A053585 -id:A053585 - cp's OEIS frontend

A065202 Characteristic function of A065201: a(n) = if A065201(k) = n for some k then 1 else 0.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Oct 21 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for characteristic functions.

Cf. A051119, A053585, A065201, A065200, A070003, A107078.

a[n_] := If[Max[Most[FactorInteger[n][[;;, 2]]]] > 1, 1, 0]; Array[a, 100] (* Amiram Eldar, Mar 19 2025 *)

A053585(n) = if(n>1, my(f=factor(n)); f[#f~, 1]^f[#f~, 2], 1) \\ Charles R Greathouse IV, Nov 10 2015
A051119(n) = (n/A053585(n));
A065202(n) = (1-abs(moebius(A051119(n)))); \\ Antti Karttunen, Aug 27 2017

a(n) = {my(e = factor(n)[,2]); #e > 1 && vecmax(e[1..#e-1]) > 1;} \\ Amiram Eldar, Mar 19 2025

a(A065200(n)) = 0 and a(A065201(n)) = 1.

a(n) = A107078(A051119(n)). - Antti Karttunen, Aug 27 2017

Edited by Franklin T. Adams-Watters, Oct 27 2006

A309243 Completely multiplicative with a(p) = p * a(p-1) for any prime number p.

Original entry on oeis.org

1, 2, 6, 4, 20, 12, 84, 8, 36, 40, 440, 24, 312, 168, 120, 16, 272, 72, 1368, 80, 504, 880, 20240, 48, 400, 624, 216, 336, 9744, 240, 7440, 32, 2640, 544, 1680, 144, 5328, 2736, 1872, 160, 6560, 1008, 43344, 1760, 720, 40480, 1902560, 96, 7056, 800, 1632, 1248

Offset: 1

Rémy Sigrist, Jul 17 2019

All terms are distinct and belong to A064522.

			a(2) = 2 * a(1) = 2.
a(5) = 5 * a(4) = 5 * a(2)^2 = 5 * 2^2 = 20.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A000010, A006530, A053585, A064415, A064522.

a(n) = my (f=factor(n), p=f[,1]~, e=f[,2]~); prod (i=1, #p, (p[i] * a(p[i] - 1))^e[i])

a(n) >= n with equality iff n is a power of 2.
a(n) is a multiple of n.
a(n) is a multiple of A000010(n).
A006530(a(n)) = A006530(n).
A053585(a(n)) = A053585(n).
Apparently, A007814(a(n)) = A064415(n).

A387406 Numbers k such that sigma(A253560(k)) / A253560(k) is equal to (sigma(k)+1) / k, where A253560(k) = k multiplied by its largest prime factor.

Original entry on oeis.org

6, 18, 28, 54, 117, 162, 196, 486, 496, 775, 1372, 1458, 1521, 4374, 8128, 9604, 13122, 15376, 19773, 24025, 39366, 67228, 88723, 118098, 257049, 354294, 470596, 476656, 744775, 796797, 1032256, 1062882, 2896363, 3188646, 3294172, 3341637, 6725201, 9565938, 12326221, 14776336, 23059204, 23088025, 25774633, 27237961

Offset: 1

Antti Karttunen, Aug 30 2025

Terms k for which sigma(k/A053585(k)) = A006530(k). This further entails that A001221(k) = 2 [See A023194].

Antti Karttunen, Table of n, a(n) for n = 1..58 (larger b-file needed)
C. A. Holdener and J. A. Holdener, Characterizing Quasi-Friendly Divisors, Journal of Integer Sequences, Vol. 23 (2020), Article 20.8.4.
Index entries for sequences related to sigma(n)

Cf. A000203, A001221, A006530, A023194, A053585, A253560, A387405.

Subsequences: A000396 (even terms only), A240991 (conjectured, if true, then A000396 has only even terms).

fk[k_]:=k*FactorInteger[k][[-1,1]];Select[Range[10^6],DivisorSigma[1,fk[#]]/fk[#]==(DivisorSigma[1,#]+1)/#&] (* James C. McMahon, Aug 31 2025 *)

A253560(n) = if (n==1, 1, n*vecmax(factor(n)[, 1]));
isA387406(n) = { my(x=A253560(n)); ((sigma(x)/x) == ((sigma(n)+1)/n)); };

A304180 If n = Product (p_j^k_j) then a(n) = max{p_j}^max{k_j}.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 9, 13, 7, 5, 16, 17, 9, 19, 25, 7, 11, 23, 27, 25, 13, 27, 49, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 125, 41, 7, 43, 121, 25, 23, 47, 81, 49, 25, 17, 169, 53, 27, 11, 343, 19, 29, 59, 25, 61, 31, 49, 64, 13, 11, 67, 289, 23, 7, 71, 27, 73, 37, 25

Offset: 1

Ilya Gutkovskiy, May 07 2018

nonn

			a(40) = 125 because 40 = 2^3*5^1, max{2,5} = 5, max{3,1} = 3 and 5^3 = 125.

Ilya Gutkovskiy, Scatter plot of a(n) up to n=100000.
Eric Weisstein's World of Mathematics, Greatest Prime Factor.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000040, A000961 (fixed points), A002110, A005117, A006530, A034699, A051903, A053585, A073482, A081810, A304181.

