A023523 -id:A023523 - 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]

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).

A286624 a(n) = (prime(1+n)*prime(n)) + prime(n) + 1.

Original entry on oeis.org

9, 19, 41, 85, 155, 235, 341, 457, 691, 929, 1179, 1555, 1805, 2065, 2539, 3181, 3659, 4149, 4825, 5255, 5841, 6637, 7471, 8723, 9895, 10505, 11125, 11771, 12427, 14465, 16765, 18079, 19181, 20851, 22649, 23859, 25749, 27385, 29059, 31141, 32579, 34753, 37055, 38215, 39401, 42189, 47265, 50845, 52211

Offset: 1

Antti Karttunen, Jun 28 2017

nonn

9 is the only perfect square in this sequence. - Altug Alkan, Jul 01 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000

Row 6 of A286625 (column 6 of A286623). Column 4 of A328464.
One more than A123134.
Cf. A000040, A023523, A180932 (primes in this sequence).

Array[(#1 #2) + #1 + 1 & @@ Prime[# + {0, 1}] &, 49] (* Michael De Vlieger, Mar 14 2021 *)

lista(nn) = forprime(p=2, nn, print1(p*(nextprime(p+1)+1)+1, ", ")); \\ Altug Alkan, Jul 01 2017

(define (A286624 n) (+ (* (A000040 (+ 1 n)) (A000040 n)) (A000040 n) 1))
(define (A286624 n) (+ 1 (* (+ 1 (A000040 (+ 1 n))) (A000040 n))))

a(n) = (A000040(1+n)*A000040(n)) + A000040(n) + 1.
a(n) = 1 + A123134(n).
a(n) = A000040(n) + A023523(1+n).

A079080 a(n) = gcd((prime(n)+1)(prime(n+1)+1)/4, prime(n)prime(n+1)+1).

Original entry on oeis.org

1, 2, 12, 6, 6, 3, 18, 6, 4, 60, 4, 3, 42, 6, 4, 2, 30, 2, 6, 36, 8, 6, 2, 3, 3, 204, 6, 54, 3, 48, 6, 2, 138, 6, 300, 4, 2, 6, 4, 2, 90, 12, 96, 3, 396, 10, 14, 6, 114, 3, 8, 120, 6, 2, 4, 4, 540, 4, 3, 282, 6, 6, 6, 156, 3, 6, 2, 6, 174, 3, 4, 6, 4, 2, 6, 4, 3, 3, 15, 6, 210, 12, 216, 4, 6, 2

Offset: 1

Reinhard Zumkeller, Dec 22 2002

nonn

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

Cf. A023523, A079079, A079081, A079082.

a079080 n = a079079 n `gcd` a023523 (n + 1)
-- Reinhard Zumkeller, Oct 09 2012

a[n_] := Module[{p = Prime[n], q}, q = NextPrime[p]; GCD[(p+1) * (q+1) / 4, p*q + 1]]; Array[a, 100] (* Amiram Eldar, Apr 06 2025 *)

a(n) = my(p = prime(n), q = nextprime(p+1)); gcd((p+1)*(q+1)/4, p*q+1); \\ Amiram Eldar, Apr 06 2025

a(n) = gcd(A079079(n), A023523(n+1)).

A079081 Numerator of (prime(n)+1)(prime(n+1)+1)/(4(prime(n)*prime(n+1)+1)).

Original entry on oeis.org

3, 3, 1, 4, 7, 21, 5, 20, 45, 4, 76, 133, 11, 88, 162, 405, 31, 527, 204, 37, 185, 280, 945, 735, 833, 13, 468, 55, 1045, 76, 704, 2277, 35, 875, 19, 1501, 3239, 1148, 1827, 3915, 91, 728, 97, 3201, 25, 1060, 848, 2128, 115, 4485, 1755, 121, 2541, 8127, 4257, 4455

Offset: 1

Reinhard Zumkeller, Dec 22 2002

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

Denominator = A079082.

Cf. A023523, A079079, A079080.

a079081 n = a079081_list !! (n-1)
a079081_list = zipWith div a079079_list a079080_list
-- Reinhard Zumkeller, Oct 09 2012

a[n_] := Module[{p = Prime[n], q}, q = NextPrime[p]; Numerator[(p+1) * (q+1) / (4 * (p*q + 1))]]; Array[a, 100] (* Amiram Eldar, Apr 06 2025 *)

a(n) = my(p = prime(n), q = nextprime(p+1)); numerator((p+1) * (q+1) / (4 * (p*q + 1))); \\ Amiram Eldar, Apr 06 2025

a(n) = numerator(A079079(n)/A023523(n+1)).

a(n) = A079079(n)/A079080(n).

