A025475 -id:A025475 - cp's OEIS frontend

A071139 Numbers k such that the sum of distinct primes dividing k is divisible by the largest prime dividing k.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 70, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 107, 109, 113, 120, 121, 125, 127, 128, 131, 137, 139, 140, 149, 150, 151, 157, 163, 167, 169, 173, 179, 180

Offset: 1

Labos Elemer, May 13 2002

nonn

All primes and prime powers are terms, as are certain other composites (see Example section).
If k is a term then every multiple of k having no prime factors other than those of k are also terms. E.g., since 286 = 2*11*13 is a term, so are 572 = 286*2 and 3146 = 286*11.
If k = 2*p*q where p and q are twin primes, then sum = 2+p+q = 2q is divisible by q, the largest prime factor, so 2*A037074 is a subsequence.

			30 = 2*3*5; sum of distinct prime factors is 2+3+5 = 10, which is divisible by 5, so 30 is a term;
2181270 = 2*3*5*7*13*17*47; sum of distinct prime factors is 2+3+5+7+13+17+47 = 94, which is divisible by 47, so 2181270 is a term.

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

Cf. A008472, A006530, A000961, A025475, A037074.

a071139 n = a071139_list !! (n-1)
a071139_list = filter (\x -> a008472 x `mod` a006530 x == 0) [2..]
-- Reinhard Zumkeller, Apr 18 2013

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=sb[n]/ma[n]; If[IntegerQ[s], Print[{n, ba[n]}]], {n, 2, 1000000}]

isok(n) = if (n != 1, my(f=factor(n)[,1]); (sum(k=1, #f~, f[k]) % vecmax(f)) == 0); \\ Michel Marcus, Jul 09 2018

A008472(k)/A006530(k) is an integer.

Edited by Jon E. Schoenfield, Jul 08 2018

A134619 Numbers such that the arithmetic mean of the cubes of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

20, 44, 188, 297, 336, 400, 425, 540, 575, 605, 704, 752, 764, 908, 912, 1025, 1053, 1124, 1172, 1183, 1365, 1380, 1412, 1420, 1452, 1475, 1484, 1519, 1604, 1625, 1809, 1844, 1856, 1936, 1953, 2107, 2192, 2205, 2255, 2320, 2325, 2348, 2368, 2372, 2468

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

			a(10)=605, since 605=5*11*11 and (5^3+11^3+11^3)/3=929 which is prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134600, A134602, A134605, A134608, A134613, A134617, A134621.

amcpfQ[n_]:=PrimeQ[Mean[Flatten[PadRight[{},#[[2]],#[[1]]]&/@FactorInteger[n]]^3]]; Select[ Range[ 2500],amcpfQ] (* Harvey P. Dale, Jun 06 2023 *)

lista(m) = {for (i=2, m, f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); s /= bigomega(i); if (type(s) == "t_INT" && isprime(s), print1(i, ", ")););} \\ Michel Marcus, Apr 14 2013

Minor edits by Hieronymus Fischer, May 06 2013

A376308 Run-compression of the sequence of first differences of prime-powers.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 4, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 4, 2, 6, 2, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 6, 4, 2, 4, 6, 2, 10, 2, 4, 2, 12, 4, 2, 4, 6, 2, 8, 5, 1, 6, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 14, 6, 4, 2, 4, 6, 2, 6, 4, 6, 8, 4, 8, 10, 2, 10

Offset: 1

Gus Wiseman, Sep 20 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The sequence of prime-powers (A246655) is:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...
The sequence of first differences (A057820) of prime-powers is:
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, ...
The run-compression is A376308 (this sequence).

For primes instead of prime-powers we have A037201, halved A373947.
For squarefree numbers instead of prime-powers we have A376305.
For run-lengths instead of compression we have A376309.
For run-sums instead of compression we have A376310.
For positions of first appearances we have A376341, sorted A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A003242 counts compressed compositions, ranks A333489.
A024619 and A361102 list non-prime-powers, differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A007916, A025475, A034296, A053289, A076259, A110969, A120430, A124767, A174965, A374251.

First/@Split[Differences[Select[Range[100],PrimePowerQ]]]

A134611 Nonprime numbers such that the root mean cube of their prime factors is an integer (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1512, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

The prime factors are taken with multiplicity.
All perfect prime powers (A025475) are included. First term not included in A025475 is a(30) = 1512 = A134613(2) = A134613(1).
Most terms have a last digit of 1 or 9 (i.e., 8326 out of 9000 terms). Mainly, this comes from the fact that all squares of primes are included. Since each prime > 10 has a last digit of 1, 3, 7 or 9, its square has a last digit of 1 or 9. In addition, m-th powers of primes have a last digit of 1, if m == 0 (mod 4), and have a last digit of 1 or 9 if m == 2 (mod 4), and have a 50% chance, roughly, for a last digit of 1 or 9, if m == 1 (mod 4) or m == 3 (mod 4). Since the number of terms <= N which are squares of primes is PrimePi(sqrt(N)) = A000720(sqrt(N)), it follows that the number of terms <= N which have a last digit of 1 or 9 is greater than PrimePi(sqrt(N)). This can be estimated as 2*N^(1/2)/log(N), approximately.

			a(6) = 25, since 25 = 5*5 and ((5^3+5^3)/2)^(1/3) = 5.
a(30) = 1512, since 1512 = 2*2*2*3*3*3*7 and ((3*2^3+3*3^3+7^3)/7)^(1/3) = 64^(1/3) = 4.

