A286625 -id:A286625 - cp's OEIS frontend

A100484 The primes doubled; Even semiprimes.

4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 1

Reinhard Zumkeller, Nov 22 2004

Essentially the same as A001747.
Right edge of the triangle in A065342. - Reinhard Zumkeller, Jan 30 2012
A253046(a(n)) > a(n). - Reinhard Zumkeller, Dec 26 2014
Apart from first term, these are the tau2-primes as defined in [Anderson, Frazier] and [Lanterman]. - Michel Marcus, May 15 2019
For every positive integer b and each m in this sequence b^(m-1) == b (mod m). - Florian Baur, Nov 26 2021

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. D. Anderson and Andrea M. Frazier, On a general theory of factorization in integral domains, Rocky Mountain J. Math., Volume 41, Number 3 (2011), 663-705. See pp. 698, 699, 702.
James Lanterman, Irreducibles in the Integers modulo n, arXiv:1210.2991 [math.NT], 2012.
Eric Weisstein's World of Mathematics, Semiprime

Subsequence of A091376. After the initial 4 also a subsequence of A039956.
Cf. A046315, A152099, A179740.
Cf. A001748, A253046, A353478 (characteristic function).
Row 3 of A286625, column 3 of A286623.

2*Filtered([1..300],IsPrime); # Muniru A Asiru, Oct 05 2018

List([1..70], n-> 2*Primes[n]); # G. C. Greubel, May 18 2019

a100484 n = a100484_list !! (n-1)
a100484_list = map (* 2) a000040_list
-- Reinhard Zumkeller, Jan 31 2012

[2*p: p in PrimesUpTo(350)]; // Vincenzo Librandi, Mar 27 2014

A100484:=n->2*ithprime(n); seq(A100484(n), n=1..70); # Wesley Ivan Hurt, Mar 27 2014

2*Prime[Range[70]] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

2*primes(70) \\ Charles R Greathouse IV, Aug 21 2011

[2*nth_prime(n) for n in (1..70)] # G. C. Greubel, May 18 2019

a(n) = 2 * A000040(n).
a(n) = A001747(n+1).
n>1: A000005(a(n)) = 4; A000203(a(n)) = 3*A008864(n); A000010(a(n)) = A006093(n); intersection of A001358 and A005843.
a(n) = A116366(n-1, n-1) for n>1. - Reinhard Zumkeller, Feb 06 2006
a(n) = A077017(n+1), n>1. - R. J. Mathar, Sep 02 2008
A078834(a(n)) = A000040(n). - Reinhard Zumkeller, Sep 19 2011
a(n) = A087112(n, 1). - Reinhard Zumkeller, Nov 25 2012
A000203(a(n)) = 3*n/2 + 3, n > 1. - Wesley Ivan Hurt, Sep 07 2013

Simpler definition.

A072055 a(n) = 2*prime(n)+1.

Original entry on oeis.org

5, 7, 11, 15, 23, 27, 35, 39, 47, 59, 63, 75, 83, 87, 95, 107, 119, 123, 135, 143, 147, 159, 167, 179, 195, 203, 207, 215, 219, 227, 255, 263, 275, 279, 299, 303, 315, 327, 335, 347, 359, 363, 383, 387, 395, 399, 423, 447, 455, 459, 467, 479

Offset: 1

Reinhard Zumkeller, Jun 11 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. P. Boas & N. J. A. Sloane, Correspondence, 1974

Cf. A000040, A005385, A023589, A023590, A023591, A023592, A023593, A072059, A023595, A072060, A072056, A072057, A072058, A072192, A165286, A278230.
One less than A089241. After the initial term equal to A166496.
Row 4 of A286625, column 4 of A286623.

a072055 = (+ 1) . (* 2) . a000040  -- Reinhard Zumkeller, Oct 10 2013

2*Prime[Range[60]]+1  (* Harvey P. Dale, Mar 31 2011 *)

a(n) = A089241(n)-1.