A079082 Denominator of (prime(n)+1)(prime(n+1)+1)/(4(prime(n)*prime(n+1)+1)).

Original entry on oeis.org

7, 8, 3, 13, 24, 74, 18, 73, 167, 15, 287, 506, 42, 337, 623, 1564, 120, 2044, 793, 144, 721, 1093, 3694, 2878, 3266, 51, 1837, 216, 4106, 299, 2773, 8974, 138, 3452, 75, 5927, 12796, 4537, 7223, 15484, 360, 2881, 384, 12674, 99, 4199, 3361, 8437, 456

Offset: 1

Reinhard Zumkeller, Dec 22 2002

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

Numerator = A079081.

Cf. A023523, A079079, A079080.

a079082 n = a079082_list !! (n-1)
a079082_list = zipWith div (tail a023523_list) a079080_list
-- Reinhard Zumkeller, Oct 09 2012

((#[[1]]+1)(#[[2]]+1))/(4(Times@@#+1))&/@Partition[Prime[Range[50]],2,1]//Denominator (* Harvey P. Dale, Jan 01 2018 *)

a(n) = my(p = prime(n), q = nextprime(p+1)); denominator((p+1) * (q+1) / (4 * (p*q + 1))); \\ Amiram Eldar, Apr 06 2025

a(n) = denominator(A079079(n)/A023523(n+1)).

a(n) = A023523(n+1)/A079080(n).

A072565 a(n) = prime(n+1)*prime(n+2)+1 mod prime(n), where prime(n) is the n-th prime.

Original entry on oeis.org

0, 0, 3, 4, 2, 12, 13, 3, 3, 17, 30, 25, 13, 41, 26, 49, 17, 0, 25, 17, 61, 41, 2, 8, 25, 13, 25, 13, 73, 27, 41, 49, 25, 121, 17, 73, 61, 41, 73, 49, 25, 121, 13, 25, 29, 90, 193, 25, 13, 41, 49, 25, 161, 73, 73, 49, 17, 61, 25, 25, 241, 253, 25, 13, 73, 281, 97, 121, 13

Offset: 1

G. L. Honaker, Jr., Aug 06 2002

nonn

			a(18) = prime(19)*prime(20)+1 mod prime(18) = 67*71+1 mod 61 = 0.

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer-Verlag, NY, (2002 printing), Research problem 1.85, p. 73.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040, A023523, A022461.

P:=Filtered([1..1000],IsPrime);;
List([1..70],n->(P[n+1]*P[n+2]+1) mod P[n]); # Muniru A Asiru, Mar 09 2018

[(NthPrime(n+1)*NthPrime(n+2)+1) mod NthPrime(n): n in [1..100]]; // Vincenzo Librandi, Feb 28 2018

p:=ithprime; seq((p(n+1)*p(n+2)+1) mod p(n),n=1..70); # Muniru A Asiru, Mar 09 2018

a[n_] := Mod[Prime[n+1] Prime[n+2] + 1, Prime[n]]
Mod[#[[2]]#[[3]]+1,#[[1]]]&/@Partition[Prime[Range[80]],3,1] (* Harvey P. Dale, Dec 19 2018 *)

a(n) = (prime(n+1)*prime(n+2) + 1) % prime(n); \\ Michel Marcus, Feb 28 2018

a(n) = A023523(n+1) mod A000040(n). - Michel Marcus, Feb 28 2018

Edited by Dean Hickerson and Robert G. Wilson v, Aug 10 2002

A023525 Greatest prime divisor of prime(n)*prime(n-1) + 1.

Original entry on oeis.org

3, 7, 2, 3, 13, 3, 37, 3, 73, 167, 5, 41, 23, 7, 337, 89, 23, 5, 73, 61, 3, 103, 1093, 1847, 1439, 71, 17, 167, 3, 2053, 23, 59, 641, 23, 863, 5, 5927, 457, 349, 233, 79, 5, 67, 3, 6337, 11, 19, 3361, 59, 19, 8893, 6961, 5, 71, 16127, 71, 769, 5, 383

Offset: 1

Clark Kimberling

nonn

This sequence assumes prime(0) = 1.

