A047201 -id:A047201 - cp's OEIS frontend

A236207 Numbers not divisible by 5 or 11.

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 67, 68, 69, 71, 72, 73, 74, 76, 78, 79, 81, 82, 83, 84, 86, 87, 89, 91, 92, 93, 94, 96, 97

Offset: 1

Oleg P. Kirillov, Jan 20 2014

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1).

Intersection of A047201 and A160542.

Select[Range[100], Mod[#, 5] > 0 && Mod[#, 11] > 0 &] (* or *) Select[Range[100], Or @@ Divisible[#, {5, 11}] == False &] (* Bruno Berselli, Mar 24 2014 *)

A236217 Numbers not divisible by 3, 5 or 11.

Original entry on oeis.org

1, 2, 4, 7, 8, 13, 14, 16, 17, 19, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 119, 122, 124, 127, 128

Offset: 1

Oleg P. Kirillov, Jan 20 2014

Numbers coprime to 165. The asymptotic density of this sequence is 16/33. - Amiram Eldar, Oct 23 2020

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Intersection of: A160542 and A229829; A047201 and A229968; A001651, A047201 and A160542.

Cf. A236206, A236208.

Select[Range[200], Mod[#, 3] > 0 && Mod[#, 5] > 0 && Mod[#, 11] > 0 &] (* or *) Select[Range[200], Or @@ Divisible[#, {3, 5, 11}] == False &] (* Bruno Berselli, Mar 24 2014 *)
Select[Range[130], CoprimeQ[165, #] &] (* Amiram Eldar, Oct 23 2020 *)

a(n) = a(n-1) + a(n-80) - a(n-81) for n > 81. - Bruno Berselli, Mar 25 2014

A344320 Number of partitions of n into consecutive parts not divisible by 5.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 3, 1, 1, 2, 2, 2, 3, 2, 2, 1, 2, 3, 1, 3, 2, 2, 2, 2, 2, 2, 3, 1, 3, 3, 2, 1, 3, 1, 3, 1, 4, 2, 1, 3, 3, 1, 4, 2, 2, 1, 3, 2, 2, 4, 1, 3, 3, 1, 2, 4, 3, 1, 2, 3, 1, 2, 4, 2, 4, 2, 3, 2, 1, 3, 2, 1, 4, 4, 2, 1, 3, 2, 2, 5

Offset: 1

Ilya Gutkovskiy, May 14 2021

nonn

			a(13) = 3 because we have [13], [7, 6] and [6, 4, 3].

Cf. A001227, A034178, A035959, A047201, A344318, A344319.

A374290 7-rough powerful numbers: numbers k coprime to 30 such that if a prime p divides k then p^2 also divides k.

Original entry on oeis.org

1, 49, 121, 169, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2197, 2209, 2401, 2809, 3481, 3721, 4489, 4913, 5041, 5329, 5929, 6241, 6859, 6889, 7921, 8281, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 14161, 14641, 16129, 16807, 17161, 17689, 18769

Offset: 1

Amiram Eldar, Jul 02 2024

This sequence is closed under multiplication.

The least term that is not a power of a prime (A000961) is a(25) = 7^2*11^2 = 5929.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Rough Number.
Index entries for sequences related to powerful numbers.

Intersection of A007775 and A001694.
Intersection of A229829 and A062739.
Intersection of A047201 and A374289.
Cf. A000961, A082695.

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[20000], CoprimeQ[#, 30] && powQ[#] &]

is(k) = gcd(k, 30) == 1 && ispowerful(k);

Sum_{n>=1} 1/a(n) = 80*zeta(2)*zeta(3)/(147*zeta(6)) = (80/147) * A082695 = 1.05773955745... .

In general, the sum of reciprocals of the p-rough powerful numbers is (zeta(2)*zeta(3)/zeta(6)) * Product_{prime q < p} ((q-1)*q/(q^2-q+1)).

