A000430 -id:A000430 - cp's OEIS frontend

A109810 Number of permutations of the positive divisors of n, where every element is coprime to its adjacent elements.

1, 2, 2, 2, 2, 4, 2, 0, 2, 4, 2, 0, 2, 4, 4, 0, 2, 0, 2, 0, 4, 4, 2, 0, 2, 4, 0, 0, 2, 0, 2, 0, 4, 4, 4, 0, 2, 4, 4, 0, 2, 0, 2, 0, 0, 4, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 4, 4, 2, 0, 2, 4, 0, 0, 4, 0, 2, 0, 4, 0, 2, 0, 2, 4, 0, 0, 4, 0, 2, 0, 0, 4, 2, 0, 4, 4, 4, 0, 2, 0, 4, 0, 4, 4, 4, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0

Offset: 1

Leroy Quet, Aug 16 2005

nonn

Depends only on prime signature. - Reinhard Zumkeller, May 24 2010

			The divisors of 6 are 1, 2, 3 and 6. Of the permutations of these integers, only (6,1,2,3), (6,1,3,2), (2,3,1,6) and (3,2,1,6) are such that every pair of adjacent elements is coprime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A178254. - Reinhard Zumkeller, May 24 2010

a(1)=1, a(p) = 2, a(p^2) = 2, a(p*q) = 4 (where p and q are distinct primes), all other terms are 0.

a(A033942(n))=0; a(A037143(n))>0; a(A000430(n))=2; a(A006881(n))=4. - Reinhard Zumkeller, May 24 2010

Terms 17 to 59 from Diana L. Mecum, Jul 18 2008

More terms from David Wasserman, Oct 01 2008

A284288 Numbers n such that the average of the strong divisors of n is an integer.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 28, 29, 31, 37, 41, 43, 47, 49, 53, 54, 56, 59, 61, 64, 67, 68, 71, 73, 79, 81, 83, 89, 91, 97, 98, 99, 100, 101, 103, 107, 109, 113, 121, 127, 131, 133, 137, 138, 139, 148, 149, 151, 154, 157, 163, 165, 167, 169, 173, 179, 181, 188, 191, 192, 193, 197, 199

Offset: 1

Ilya Gutkovskiy, Mar 24 2017

nonn

We say d is a strong divisor of n iff d is a divisor of n and d > 1.
Numbers n such that A032741(n) divides A039653(n).
All primes and squares of primes are in this sequence.
Positions of ones in A296082 and A296084. - Antti Karttunen, Dec 05 2017

			28 is in the sequence because 28 has 6 divisors {1, 2, 4, 7, 14, 28} therefore 5 strong divisors {2, 4, 7, 14, 28}, 2 + 4 + 7 + 14 + 28 = 55 and 5 divides 55.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors

Cf. A000203, A000430, A003601, A023884, A023886, A032741, A039653, A296082, A296084 (characteristic function).

filter:= proc(n) local d,t;
  d:= numtheory:-divisors(n) minus {1};
  convert(d,`+`) mod nops(d) = 0
end proc:
select(filter, [$2..1000]); # Robert Israel, Mar 27 2017

Select[Range[2, 200], Mod[DivisorSigma[1, #1] - 1, DivisorSigma[0, #1] - 1] == 0 &]

for(n=2, 200, if((sigma(n) - 1)%(numdiv(n) - 1)==0, print1(n,", "))) \\ Indranil Ghosh, Mar 24 2017

from sympy.ntheory import divisor_sigma, divisor_count
print([n for n in range(2, 201) if (divisor_sigma(n) - 1)%(divisor_count(n) - 1) == 0]) # Indranil Ghosh, Mar 24 2017

A087797 Primes, squares of primes and cubes of primes.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Benoit Cloitre, Oct 10 2003

nonn

Union of A000040 and A168363. - Chai Wah Wu, Aug 09 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A000430, A002760, A168363.

m=400;Union[Prime[Range[PrimePi[m]]],Prime[Range[PrimePi[m^(1/2)]]]^2,Prime[Range[PrimePi[m^(1/3)]]]^3] (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)
With[{nn=70},Take[Union[Flatten[{#,#^2,#^3}&/@Prime[Range[nn]]]],nn]] (* Harvey P. Dale, Oct 16 2012 *)

is(n)=my(t=isprimepower(n)); t && t<4 \\ Charles R Greathouse IV, Oct 19 2015

from math import isqrt
from sympy import primepi, integer_nthroot
def A087797(n):
    def f(x): return n+x-primepi(x)-primepi(isqrt(x))-primepi(integer_nthroot(x,3)[0])
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m) # Chai Wah Wu, Aug 09 2024

a(n) ~ n log n. - Charles R Greathouse IV, Oct 19 2015

A088382 Numbers not exceeding the 4th power of their smallest prime factor.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133

Offset: 1

Reinhard Zumkeller, Sep 28 2003

nonn

a(n) <= A020639(a(n))^4 = A088379(a(n)); complement of A088383;

a(n) < A088383(k) for n <= 67, a(n) > A088383(k) for n > 67.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A088380, A000430, A020639, A088379, A088383.

