A084968 -id:A084968 - cp's OEIS frontend

A332799 Numbers whose smallest prime factor is 17.

17, 289, 323, 391, 493, 527, 629, 697, 731, 799, 901, 1003, 1037, 1139, 1207, 1241, 1343, 1411, 1513, 1649, 1717, 1751, 1819, 1853, 1921, 2159, 2227, 2329, 2363, 2533, 2567, 2669, 2771, 2839, 2941, 3043, 3077, 3247, 3281, 3349, 3383, 3587, 3791, 3859, 3893

Offset: 1

Frank Ellermann, Feb 24 2020

The asymptotic density of this sequence is 192/17017. - Amiram Eldar, Dec 06 2020

			a(2) = 17*17, a(3) = 17*19.

Emmanuel Desurvire, Classical and Quantum Information Theory: An Introduction for the Telecom Scientist, Cambridge University Press, 2009, table 20.5 p. 421.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A084967 (5), A084968 (7), A084969 (11), A084970 (13), A332798 (19), A332797 (23), A008366 (17-rough numbers).

17 * Select[Range[230], CoprimeQ[#, 30030] &] (* Amiram Eldar, Feb 24 2020 *)

P = 17         ;  S = P
do N = P by 2 while length( S ) < 255
   do I = 1 until P = X
      X = PRIME( I )
      if P = X       then  leave I
      if N // X = 0  then  iterate N
   end I
   S = S || ',' P*N
end N
say S          ;  return S

a(n) = 17*A008366(n).

A145011 First differences of A007775.

Original entry on oeis.org

6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2

Offset: 1

Ki Punches, Feb 25 2009

Also the first differences of A084968 divided by 7. - Antti Karttunen, May 01 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A007775, A084968.

Multiplied by 7: row 4 of A257251.

a145011 n = a145011_list !! (n-1)
a145011_list = zipWith (-) (tail a007775_list) a007775_list
-- Reinhard Zumkeller, Jan 06 2013

Differences[Select[Range[400],GCD[#,30]==1&]] (* Harvey P. Dale, Dec 07 2011 *)

a(n)=[4,6,4,2,4,2][n%8+1] \\ Charles R Greathouse IV, Oct 20 2013

Period 8: repeat 6,4,2,4,2,4,6,2.
a(n) = 2*((abs(abs((n mod 8) - 3) - 1) mod 3) + 1). - Pieter Stadhouders, Mar 09 2010
G.f.: x*(-2*x^7 - 6*x^6 - 4*x^5 - 2*x^4 - 4*x^3 - 2*x^2 - 4*x - 6)/(x^8 - 1). - Chai Wah Wu, Feb 16 2021

Edited by Omar E. Pol, Mar 02 2009

Offset corrected by Reinhard Zumkeller, Jan 06 2013

A251758 Let n>=2 be a positive integer with divisors 1 = d_1 < d_2 < ... < d_k = n, and s = d_1d_2 + d_2d_3 + ... + d_(k-1)*d_k. The sequence lists the values a(n) = floor(n^2/s).

Original entry on oeis.org

2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 13, 1, 2, 1, 17, 1, 19, 1, 2, 1, 23, 1, 4, 1, 2, 1, 29, 1, 31, 1, 2, 1, 4, 1, 37, 1, 2, 1, 41, 1, 43, 1, 2, 1, 47, 1, 6, 1, 2, 1, 53, 1, 4, 1, 2, 1, 59, 1, 61, 1, 2, 1, 4, 1, 67, 1, 2, 1, 71, 1, 73, 1, 2, 1, 6, 1, 79, 1, 2, 1