Table[(FactorInteger[n][[-1, 1]])^(Max @@ Last /@ FactorInteger[n]), {n, 75}]

a(n) = if(n == 1, 1, my(f = factor(n), p = f[, 1], e = f[, 2]); vecmax(p)^vecmax(e)); \\ Amiram Eldar, Sep 08 2024

a(n) = A006530(n)^A051903(n).
a(p^k) = p^k where p is a prime.
a(A005117(k)) = A073482(k).
a(A002110(k)) = A000040(k).

A323129 a(1) = 1, and for any n > 1, let p be the greatest prime factor of n, and e be its exponent, then a(n) = p^a(e).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 9, 19, 5, 7, 11, 23, 3, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 49, 25, 17, 13, 53, 27, 11, 7, 19, 29, 59, 5, 61, 31, 7, 8, 13, 11, 67, 17, 23, 7, 71, 9, 73

Offset: 1

Rémy Sigrist, Jan 05 2019

This sequence is a recursive variant of A053585.

All terms belong to A164336.

			a(1458) = a(2 * 3^6) = 3^a(6) = 3^a(2*3) = 3^3 = 27.

Robert Israel, Table of n, a(n) for n = 1..10000

See A323130 for the variant involving the least prime factor.

Cf. A006530, A053585, A071178, A164336.

f:= proc(n) option remember;
  local F,t;
  F:= ifactors(n)[2];
  t:= F[max[index](map(t -> t[1],F))];
  t[1]^procname(t[2]);
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Jan 07 2019

Nest[Append[#, Last@ FactorInteger[Length[#] + 1] /. {p_, e_} :> p^#[[e]] ] &, {1}, 72] (* Michael De Vlieger, Jan 07 2019 *)

a(n) = if (n==1, 1, my (f=factor(n)); f[#f~,1]^a(f[#f~,2]))

a(n) <= n with equality iff n belongs to A164336.

a(n) = A006530(n)^a(A071178(n)) for any n > 1.

A338668 a(n) is the rightmost prime number in prime tower factorization of n; a(1) = 1.

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Apr 23 2021

nonn

The prime tower factorization of a number is defined in A182318.

			See Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of initial terms

Cf. A006530, A182318, A338669.

a(n) = if (n==1, 1, my (f=factor(n), w=#f~); if (f[w,2]==1, f[w,1], a(f[w,2])))

a(n) <= n with equality iff n = 1 or n is a prime number.

a(n) = a(A053585(n)).

A085236 (Greatest power of greatest prime factor of n) < square root(n).

Original entry on oeis.org

12, 24, 30, 40, 45, 48, 56, 60, 63, 70, 80, 84, 90, 96, 105, 112, 120, 126, 132, 135, 140, 144, 154, 160, 165, 168, 175, 176, 180, 182, 189, 192, 195, 198, 208, 210, 220, 224, 231, 234, 240, 252, 260, 264, 270, 273, 275, 280, 286, 288, 297, 306, 308, 312

Offset: 1

Reinhard Zumkeller, Jun 22 2003

nonn

A053585(a(n))^2 < a(n).

Robert Israel, Table of n, a(n) for n = 1..10000

filter:= proc(n) local F,j;
  F:= ifactors(n)[2];
  j:= max[index](map(t->t[1],F));
  F[j][1]^(2*F[j][2]) < n
end proc:
select(filter, [$2..1000]); # Robert Israel, Nov 30 2016

A362983 Number of prime factors of n (with multiplicity) that are greater than the least.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 1, 1, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 1, 2, 0, 1, 0, 1, 0, 2, 1, 1, 1

Offset: 1

Gus Wiseman, May 18 2023

nonn

			The prime factorization of 360 is 2*2*2*3*3*5, with factors greater than the least 3*3*5, so a(360) = 3.

Positions of 0's are A000961.
Positions of numbers > 0 are A024619.
Positions of first appearances appear to be A099856.
For "less than greatest" instead of "greater than least" we have A325226.
For multiplicities instead of parts we have A363131.
A027746 lists prime factors, A112798 indices, A124010 exponents.
A047966 counts uniform partitions, ranks A072774.
A363128 counts partitions with more than one non-mode, complement A363129.
Cf. A001221, A001222, A052126, A053585, A061395, A064989, A071178, A105441, A307517, A325230.

Table[PrimeOmega[n]-If[n==1,0,FactorInteger[n][[1,2]]],{n,30}]

a(n) = A001222(n) - A067029(n).

a(n) = A001222(A028234(n)).

A085235 (Greatest power of greatest prime factor of n) > square root(n).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81

Offset: 1

Reinhard Zumkeller, Jun 22 2003

nonn

A053585(a(n))^2 > a(n).

cp's OEIS Frontend

A065202 Characteristic function of A065201: a(n) = if A065201(k) = n for some k then 1 else 0.

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A309243 Completely multiplicative with a(p) = p * a(p-1) for any prime number p.

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A387406 Numbers k such that sigma(A253560(k)) / A253560(k) is equal to (sigma(k)+1) / k, where A253560(k) = k multiplied by its largest prime factor.

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A304180 If n = Product (p_j^k_j) then a(n) = max{p_j}^max{k_j}.

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A323129 a(1) = 1, and for any n > 1, let p be the greatest prime factor of n, and e be its exponent, then a(n) = p^a(e).

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A338668 a(n) is the rightmost prime number in prime tower factorization of n; a(1) = 1.

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A085236 (Greatest power of greatest prime factor of n) < square root(n).

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A362983 Number of prime factors of n (with multiplicity) that are greater than the least.

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