A276154 a(n) = Shift primorial base representation (A049345) of n left by one digit (append one zero to the right, then convert back to decimal).

Original entry on oeis.org

0, 2, 6, 8, 12, 14, 30, 32, 36, 38, 42, 44, 60, 62, 66, 68, 72, 74, 90, 92, 96, 98, 102, 104, 120, 122, 126, 128, 132, 134, 210, 212, 216, 218, 222, 224, 240, 242, 246, 248, 252, 254, 270, 272, 276, 278, 282, 284, 300, 302, 306, 308, 312, 314, 330, 332, 336, 338, 342, 344, 420, 422, 426, 428, 432, 434, 450, 452, 456, 458, 462, 464, 480, 482, 486, 488

Offset: 0

Antti Karttunen, Aug 24 2016

			   n   A049345  with one zero           converted back
                appended to the right   to decimal = a(n)
---------------------------------------------------------
   0       0            00                     0
   1       1            10                     2
   2      10           100                     6
   3      11           110                     8
   4      20           200                    12
   5      21           210                    14
   6     100          1000                    30
   7     101          1010                    32
   8     110          1100                    36
   9     111          1110                    38
  10     120          1200                    42
  11     121          1210                    44
  12     200          2000                    60
  13     201          2010                    62
  14     210          2100                    66
  15     211          2110                    68
  16     220          2200                    72

Antti Karttunen, Table of n, a(n) for n = 0..30029
Index entries for sequences related to primorial base

Complement: A276155.
Cf. A002110, A003961, A049345, A276085, A276086, A276151, A276152, A286629 [= a(A061720(n-1))], A324384 [= gcd(n, a(n))], A323879, A328770 (a subsequence).
Cf. also A276156, A328461, A328464.
Dispersion array and its transpose: A276943, A276945, with primorials divided out: A286623, A286625.
Analogous to A153880.

nn = 75; b = MixedRadix[Reverse@ Prime@ NestWhileList[# + 1 &, 1, Times @@ Prime@ Range[#] <= nn &]]; Table[FromDigits[#, b] &@ Append[IntegerDigits[n, b], 0], {n, 0, nn}] (* Version 10.2, or *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ # - 1]], Reverse@ #}] &@ Append[f@ n, 0], {n, 0, 75}] (* Michael De Vlieger, Aug 26 2016 *)

A276154(n) = A276085(A003961(A276086(n))); \\ Antti Karttunen, Mar 15 2021

A276151(n) = { my(s=1); forprime(p=2, , if(n%p, return(n-s), s *= p)); };
A276152(n) = { my(s=1); forprime(p=2, , if(n%p, return(s*p), s *= p)); };
A276154(n) = if(!n,n,(A276152(n) + A276154(A276151(n)))); \\ Antti Karttunen, Mar 15 2021

(definec (A276154 n) (if (zero? n) n (+ (A276152 n) (A276154 (A276151 n)))))

a(0) = 0; for n >= 1, a(n) = A276152(n) + a(A276151(n)).

a(n) = A276085(A003961(A276086(n))). - Antti Karttunen, Mar 15 2021

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

Original entry on oeis.org

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]

A276945 Square array A(row,col): A(row,1) = A276155(row), and for col > 1, A(row,col) = A276154(A(row,col-1)); Dispersion of primorial base left shift A276154.