Offset: 2

Michel Lagneau, Dec 08 2014

nonn

s is always less than n^2 and if n is a prime number then s divides n^2.
For n >= 2, the sequence has the following properties:
a(n) = n if n is prime.
a(n) = 1 if n is in A005843 and > 2;
a(n) <= 2 if n is in A016945 and > 3;
a(n) <= 4 if n is in A084967 and > 5;
a(n) <= 6 if n is in A084968 and > 7;
a(n) = 8: <= 35336848261, ...;
a(n) <= 10 if n is in A084969 and > 11;
a(n) <= 12 if n is in A084970 and > 13;
a(n) = 14: 6678671, ...;
This is different from A250480 (a(n) = n for all prime n, and a(n) = A020639(n) - 1 for all composite n), which thus satisfies the above conditions exactly, while with this sequence A020639(n)-1 gives only the guaranteed upper limit for a(n) at composite n. Note that the first different term does not occur until at n = 2431 = 11*13*17, for which a(n) = 9. (See the example below.)
Conjecture: Terms x, where a(x)=n, x=p#k/p#j, p#i is the i-th primorial, k>j is suitable large k and j is the number of primes less than n. As an example, n=9, x = p#7/p#4 = 2431. For n=10, x = p#6/p#4 = 143 although 121 = 11^2 is the least x where a(x)=10 (see formula section). For n=8, x = p#12/p#4, p#13/p#4, p#14/p#4, p#15/p#4, p#16/p#4, etc. But is p#12/p#4 the least such x? - Robert G. Wilson v, Dec 18 2014
n^2/s is only an integer iff n is prime. - Robert G. Wilson v, Dec 18 2014
First occurrence of n >= 1: 4, 2, 3, 25, 5, 49, 7, ??? <= 35336848261, 2431, 121, 11, 169, 13, 6678671, 7429, 289, 17, 361, 19, 31367009, 20677, 529, 23, ..., . - Robert G. Wilson v, Dec 18 2014

			For n = 2431 = 11*13*17, we have (as the eight divisors of 2431 are [1, 11, 13, 17, 143, 187, 221, 2431]) a(n) = floor((2431*2431) / ((1*11)+(11*13)+(13*17)+(17*143)+(143*187)+(187*221)+(221*2431))) = floor(5909761/608125) = floor(9.718) = 9.

Michel Lagneau, Table of n, a(n) for n = 2..10000
International Mathematical Olympiad, Problems, IMO-2002, Problem 4.

Cf. A000040 (prime numbers), A005843 (even numbers), A016945 (6n+3), A084967 (GCD( 5k, 6) =1), A084968 (GCD( 7k, 30) =1), A084969 (GCD( 11k, 30) =1), A084970 (Numbers whose smallest prime factor is 13).
Cf. also A020639 (the smallest prime divisor), A055396 (its index) and arrays A083140 and A083221 (Sieve of Eratosthenes).
Differs from A250480 for the first time at n = 2431, where a(2431) = 9, while A250480(2431) = 10.
Cf. A078730 (sum of products of two successive divisors of n).

with(numtheory):nn:=100:
for n from 2 to nn do:
   x:=divisors(n):n0:=nops(x):s:=sum('x[i]*x[i+1]','i'=1..n0-1):
   z:=floor(n^2/s):printf(`%d, `,z):
od:

f[n_] := Floor[ n^2/Plus @@ Times @@@ Partition[ Divisors@ n, 2, 1]]; Array[f, 81, 2] (* Robert G. Wilson v, Dec 18 2014 *)

a(n) <= A250480(n), and especially, for all composite n, a(n) < A020639(n). [Cf. the Comments section above.] - Antti Karttunen, Dec 09 2014
From Robert G. Wilson v, Dec 18 2014: (Start)
a(n) = floor(n^2/A078730(n));
a(n) = n iff n is prime. (End)

Comments section edited by Antti Karttunen, Dec 09 2014

Instances of n for which a(n) = 8 and 14 found by Robert G. Wilson v, Dec 18 2014