A053448 Multiplicative order of 5 mod m, where gcd(m, 5) = 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 2, 6, 5, 2, 4, 6, 4, 16, 6, 9, 6, 5, 22, 2, 4, 18, 6, 14, 3, 8, 10, 16, 6, 36, 9, 4, 20, 6, 42, 5, 22, 46, 4, 42, 16, 4, 52, 18, 6, 18, 14, 29, 30, 3, 6, 16, 10, 22, 16, 22, 5, 6, 72, 36, 9, 30, 4, 39, 54, 20, 82, 6, 42, 14, 10, 44, 12, 22, 6, 46, 8, 96, 42, 30, 25, 16

Offset: 1

David W. Wilson

nonn

Essentially the same as A050977. - R. J. Mathar, Oct 21 2012

Cf. A047201, A002326 (order of 2), A053446 (order of 3), A053447 (order of 4).

MultiplicativeOrder[5, #] & /@ Select[ Range@ 100, GCD[5, #] == 1 &] (* Robert G. Wilson v, Apr 05 2011 *)

lista(nn) = {for(n=1, nn, if (gcd(n, 5) == 1, print1(znorder(Mod(5, n)), ", ")););} \\ Michel Marcus, Feb 09 2015

a(n) = multiplicative order of 5 modulo floor((5*n-1)/4), for n >= 1. This modulus is A047201(n). - Wolfdieter Lang, Sep 30 2020

A336450 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2z^3 = 25^6.

Original entry on oeis.org

1, -3, 4, 9, 16, 25, 36, 49, -56, 64, 81, 88, -104, 121, 144, -167, 169, 177, 196, -203, -243, -255, 256, 277, 289, 324, 361, -363, 373, -395, -411, 441, 484, 529, 576, 676, 709, -719, 729, 784, 841, 961, 1017, 1024, -1028, 1080, 1089, -1091, 1156, 1296, 1369

Offset: 1

XU Pingya, Aug 08 2020

sign

Terms are arranged in order of increasing absolute value (if equal, the negative number comes first).

Let x = a^(2*m) - (a^m)*t - t^2, y = a^(2*m) + (a^m)*t - t^2, z = t^2; then x^3 + y^3 + 2*z^3 = 2*a^(6*m). When a = 5, m = 1, t = 5*n + k(k = {1, 2, 3, 4}); (x, y, z) are primitive solutions of equation. Thus, A047201(n)^2 are terms of the sequence.

			(-20)^3 + 34^3 + 2*(-3)^3 = 31250, -3 is a term.
(-11)^3 + 29^3 + 2*16^3 = 15^3 + 27^3 + 2*16^3 = 31250, 16 is a term.

R. K. Guy, Unsolved Problems in Number Theory, D5.

Cf. A000290, A000578, A003215, A004825, A004826, A047201, A050791, A130472, A195006, A336449.

Clear[t]
t = {};
Do[y = (31250 - x^3 - 2z^3)^(1/3) /. (-1)^(1/3) -> -1;
If[IntegerQ[y] && GCD[x, y, z] == 1, AppendTo[t, z]], {z, -1369, 1369}, {x, -Round[(Abs[31250 - 2z^3]/3)^(1/2)], Round[(Abs[31250 - 2z^3]/3)^(1/2)]}]
u = Union@t;
v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 2739}];
Select[v, MemberQ[u, #] &]

A348306 List of Agathokakological Numbers "k": string of digits of the juxtaposition of the prime factors of k has the same length as k but these digits do not appear in k.

Original entry on oeis.org

10, 14, 21, 49, 106, 111, 118, 129, 134, 146, 158, 161, 166, 177, 201, 219, 249, 259, 267, 329, 343, 413, 511, 553, 623, 1011, 1029, 1046, 1077, 1081, 1101, 1106, 1114, 1119, 1138, 1149, 1167, 1186, 1227, 1299, 1318, 1354, 1358, 1363, 1418, 1454, 1466, 1538, 1541, 1546, 1561, 1589, 1591