Positions of numbers less than 5 in A307908.

a088382 n = a088382_list !! (n-1)
a088382_list = [x | x <- [1..], x <= a020639 x ^ 4]
-- Reinhard Zumkeller, Feb 06 2015

Select[Range[200],#<=FactorInteger[#][[1,1]]^4&] (* Harvey P. Dale, Jan 25 2015 *)

A088380 Numbers not exceeding the cube of their smallest prime factor.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169

Offset: 1

Reinhard Zumkeller, Sep 28 2003

nonn

a(n) <= A020639(a(n))^3 = A088378(a(n)); complement of A088381;

a(n) < A088381(k) for n <= 28, a(n) > A088381(k) for n > 28.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000430, A088382, A020639, A088378, A088381.

Positions of numbers less than 4 in A307908.

a088380 n = a088382_list !! (n-1)
a088380_list = [x | x <- [1..], x <= a020639 x ^ 3]
-- Reinhard Zumkeller, Feb 06 2015

Select[Range[200],#<=FactorInteger[#][[1,1]]^3&] (* Harvey P. Dale, Apr 28 2022 *)

A371734 Maximal length of a factorization of n into factors > 1 all having different sums of prime indices.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 13 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Sum of prime indices is given by A056239.

Factorizations into factors > 1 all having different sums of prime indices are counted by A321469.

			The factorizations of 90 of this type are (2*3*15), (2*5*9), (2*45), (3*30), (5*18), (6*15), (90), so a(90) = 3.

For set partitions of binary indices we have A000120, same sums A371735.
Positions of 1's are A000430.
Positions of terms > 1 are A080257.
Factorizations of this type are counted by A321469, same sums A321455.
For same instead of different sums we have A371733.
A001055 counts factorizations.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
Cf. A035470, A279787, A305551, A321142, A322794, A326515, A326518, A326534, A336137, A371783, A371789, A371791.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Table[Max[Length/@Select[facs[n],UnsameQ@@hwt/@#&]],{n,100}]

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
all_have_different_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == #facs));
A371734(n, m=n, facs=List([])) = if(1==n, if(all_have_different_sum_of_pis(facs),#facs,0), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s = max(s,A371734(n/d, d, newfacs)))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A232175 Least positive k such that n^3 + k^2 is a square, or 0 if there is no such k.

Original entry on oeis.org

0, 1, 3, 6, 10, 3, 21, 8, 36, 15, 55, 6, 78, 35, 15, 48, 136, 27, 171, 10, 42, 99, 253, 10, 300, 143, 81, 42, 406, 15, 465, 64, 88, 255, 35, 63, 666, 323, 91, 3, 820, 21, 903, 55, 66, 483, 1081, 48, 1176, 125, 85, 39, 1378, 81, 165, 28, 76, 783, 1711, 15, 1830, 899, 63

Offset: 1

Alex Ratushnyak, Nov 19 2013

Numbers n such that a(n) = n*(n-1)/2 appear to be A000430.

n = 1 is the only number for which a(n) = 0. - T. D. Noe, Nov 21 2013

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from T. D. Noe).
StackExchange, The cube of integer can be written as the difference of two square.

Cf. A000290, A000430, A000578, A038202, A055527, A232176.

Join[{0}, Table[k = 1; While[! IntegerQ[Sqrt[n^3 + k^2]], k++]; k, {n, 2, 100}]] (* T. D. Noe, Nov 21 2013 *)

a(n) = {k = 1; while (!issquare(n^3+k^2), k++); k;} \\ Michel Marcus, Nov 20 2013

import math
for n in range(77):
   n3 = n*n*n
   y=1
   for k in range(1, 10000001):
     s = n3 + k*k
     r = int(math.sqrt(s))
     if r*r == s:
       print(k, end=', ')
       y=0
       break
   if y: print(end='-, ')

from _future_ import division
from sympy import divisors
def A232175(n):
    n3 = n**3
    ds = divisors(n3)
    for i in range(len(ds)//2-1,-1,-1):
        x = ds[i]
        y = n3//x
        a, b = divmod(y-x,2)
        if not b:
            return a
    return 0 # Chai Wah Wu, Sep 12 2017

A342657 The difference between floor(log_2(.)) of and the number of prime factors in A156552(n) (when counted with multiplicity).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 1, 1, 0, 2, 0, 3, 1, 3, 0, 3, 0, 4, 1, 3, 0, 2, 0, 3, 3, 5, 1, 1, 0, 7, 3, 3, 0, 4, 0, 5, 2, 5, 0, 4, 0, 2, 4, 6, 0, 3, 1, 5, 5, 7, 0, 4, 0, 9, 3, 2, 3, 4, 0, 6, 7, 4, 0, 3, 0, 10, 2, 7, 1, 5, 0, 5, 1, 11, 0, 3, 3, 11, 5, 3, 0, 2, 1, 8, 7, 11, 4, 4, 0, 3, 3, 3, 0, 5, 0, 7, 2

Offset: 2

Antti Karttunen, Mar 18 2021

nonn

Cf. A000430 (positions of 0's), A000523, A001222, A003961, A061395, A070939, A156552, A246277, A322993, A325134, A339893, A342655, A342656.