A376839 a(1) = 1. For n > 1 if A007947(a(n-1)) is in A002110, a(n) is the smallest prime not already a term. Otherwise a(n) is the least novel multiple of the smallest non divisor prime of a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 10, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 23, 26, 27, 28, 30, 29, 32, 31, 34, 33, 36, 37, 38, 39, 40, 42, 25, 44, 45, 46, 48, 41, 50, 51, 52, 54, 43, 56, 57, 58, 60, 47, 62, 63, 64, 53, 66, 35, 68, 69, 70, 72, 59, 74, 75

Offset: 1

David James Sycamore, Oct 06 2024

nonn

A non divisor prime of a(n-1) is any prime p < Gpf(a(n-1)) which does not divide a(n-1). A007947(a(n-1)) is in A002110 iff a(n-1) is a term in A055932. Sequence is conjectured to be a permutation of the natural numbers (A000027) with primes in order.

Scatterplot shows trajectories of numbers whose smallest prime factor is prime p, e.g., for p = 5, numbers in A084967, p = 7, those in A084968, p = 11 those in A084969, etc. - Michael De Vlieger, Oct 09 2024

			a(1) = 1 = A002110(0) so a(2) = 2 (smallest prime not already a term).
a(2) = 2 = A002110(1) so a(3) = 3.
a(3) = 3 not a term in A002110 so a(4) is least novel multiple of 2, the least non divisor prime of 3. Therefore a(4) = 4 since 2 has occurred earlier.
a(39) = 42, not a term in A002110 so a(40) = 25, the least novel multiple of 5, the smallest non divisor prime of 42.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue or purple, with the latter additionally representing powerful numbers that are not prime powers.

Cf. A000040, A000027, A002110, A007947, A055932, A084967, A084968, A084969, A372368, A376865.

nn = 120; c[] := False; m[] := 1; f[x_] := FactorInteger[x][[All, 1]];
  Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; j = 2; v = 3;
  Do[If[Or[IntegerQ@ Log2[j], And[EvenQ[j], Union@ Differences@ PrimePi[#] == {1}]],
     k = v; While[c[k*m[k]], m[k]++]; k *= m[k],
     k = 2; While[Divisible[j, k], k = NextPrime[k]];
     While[c[k*m[k]], m[k]++]; k *= m[k]] &[f[j]];
  Set[{a[n], c[k], j}, {k, True, k}];
  If[k == v, While[c[v], v = NextPrime[v]]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Oct 09 2024 *)

More terms from Michael De Vlieger, Oct 09 2024

A063163 Composite numbers which in base 7 contain their largest proper factor as a substring.

Original entry on oeis.org

49, 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 343, 371, 413, 427, 469, 497, 511, 539, 551, 553, 581, 623, 637, 679, 707, 721, 749, 763, 791, 833, 847, 889, 917, 931, 959, 973, 989, 1001, 1043, 1057, 1099, 1127, 1141, 1169, 1183, 1211, 1253

Offset: 1

Robert G. Wilson v, Aug 08 2001

Sequence contains every term of A084968 except 7. - Bill McEachen, Dec 29 2020

			91 = 160_7 and its largest proper factor is 13 = 16_7 where 16 is a substring of 160. - _Bill McEachen_, Dec 30 2020

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A062238, A032742, A084968.

Do[ If[ !PrimeQ[ n ] && StringPosition[ ToString[ FromDigits[ IntegerDigits[ n, 7 ] ] ], ToString[ FromDigits[ IntegerDigits[ Divisors[ n ] [ [ -2 ] ], 7 ] ] ] ] != {}, Print[ n ] ], {n, 2, 2000} ]
Select[Range[1300],CompositeQ[#]&&SequenceCount[IntegerDigits[#,7],IntegerDigits[ Divisors[#][[-2]],7]]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 21 2021 *)

isok(n)={mystr=digits(n,7);d=divisors(n);gpf=d[#d-1];seek=digits(gpf,7);ls=#seek;for(w=1,#mystr-ls+1,if(mystr[w]!=seek[1],next);for(h=1,ls-1,if(mystr[w+h]!=seek[h+1],break);if(h==ls-1,return(1))));return(0);} \\ Bill McEachen, Dec 31 2020

A120321 RF(7): refactorable numbers with 7 as smallest prime factor.