Heuristically, we should expect a(n) -> infinity as n -> infinity, i.e. for any k there should be only finitely many terms <= k. It seems likely that 3 is the only n for which a(n) = 2. There are at least 14 occurrences of a(n) = 3, including 1, 4, 6, 8, 21, 29, 44, 84, 191, 378, 13006, 39420, 62947, 78156. - Robert Israel, Aug 14 2015

Harvey P. Dale and Robert Israel, Table of n, a(n) for n = 1..10000(1..1000 from Harvey P. Dale)

Cf. A006530, A023523.

Primes:= select(isprime,[2,(2*i+1 $ i=1..1000)]):
3, seq(max(numtheory:-factorset(Primes[i]*Primes[i+1]+1)), i=1..nops(Primes)-1); # Robert Israel, Aug 14 2015

Join[{3},FactorInteger[#][[-1,1]]&/@(Times@@@Partition[Prime[ Range[ 60]], 2,1]+1)] (* Harvey P. Dale, Apr 12 2013 *)

gpf(n)=my(f=factor(n)[, 1]~);f[#f];
myprime(n)=if(n==0,1,prime(n));
first(m)=vector(m,i,gpf(1+myprime(i)*myprime(i-1))); \\ Anders Hellström, Aug 13 2015

a(n) = A006530(A023523(n)). - Michel Marcus, Aug 12 2015

A023528 Exponent of 2 in prime factorization of prime(n)*prime(n-1) + 1.

Original entry on oeis.org

0, 0, 4, 2, 1, 4, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 3, 4, 3, 1, 6, 3, 1, 2, 1, 1, 2, 1, 4, 1, 4, 1, 2, 2, 3, 2, 2, 3, 1, 2, 3, 4, 2, 12, 1, 2, 1, 1, 1, 4, 1, 3, 8, 2, 2, 3, 2, 2, 2, 1, 2, 3, 5, 1, 6, 1, 5, 2, 2, 4, 1, 3, 1, 2, 3, 1, 2, 1, 1, 1

Offset: 1

Clark Kimberling

nonn

Assumes the not generally accepted convention prime(0) = 1. - Michel Marcus, Jun 06 2019

Marius A. Burtea, Table of n, a(n) for n = 1..6001

Cf. A007814, A023523.

p:=PrimesUpTo(10000); sol:=[];sol[1]:=0; for n in [2..80] do sol[n]:=Valuation(1+p[n]*p[n-1],2);end for; sol; // Marius A. Burtea, Jun 06 2019

Join[{0},FactorInteger[#][[1,2]]&/@(Times@@@Partition[Prime[Range[ 80]], 2,1]+1)] (* Harvey P. Dale, Dec 25 2011 *)

p(n) = if (n==0, 1, prime(n));
a(n) = valuation(p(n)*p(n-1) + 1, 2); \\ Michel Marcus, Jun 06 2019

from sympy import prime
def A023528(n): return 0 if n == 1 else (~(m:=prime(n)*prime(n-1)+1)& m-1).bit_length() # Chai Wah Wu, Jul 07 2022

a(n) = A007814(A023523(n)). - Michel Marcus, Jun 06 2019

a(1)=a(2)=0 corrected by Sean A. Irvine, Jun 05 2019

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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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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A286624 a(n) = (prime(1+n)*prime(n)) + prime(n) + 1.

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A079080 a(n) = gcd((prime(n)+1)*(prime(n+1)+1)/4, prime(n)*prime(n+1)+1).

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A079081 Numerator of (prime(n)+1)*(prime(n+1)+1)/(4*(prime(n)*prime(n+1)+1)).

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A079082 Denominator of (prime(n)+1)*(prime(n+1)+1)/(4*(prime(n)*prime(n+1)+1)).

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A072565 a(n) = prime(n+1)*prime(n+2)+1 mod prime(n), where prime(n) is the n-th prime.

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A079080 a(n) = gcd((prime(n)+1)(prime(n+1)+1)/4, prime(n)prime(n+1)+1).

A079081 Numerator of (prime(n)+1)(prime(n+1)+1)/(4(prime(n)*prime(n+1)+1)).

A079082 Denominator of (prime(n)+1)(prime(n+1)+1)/(4(prime(n)*prime(n+1)+1)).