A017473 -id:A017473 - cp's OEIS frontend

A017483 a(n) = (11*n + 7)^11.

1977326743, 64268410079232, 12200509765705829, 419430400000000000, 6071163615208263051, 52036560683837093888, 313726685568359708377, 1469170321634239709184, 5688000922764599609375, 18982985583354248390656, 56239892154164025151533, 151115727451828646838272

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Powers of the form (11*n+7)^m: A017473 (m=1), A017474 (m=2), A017475 (m=3), A017476 (m=4), A017477 (m=5), A017478 (m=6), A017479 (m=7), A017480 (m=8), A017481 (m=9), A017482 (m=10), this sequence (m=11), A017484 (m=12).

List([0..20], n-> (11*n+7)^11); # G. C. Greubel, Sep 19 2019

[(11*n+7)^11: n in [0..20]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+7)^11, n=0..20); # G. C. Greubel, Sep 19 2019

(11*Range[21] -4)^11 (* G. C. Greubel, Sep 19 2019 *)

vector(20, n, (11*n-4)^11) \\ G. C. Greubel, Sep 19 2019

[(11*n+7)^11 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (1977326743 +64244682158316*x +11429419348320083*x^2 + 277265562864875904*x^3 +1829094388304154510*x^4 +4212702849829094280*x^5 +3698421546351487230*x^6 +1221731311784947392*x^7 +134444370899578971* x^8 +3566547693499340*x^9 +8649705527727*x^10 +4194304*x^11)/(1-x)^12.
E.g.f.: (1977326743 +64266432752489*x +6035987461437054*x^2 + 63836945659298911*x^3 +186099500089146160*x^4 +214611357085098248*x^5 + 117178874627032680*x^6 +33290649534885897*x^7 +5128288643417445*x^8 + 425762824825415*x^9 +17689323577882*x^10 +285311670611*x^11)*exp(x). (End)

A017484 a(n) = (11*n + 7)^12.

Original entry on oeis.org

13841287201, 1156831381426176, 353814783205469041, 16777216000000000000, 309629344375621415601, 3226266762397899821056, 22902048046490258711521, 123410307017276135571456, 540360087662636962890625, 2012196471835550329409536, 6580067382037190942729361

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

List([0..20], n-> (11*n+7)^12); # G. C. Greubel, Sep 19 2019

[(11*n+7)^12: n in [0..20]]; // Vincenzo Librandi, Sep 04 2011

seq((11*n+7)^12, n=0..20); # G. C. Greubel, Sep 19 2019

(11*Range[21] -4)^12 (* G. C. Greubel, Sep 19 2019 *)

makelist((11*n+7)^12, n,0,20); /* Martin Ettl, Oct 21 2012 */

vector(20, n, (11*n-4)^12) \\ G. C. Greubel, Sep 19 2019

[(11*n+7)^12 for n in (0..20)] # G. C. Greubel, Sep 19 2019

From G. C. Greubel, Sep 19 2019: (Start)
G.f.: (13841287201 +1156651444692563*x +338777054867330431*x^2 + 12267852707472004709*x^3 +118792245587080463178*x^4 + 409344222142040360670*x^5 +564873972371695167390*x^6 + 320832301424673327498*x^7 +71415318201137477061*x^8 + 5352495778795351967*x^9 +93742255577726899*x^10 129746119786817*x^11 - 16777216*x^12)/(1-x)^13.
E.g.f.: (13841287201 +1156817540138975*x + 175750567141951945*x^2 + 2619873688447764034*x^3 +10193280906798742181*x^4 +15352998640256699136 *x^5 + 10914782775709466368*x^6 +4085382181827774828*x^7 + 853384566008402142*x^8 +101542034509085885*x^9 +6753041931691759*x^10 + 231102453194910*x^11 +3138428376721*x^12)*exp(x). (End)

A131191 Numbers n>=0 such that d(n) = (n^1 + 1) (n^2 + 2) ... (n^22 + 22) / 22!, e(n) = (n^1 + 1) (n^2 + 2) ... (n^23 + 23) / 23!, and f(n) = (n^1 + 1) (n^2 + 2) ... (n^24 + 24) / 24! take nonintegral values.

Original entry on oeis.org

7, 18, 29, 40, 51, 62, 73, 84, 95, 106, 128, 139, 150, 161, 172, 183, 194, 205, 216, 227, 249, 260, 271, 282, 293, 304, 315, 326, 337, 348, 370, 381, 392, 403, 414, 425, 436, 447, 458, 469, 491, 502, 513, 524, 535, 546, 557, 568, 579, 590, 612, 623, 634, 645, 656, 667, 678, 689, 700, 711, 733, 744

Offset: 1

Alexander R. Povolotsky, Sep 25 2007

nonn

If n is in this sequence, then so is n+121. - Max Alekseyev, Feb 02 2015

Cf. A017473, A129995

Notice that 22! = 2^19 * 3^9 * 5^4 * 7^3 * 11^2 * 13 * 17 * 19. All these prime powers divide (n^1 + 1)*(n^2 + 2)*(n^3 +3)*...*(n^22 + 22), except for 11^2. 11^2 does not divide (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^22 + 22) for n = 7, 18, 29, 40, 51, 62, 73, 84, 95, 106 modulo 121. That is, d(n) is nonintegral for n the form 11m+7 but not 121m+117, and so are e(n) and f(n). - Max Alekseyev, Nov 10 2007

Initial terms were calculated by Peter J. C. Moses; see comment in A129995.

