n)*(prime(n+1) + 1).

Original entry on oeis.org

8, 18, 40, 84, 154, 234, 340, 456, 690, 928, 1178, 1554, 1804, 2064, 2538, 3180, 3658, 4148, 4824, 5254, 5840, 6636, 7470, 8722, 9894, 10504, 11124, 11770, 12426, 14464, 16764, 18078, 19180, 20850, 22648, 23858, 25748, 27384, 29058, 31140, 32578

Offset: 1

Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 30 2006

nonn

All terms are even. - Michel Marcus, Apr 02 2017

			a(1) = 2*(3+1) = 8, a(2) = 3*(5+1) = 18, a(3) = 5*(7+1) = 40, ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
A. Frank & P. Jacqueroux, International Contest, 2001. Item 21

Cf. A000040, A008864, A286624.

[NthPrime(n)*(NthPrime(n+1) +1): n in [1..40]]; // G. C. Greubel, Aug 04 2021

a[n_]:=Prime[n](Prime[n+1]+1); Array[a, 80] (* Giovanni Resta, Apr 02 2017 *)
#[[1]](#[[2]]+1)&/@Partition[Prime[Range[50]],2,1] (* Harvey P. Dale, Jan 06 2019 *)

for(n=1,100,print1(prime(n)*(prime(n+1)+1),","))

from sympy import prime
def a(n): return prime(n) * (prime(n + 1) + 1) # Indranil Ghosh, Apr 02 2017

a(n) = A000040(n)*A008864(n+1). - Zak Seidov, Apr 02 2017

a(n) = A286624(n) - 1. - Antti Karttunen, Jul 06 2017

A180932 Primes of the form pq + p + 1 where p < q are adjacent primes.

Original entry on oeis.org

19, 41, 457, 691, 929, 2539, 3181, 3659, 6637, 19181, 29059, 32579, 55921, 57839, 60733, 71011, 83203, 123547, 127081, 132113, 149371, 154823, 176819, 181873, 194917, 245501, 320911, 382541, 430981, 497711, 510481, 523433, 549817, 595207

Offset: 1

Carmine Suriano, Sep 26 2010

nonn

In the sequence of first differences D(9) = 12544 is the square of 112.

In the sequence of second differences DD(3) = 4 and DD(7) = 2500 are squares.

			a(7) = 3181 since 3181 = 53*(59+1) + 1 where 53 and 59 are successive primes.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A180931.

Primes in A286624.

[a: n in [0..200] | IsPrime(a) where a is NthPrime(n)*NthPrime(n+1)+NthPrime(n)+1]; // Vincenzo Librandi, Jul 07 2017

Select[Times@@#+#[[1]]+1&/@Partition[Prime[Range[200]],2,1],PrimeQ] (* Harvey P. Dale, Apr 26 2014 *)

p=2;forprime(q=3,1e3,k=p*(q+1)+1;if(isprime(k),print1(k","));p=q) \\ Charles R Greathouse IV, Sep 27 2010

New description and editing from Charles R Greathouse IV, Sep 27 2010

cp's OEIS Frontend

A328464 Square array A(n,k) = A276156((2^(n-1)) * (2k-1)) / A002110(n-1), read by descending antidiagonals.

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A060389 a(1)=p_1, a(2)=p_1 + p_1*p_2, a(3)=p_1 + p_1*p_2 + p_1*p_2*p_3, ... where p_i is the i-th prime.

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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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A286625 Square array A(n,k) = A276945(n,k)/A002110(k-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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A123134 a(n) = prime(n)*(prime(n+1) + 1).

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Magma

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A180932 Primes of the form pq + p + 1 where p < q are adjacent primes.

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Examples

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Magma

Mathematica

PARI

Extensions

A060389 a(1)=p_1, a(2)=p_1 + p_1p_2, a(3)=p_1 + p_1p_2 + p_1p_2p_3, ... where p_i is the i-th prime.