Cf. also A339895.

A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
A342657(n) = { my(u=A156552(n)); (#binary(u)-bigomega(u))-1; };

a(n) = A339893(A156552(n)) = A339893(A322993(n)) = A000523(A156552(n)) - A001222(A156552(n)).
a(n) = (A252464(n)-A342655(n))-1 = (A325134(n)-A342655(n)) - 2.
a(p) = a(p^2) = 0 for all primes p. (Second part added Jul 27 2023)
a(A003961(n)) = a(2*A246277(n)) = a(n).

A381747 a(n) is the number of solutions to tau(x) + tau(n-x) = tau(n) where 1 <= x <= floor(n/2).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 1, 2, 0, 2, 0, 1, 1, 3, 0, 1, 0, 4, 0, 3, 0, 3, 0, 3, 1, 4, 0, 1, 0, 2, 1, 2, 0, 3, 0, 5, 6, 4, 0, 0, 0, 5, 0, 5, 0, 5, 1, 4, 0, 4, 0, 2, 0, 3, 6, 4, 0, 5, 0, 8, 1, 5, 0, 3, 0, 5, 8, 8, 0, 5, 0, 3, 0, 5, 0, 3, 1, 5, 0, 6

Offset: 1

Felix Huber, Mar 30 2025

Observation: For even positive multiples of 48, k <= 17000, a(k) = 0 only for k = 1*48, 2304 = 48*48 and 3600 = 75*48. The next numbers k which are multiples of 48 and for which a(k) = 0 are 46656, 63504, 233280, 513216, 793152, 2286144, 3111696.

			a(10) = 3 because tau(x) + tau(10-x) = tau(10) has 3 solutions for 0 <= x <= 5:
  x = 1: tau(1) + tau(9) = 1 + 3 = 4 = tau(10);
  x = 3: tau(3) + tau(7) = 2 + 2 = 4 = tau(10);
  x = 5: tau(5) + tau(5) = 2 + 2 = 4 = tau(10).

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A000005, A000430, A211225, A382074.

with(NumberTheory):
A381747:=proc(n)
    local a,x;
    a:=0;
    for x to n/2 do
        if tau(x)+tau(n-x)=tau(n) then
            a:=a+1
        fi
    od;
    return a
end proc;
seq(A381747(n),n=1..88);

a(n) = my(nd=numdiv(n)); sum(x=1, n\2, numdiv(x)+numdiv(n-x) == nd); \\ Michel Marcus, Apr 26 2025

a(A000430(n)) = 0 for n > 3.

A385378 The maximum possible number of distinct factors in the factorization of n into prime powers (A246655).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 27 2025

Differs from A376885 and A384422 at n = 32, 64, 96, 128, 160, 192, ... .
Differs from A086435 at n = 36, 100, 144, 180, 196, 225, ... .
Differs from A375272 at n = 128, 384, 640, 896, 1024, 1152, ...	.
a(n) depends only on the prime signature of n (A118914).
The indices of records in this sequence are the partial products of the sequence of powers of primes (A000961), i.e., the terms in A024923.
The least index n such that a(n) = k, for k = 0, 1, 2, ..., is A024923(k+1).

			      n | a(n) | factorization
  ------+------+--------------------------------
      2 |  1   | 2
      6 |  2   | 2 * 3
     24 |  3   | 2 * 3 * 2^2
    120 |  4   | 2 * 3 * 2^2 * 5
    840 |  5   | 2 * 3 * 2^2 * 5 * 7
   6720 |  6   | 2 * 3 * 2^2 * 5 * 7 * 2^3
  60480 |  7   | 2 * 3 * 2^2 * 5 * 7 * 2^3 * 3^2

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

Cf. A000430, A000961, A001221, A003056, A004709, A024923, A077761, A086435, A118914, A246655, A254578, A375272, A376885, A384422, A385379.

f[p_, e_] := Floor[(Sqrt[8*e + 1] - 1)/2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> (sqrtint(8*x+1)-1)\2 , factor(n)[, 2]));

Additive with a(p^e) = A003056(e).
a(n) >= A001221(n), with equality if and only if n is cubefree (A004709).
a(n) >= 1 for n >= 2, with equality if and only if n is a prime or a square of a prime (A000430).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{k>=2} P(k*(k+1)/2) = 0.19285739770001405035..., and P is the prime zeta function.

cp's OEIS Frontend

A109810 Number of permutations of the positive divisors of n, where every element is coprime to its adjacent elements.

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A284288 Numbers n such that the average of the strong divisors of n is an integer.

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Maple

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A087797 Primes, squares of primes and cubes of primes.

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Mathematica

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A088382 Numbers not exceeding the 4th power of their smallest prime factor.

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Haskell

Mathematica

A088380 Numbers not exceeding the cube of their smallest prime factor.

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Haskell

Mathematica

A371734 Maximal length of a factorization of n into factors > 1 all having different sums of prime indices.

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Mathematica

PARI

Extensions

A232175 Least positive k such that n^3 + k^2 is a square, or 0 if there is no such k.

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Mathematica

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