Offset: 1

Samuel Harkness, Oct 11 2021

Theorem: (See PDF "PROOFS" in Links)
Of Agathokakological Numbers k,
No k have a leading 9.
No k end in 2 or 5.
10 is the only k to end in 0. It is also the only k with 5 as a prime factor.
Can only be square terms when k is of the order 10^m where m is odd.
For k written as a*10^m, k can only be even when 1<=a<1.888...
Empirical observation: When graphed with the log of the n-th term on x axis and the log of the n-th term's value on the y axis a pattern appears with a similar shape for each new power of ten (see figure "LogLogGraph" in Links)
Special cases 28651 = 7*4093 and 65821 = 7*9043 use all digits 0-9 once.
"Agathokakological" is a Greek word meaning "composed of both good and evil." (Merriam-Webster) The composition (prime factorization) of Agathokakological Numbers is both good (same length) and evil (no common digits).

			158 = 2 * 79 since {2,7,9} do not appear in {1,5,8} and both have 3 digits.

Samuel Harkness, Table of n, a(n) for n = 1..6388
Samuel Harkness, MATLAB
Samuel Harkness, LogLogGraph
Samuel Harkness, PROOFS

Cf. A055642, A076649, A280928.
Intersection of A035139 and A109608.
Subsequence of A047201 from n=2.

q[n_] := Module[{d = IntegerDigits[n], f = FactorInteger[n]}, Length[d] == Plus @@ ((Last[#]*IntegerLength[First[#]]) & /@ f ) && Intersection[d, Join @@ IntegerDigits[f[[;; , 1]]]] == {}]; Select[Range[1600], q] (* Amiram Eldar, Oct 12 2021 *)

digsf(n) = my(f=factor(n), list=List()); for (k=1, #f~, my(dk=digits(f[k,1])); for (i=1, f[k,2], for (j=1, #dk, listput(list, dk[j])))); Vec(list);
isokd(m) = my(df=digsf(m), d=digits(m)); (#df == #d) && (#setintersect(Set(df), Set(d)) == 0); \\ Michel Marcus, Oct 11 2021

from sympy import factorint
def ok(n):
    s, f = str(n), factorint(n)
    pfd = set("".join(str(p) for p in f))
    if set(s) & pfd != set(): return False
    return len(s) == sum(len(str(p))*f[p] for p in f)
print(list(filter(ok, range(1601)))) # Michael S. Branicky, Oct 11 2021

A361913 a(n) is the number of steps in the main loop of the Pollard rho integer factorization algorithm for n, with x=2, y=2 and g(x)=x^2-1.

Original entry on oeis.org

2, 2, 2, 1, 2, 4, 2, 2, 1, 2, 2, 3, 2, 1, 2, 4, 2, 3, 1, 2, 2, 8, 2, 1, 2, 2, 2, 4, 1, 4, 2, 2, 2, 1, 2, 6, 2, 2, 1, 5, 2, 6, 2, 1, 2, 11, 2, 4, 1, 2, 2, 5, 2, 1, 2, 2, 2, 7, 1, 3, 2, 2, 2, 1, 2, 4, 2, 2, 1, 3, 2, 8, 2, 1, 2, 2, 2, 10, 1, 2, 2, 12, 2, 1, 2, 2

Offset: 2

Darío Clavijo, Mar 29 2023

nonn

x=2 and y=2 are the minimum effective values for Pollard rho, but any x = y > 2 would give the same answer.
n is in A217562 if gcd(n, a(n)) > 1.
n is in A047201 if gcd(phi(n), a(n)) > 1, where phi is Euler's totient function.

Eric Weisstein's World of Mathematics, Pollard rho Factorization Method.
Wikipedia, Pollard's rho algorithm

Cf. A005563.

from gmpy2 import *
def a(n):
  c,d,x,y,g = 0,1,2,2,lambda x:pow(x,2,n)-1
  while d == 1:
    c,x,y =c+1,g(x),g(g(y))
    d = gcd(abs(x-y), n)
  return c

A380233 Odd abundant numbers not divisible by 5 that are also doublets (cf. A020338).