Original entry on oeis.org

117649, 208422380089, 567869252041, 2839760855281, 5534900853769, 17416274304961, 69980368892329, 104413920565969, 301855146292441, 558845013849409, 743702041351801, 1268163904241521, 2607614922465721

Offset: 1

Walter Kehowski, Jun 21 2006

nonn

Numbers that are odd squares, 7 is their smallest prime factor, and are refactorable.

See A033950 for references. For any prime p, p^(p-1) is the smallest element of RF(p), the refactorable numbers whose smallest prime factor is p. Thus 7^(7-1)=117649 is the first element. Other elements would also be 7^6*17^6 or 7^16*17^6. Here are the prime factorizations for the first 49 elements of RF7: (7^6), (7^6)*(11^6), (7^6)*(13^6), (7^6)*(17^6), (7^6)*(19^6), (7^6)*(23^6), (7^6)*(29^6), (7^6)*(31^6), (7^6)*(37^6), (7^6)*(41^6), (7^6)*(43^6), (7^6)*(47^6), (7^6)*(53^6), (7^6)*(59^6), (7^6)*(61^6), (7^6)*(67^6), (7^6)*(71^6), (7^6)*(73^6), (7^6)*(79^6), (7^6)*(83^6), (7^6)*(89^6), (7^12)*(13^6), (7^6)*(97^6), (7^6)*(101^6), (7^6)*(103^6), (7^6)*(107^6), (7^6)*(109^6), (7^6)*(113^6), (7^6)*(127^6), (7^6)*(131^6), (7^6)*(137^6), (7^6)*(139^6), (7^6)*(11^6)*(13^6), (7^6)*(149^6), (7^6)*(151^6), (7^6)*(157^6), (7^6)*(163^6), (7^6)*(167^6), (7^6)*(13^12), (7^6)*(173^6), (7^6)*(179^6), (7^6)*(181^6), (7^6)*(11^6)*(17^6), (7^6)*(191^6), (7^6)*(193^6), (7^6)*(197^6), (7^6)*(199^6), (7^6)*(11^6)*(19^6), (7^6)*(211^6).

			a(1) = 7^(7-1) = 117649.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A033950 and A084968.

Cf. A036896, A036897.

with(numtheory); p:=7: RF7:=[p^(p-1)]: P:=[seq(ithprime(i),i=2..pi(p)-1)]; for w to 1 do for j from 1 to 12^3 do k:=2*j+1; if andmap(z -> k mod z <> 0, P) then for s from 2 to p-1 by 2 do #accelerate creation n:=7^6*k^s; t:=tau(n); if not n in RF7 and (n mod t = 0) then RF7:=[op(RF7),n]; print(ifactor(n)); fi; od; fi; od od; RF7:=sort(RF7);

cp's OEIS Frontend

A332799 Numbers whose smallest prime factor is 17.

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A145011 First differences of A007775.

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A251758 Let n>=2 be a positive integer with divisors 1 = d_1 < d_2 < ... < d_k = n, and s = d_1*d_2 + d_2*d_3 + ... + d_(k-1)*d_k. The sequence lists the values a(n) = floor(n^2/s).

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A376839 a(1) = 1. For n > 1 if A007947(a(n-1)) is in A002110, a(n) is the smallest prime not already a term. Otherwise a(n) is the least novel multiple of the smallest non divisor prime of a(n-1).

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A063163 Composite numbers which in base 7 contain their largest proper factor as a substring.

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Mathematica

PARI

A120321 RF(7): refactorable numbers with 7 as smallest prime factor.

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Maple

A251758 Let n>=2 be a positive integer with divisors 1 = d_1 < d_2 < ... < d_k = n, and s = d_1d_2 + d_2d_3 + ... + d_(k-1)*d_k. The sequence lists the values a(n) = floor(n^2/s).