Original entry on oeis.org

1, 2, 3, 6, 8, 4, 30, 36, 12, 5, 210, 240, 60, 14, 7, 2310, 2520, 420, 66, 32, 9, 30030, 32340, 4620, 450, 216, 38, 10, 510510, 540540, 60060, 4830, 2340, 246, 42, 11, 9699690, 10210200, 1021020, 62370, 30240, 2550, 270, 44, 13, 223092870, 232792560, 19399380, 1051050, 512820, 32550, 2730, 276, 62, 15

Offset: 1

Antti Karttunen, Sep 24 2016

The array A(row,col) is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Entries in column k are all multiples of A002110(k-1). Dividing that factor out gives array A286625. - Antti Karttunen, Jun 30 2017

			The top left corner of the array:
   1,  2,   6,   30,   210,    2310,    30030,    510510
   3,  8,  36,  240,  2520,   32340,   540540,  10210200
   4, 12,  60,  420,  4620,   60060,  1021020,  19399380
   5, 14,  66,  450,  4830,   62370,  1051050,  19909890
   7, 32, 216, 2340, 30240,  512820,  9729720, 223603380
   9, 38, 246, 2550, 32550,  542850, 10240230, 233303070
  10, 42, 270, 2730, 34650,  570570, 10720710, 242492250
  11, 44, 276, 2760, 34860,  572880, 10750740, 243002760
  13, 62, 426, 4650, 60270, 1023330, 19429410, 446696250
  15, 68, 456, 4860, 62580, 1053360, 19939920, 456395940
  16, 72, 480, 5040, 64680, 1081080, 20420400, 465585120
  17, 74, 486, 5070, 64890, 1083390, 20450430, 466095630
  18, 90, 630, 6930, 90090, 1531530, 29099070, 669278610

Inverse permutation: A276946.
Transpose: A276943. One more than A286615.
Column 1: A276155.
Row 1: A002110.
Row 2: A276939.
Row 3: A088860 (2*A002110).
Row 11: 2*A276939 (row 2) from 16, 72, 480, 5040, 64680, ... onward.
Row 13: 3*A002110, from 18, 90, 630, 6930, 90090, ... onward.
Cf. A276154.
Cf. also arrays A286625, A276955.

(define (A276945 n) (A276945bi (A002260 n) (A004736 n)))
(define (A276945bi row col) (if (= 1 col) (A276155 row) (A276154 (A276945bi row (- col 1)))))

A(row,1) = A276155(row); for row > 1, A(row,col) = A276154(A(row,col-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).

A276155 Complement of A276154; numbers that cannot be obtained by shifting left the primorial base representation (A049345) of some number.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 39, 40, 41, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 64, 65, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 108, 109

Offset: 1

Antti Karttunen, Aug 24 2016

The first 25 terms, when viewed in primorial base (A049345) look as: 1, 11, 20, 21, 101, 111, 120, 121, 201, 211, 220, 221, 300, 301, 310, 311, 320, 321, 400, 401, 410, 411, 420, 421, 1001.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Complement: A276154.
Row 1 of A276943 and A286623. Column 1 of A276945 and A286625.
Cf. A002110, A049345.
Cf. A005408, A057588, A061720, A143293, A286630 (subsequences).
For the first 17 terms coincides with A273670.

nn = 109; b = MixedRadix[Reverse@ Prime@ NestWhileList[# + 1 &, 1, Times @@ Prime@ Range[# + 1] <= nn &]]; Complement[Range@ nn, Table[FromDigits[#, b] &@ Append[IntegerDigits[n, b], 0], {n, 0, nn}]] (* Version 10.2, or *)
nn = 109; f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Complement[Range@ nn, Table[Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ # - 1]], Reverse@ #}] &@ Append[f@ n, 0], {n, 0, nn}]] (* Michael De Vlieger, Aug 26 2016 *)

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).

cp's OEIS Frontend

A100484 The primes doubled; Even semiprimes.

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A072055 a(n) = 2*prime(n)+1.

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A276154 a(n) = Shift primorial base representation (A049345) of n left by one digit (append one zero to the right, then convert back to decimal).

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

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A276945 Square array A(row,col): A(row,1) = A276155(row), and for col > 1, A(row,col) = A276154(A(row,col-1)); Dispersion of primorial base left shift A276154.

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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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A276155 Complement of A276154; numbers that cannot be obtained by shifting left the primorial base representation (A049345) of some number.

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