Original entry on oeis.org

153153, 171171, 189189, 207207, 243243, 261261, 279279, 297297, 351351, 459459, 513513, 567567, 621621, 729729, 783783, 837837, 891891, 999999, 1392313923, 1556115561, 1719917199, 1883718837, 2034920349, 2211322113, 2375123751, 2538925389, 2702727027, 3194131941, 4176941769, 4668346683

Offset: 1

Omar E. Pol, Jan 17 2025

There are 26 odd abundant numbers not divisible by 5 less than 10^6. The surprising fact is that 18 of them are doublets.

Another interesting fact is that here there are no terms with 8 digits.

Intersection of A064001 and A020338.

Cf. A005101, A005231, A047201.

doublet:= n -> n * (10^(1+ilog10(n))+1):
select(t -> numtheory:-sigma(t) > 2*t, [seq(seq(doublet(10*x+i),i=[1,3,7,9]),x=1..10000); # Robert Israel, Jan 17 2025

Select[Table[FromDigits[Join[#, #] &@ IntegerDigits[n]], {n, Select[Range[50000], CoprimeQ[#, 10] &]}], DivisorSigma[-1, #] > 2 &] (* Amiram Eldar, Jan 17 2025 *)

select(x->((x%5) && (sigma(x)>2*x)), vector(50000, n, eval(Str(2*n-1, 2*n-1)))) \\ Michel Marcus, Jan 17 2025

More terms from Michel Marcus, Jan 17 2025

A274466 Least number that is not divisible by 5 and the sum of two positive squares in exactly n ways.

Original entry on oeis.org

2, 221, 2873, 6409, 97682, 83317, 8254129, 237133, 1416389, 14080573, 2789895602, 3082729, 41075281, 82150562, 239369741, 9722453, 403139914489, 52406393, 6733146600397009, 520981201, 40453486229, 6941722489, 13883444978, 126391889, 69177855149

Offset: 1

Altug Alkan, Jun 24 2016

nonn

			a(2) = 221 because 221 = 5^2 + 14^2 = 10^2 + 11^2.

Index entries for sequences related to sums of squares

Cf. A016032, A047201.

nR[n_] := (SquaresR[2, n] + Plus @@ Pick[{-4, 4}, IntegerQ /@ Sqrt[{n, n/2}]])/8; qn[w_] := Block[{z = Floor[(1/2) Times @@ (w + 1)]}, If[And @@ (EvenQ /@ w), z + {0, 1}, {z}]]; ric[w_, t_] := Block[{u = qn[w]}, If[ MemberQ[u, t], Sow@w];If[ Min[u] < t, ric[Append[w, 1], t]; u = w; u[[-1]]++; If[Length[u] == 1 || u[[-1]] <= u[[-2]], ric[u, t]]]]; val[w_] := {1, 2} Times @@ (Take[{13, 17, 29, 37, 41, 53}, Length@w]^w); a[1] =2; a[n_] := Min@ Select[ Flatten[ val /@ Reap[ric[{1}, n]][[2, 1]]], nR[#] == n &]; Array[a, 25] (* Giovanni Resta, Jun 27 2016 *)

a(7)-a(25) from Giovanni Resta, Jun 27 2016

cp's OEIS Frontend

A236207 Numbers not divisible by 5 or 11.

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A236217 Numbers not divisible by 3, 5 or 11.

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A344320 Number of partitions of n into consecutive parts not divisible by 5.

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A374290 7-rough powerful numbers: numbers k coprime to 30 such that if a prime p divides k then p^2 also divides k.

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A053448 Multiplicative order of 5 mod m, where gcd(m, 5) = 1.

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A336450 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 2*5^6.

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A348306 List of Agathokakological Numbers "k": string of digits of the juxtaposition of the prime factors of k has the same length as k but these digits do not appear in k.

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A361913 a(n) is the number of steps in the main loop of the Pollard rho integer factorization algorithm for n, with x=2, y=2 and g(x)=x^2-1.

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A380233 Odd abundant numbers not divisible by 5 that are also doublets (cf. A020338).

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A336450 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2z^3 = 25^6.