More terms from Max Alekseyev, Feb 02 2015

A319524 a(n) is the smallest number that belongs simultaneously to the two arithmetic progressions prime(n) + mprime(n+1) and prime(n+1) + mprime(n+2), m >= 1, n >= 1.

Original entry on oeis.org

8, 33, 40, 128, 115, 302, 226, 226, 835, 401, 734, 1718, 1030, 842, 3121, 3475, 1401, 2339, 5108, 1969, 3233, 2486, 6491, 9692, 10298, 5560, 11552, 6211, 4177, 7987, 6022, 18763, 16678, 21893, 8001, 25585, 13523, 9682, 30961, 32035, 7057, 36089, 19105, 39002, 7162, 47041, 50163, 51752

Offset: 1

Alexandra Hercilia Pereira Silva, Sep 22 2018

Construct a table T in which T(m,n) = prime(n) + m*prime(n+1) as shown below. Then a(n) is defined as the smallest number appearing both in column n and column n+1, so a(1)=8, a(2)=33, a(3)=40, etc.
.
   m\n|  1     2     3     4     5     6     7     8  ...
  ----+--------------------------------------------------
    1 |  5   --8    12    18    24    30    36    42  ...
      |
    2 |  8--  13    19    29    37    47    55    65  ...
      |
    3 | 11    18    26    40    50    64    74    88  ...
      |                  /
    4 | 14    23    33  / 51    63    81    93   111  ...
      |            /   /
    5 | 17    28  / 40-   62    76    98   112   134  ...
      |          /
    6 | 20    33-   47    73    89   115   131   157  ...
      |                             /
    7 | 23    38    54    84   102 / 132   150   180  ...
      |                           /
    8 | 26    43    61    95   115   149   169   203  ...
      |
    9 | 29    48    68   106   128   166   188   226  ...
      |                       /                 /
   10 | 32    53    75   117 / 141   183   207 / 249  ...
      |                     /                 /
   11 | 35    58    82   128   154   200   226   272  ...
      |
   12 | 38    63    89   139   167   217   245   295  ...
      |
   13 | 41    68    96   150   180   234   264   318  ...
      |
   14 | 44    73   103   161   193   251   283   341  ...
      |
   15 | 47    78   110   172   206   268   302   364  ...
      |                                   /
   16 | 50    83   117   183   219   285 / 321   387  ...
      |                                 /
   17 | 53    88   124   194   232   302   340   410  ...
      |
  ... |...   ...   ...   ...   ...   ...   ...   ...  ...
Conjectures:
1. There are infinitely many pairs of consecutive equal terms. (Note that the first pair is (a(7), a(8)).)
2. There exists no N such that the sequence is monotonic for n > N.
From Amiram Eldar, Sep 22 2018: (Start)
Theorem 1: The intersection of the two mentioned arithmetic progressions is always nonempty.
Corollary: The sequence is infinite. (End)
Sequences that derive from this:
1. Positions in {s(n)} at which a(n) occurs: (2,6,5,11,8,17,19,...).
2. Positions in {s(n+1)} at which a(n) occurs: (1,4,3,9,6,15,15,...).
3. Differences between these two sequences: (1,2,2,2,2,4,...).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 600 terms from Muniru A Asiru)
Fourth International contest of logical problems, Problem 7, the Ludomind Society.
Fifth International contest of logical problems, Problem 6, the Ludomind Society, 2009.
Olivier Gérard, in reply to Zak Seidov, 11 related sequences, SeqFan list, Apr 14 2016.

Cf. A001043, A016789, A016885, A017041, A017473, A269100.

P:=Filtered([1..10000],IsPrime);;
T:=List([1..Length(P)-1],n->List([1..Length(P)-1],m->P[n]+m*P[n+1]));;
a:=List([1..50],k->Minimum(List([1..Length(T)-1],i->Intersection(T[i],T[i+1]))[k])); # Muniru A Asiru, Sep 26 2018

a[n_]:=ChineseRemainder[{Prime[n],Prime[n+1]},{Prime[n+1],Prime[n+2]} ];Array[a,44] (* Amiram Eldar, Sep 22 2018 *)

Table from Jon E. Schoenfield, Sep 23 2018

More terms from Amiram Eldar, Sep 22 2018

cp's OEIS Frontend

A017483 a(n) = (11*n + 7)^11.

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A017484 a(n) = (11*n + 7)^12.

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A131191 Numbers n>=0 such that d(n) = (n^1 + 1) (n^2 + 2) ... (n^22 + 22) / 22!, e(n) = (n^1 + 1) (n^2 + 2) ... (n^23 + 23) / 23!, and f(n) = (n^1 + 1) (n^2 + 2) ... (n^24 + 24) / 24! take nonintegral values.

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A319524 a(n) is the smallest number that belongs simultaneously to the two arithmetic progressions prime(n) + m*prime(n+1) and prime(n+1) + m*prime(n+2), m >= 1, n >= 1.

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GAP

Mathematica

Extensions

A319524 a(n) is the smallest number that belongs simultaneously to the two arithmetic progressions prime(n) + mprime(n+1) and prime(n+1) + mprime(n+2), m >= 1, n >= 1.