Hieronymus Fischer, Table of n, a(n) for n = 1..9000

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134600, A134602, A134605, A134608, A134617, A134619, A134621.

lista(m) = {for (i=2, m, if (! isprime(i), f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); s /= bigomega(i); if (type(s) == "t_INT" && ispower(s, 3), print1(i, ", "));););}  \\ Michel Marcus, Apr 14 2013

Edited by Hieronymus Fischer, May 30 2013

A188585 Moebius inversion of sequence A000688, the number of factorizations of n into prime powers greater than 1.

Original entry on oeis.org

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

Offset: 1

Marc Bogaerts, Apr 04 2011

Dirichlet convolution product of A000688 with the Moebius function.
It appears that a(n) is nonzero for n in A001694, the powerful numbers. - T. D. Noe, Apr 06 2011 [This is correct: a(n) > 0 if and only if n is in A001694. - Amiram Eldar, Jun 10 2025]
There is a similar sequence defined by b(n) = Product_{i} floor(e(i)/2) where n = Product_{p} p(i)^e(i) is the usual prime factorization, which differs from a(n) at n = 64, 128, 256, 512, 576, 729,.... - R. J. Mathar, Sep 18 2012 [This sequence is A365550. - Amiram Eldar, Jun 10 2025]
The number of unordered factorizations of n into 1 and prime powers p^e where p is prime and e >= 2 (A025475). - Amiram Eldar, Jun 10 2025

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000688, A001694, A002865, A008683, A025475, A365550.

mtrf:=function ( f, x )     # the Moebius inversion formula
    local  d;
    d := DivisorsInt( x );
    return Sum( d, function ( i )
            return f( i ) * MoebiusMu( (x / i) );
        end );
end;
nra:=function ( x )         # the number of Abelian groups of order x
    local  pp, ll;
    pp := PrimePowersInt( x );
    ll := [ 1 .. Size( pp ) / 2 ];
    return Product( List( 2 * ll, function ( i )
              return NrPartitions( pp[i] );
          end ) );
end;
a:=function ( n )
    return mtrf( nra, n );
end;

with(numtheory): with(combinat):
a:= n-> add(mobius(n/d) *mul(numbpart(i[2]),
        i=ifactors(d)[2]), d=divisors(n)):
seq(a(n), n=1..110);  # Alois P. Heinz, Apr 07 2011

MobiusTransform[a_List] := Module[{n = Length[a], b}, b = Table[0, {i, n}];Do[b[[i]] = Plus @@ (MoebiusMu[i/Divisors[i]] a[[Divisors[i]]]), {i, n}]; b]; A688[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; MobiusTransform[Array[A688, 100]] (* T. D. Noe, Apr 06 2011 *)
f[p_, e_] := PartitionsP[e] - PartitionsP[e-1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 10 2025 *)

a(n) = vecprod(apply(x -> numbpart(x)-numbpart(x-1), factor(n)[, 2])); \\ Amiram Eldar, Jun 10 2025

from math import prod
from sympy import partition, factorint
def A188585(n): return prod(partition(e)-partition(e-1) for e in factorint(n).values()) # Chai Wah Wu, Jun 10 2025

a(n) = Sum_{d|n} A008683(n/d) * A000688(d).
Dirichlet g.f.: Product_{k>=2} zeta(k*s). - Ilya Gutkovskiy, Nov 03 2020
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = Product_{k>=3} zeta(k/2) = 10.0301441966843566206076085895839492473559217336... - Vaclav Kotesovec, Apr 22 2025
Multiplicative with a(p^e) = A002865(e). - Amiram Eldar, Jun 10 2025

A376598 Points of nonzero curvature in the sequence of prime-powers inclusive (A000961).

Original entry on oeis.org

4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376596) are nonzero.

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 from all terms.

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with nonzeros at (A376598):
  4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, ...

Gus Wiseman, Points of nonzero curvature in the prime-powers inclusive.

The first differences were A057820, see also A376340.
First differences are A376309.
These are the nonzeros of A376596 (sorted firsts A376653, exclusive A376654).
The complement is A376597.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
`A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376597 (second differences), A376597 (inflections and undulations), A376653 (sorted firsts in second differences).
For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376601 (non-prime-power).
Cf. A025475, A053707, A069623, A093555, A174965, A251092, A333254, A336416, A361102.

Join@@Position[Sign[Differences[Select[Range[1000], #==1||PrimePowerQ[#]&],2]],1|-1]

A003624 Duffinian numbers: composite numbers k relatively prime to sigma(k).

Original entry on oeis.org

4, 8, 9, 16, 21, 25, 27, 32, 35, 36, 39, 49, 50, 55, 57, 63, 64, 65, 75, 77, 81, 85, 93, 98, 100, 111, 115, 119, 121, 125, 128, 129, 133, 143, 144, 155, 161, 169, 171, 175, 183, 185, 187, 189, 201, 203, 205, 209, 215, 217, 219, 221, 225, 235, 237, 242, 243, 245, 247

Offset: 1

N. J. A. Sloane, Mira Bernstein, Sep 19 1994

nonn

All prime powers greater than 1 are in the sequence. No factorial number can be a term. - Arkadiusz Wesolowski, Feb 16 2014
Even terms are in A088827. Any term also in A005153 is either an even square or twice an even square not divisible by 3. - Jaycob Coleman, Jun 08 2014
All primes satisfy the second condition since gcd(p, p+1) = 1, thus making this sequence a proper subset of A014567. - Robert G. Wilson v, Oct 02 2014

			4 is in the sequence since it is not a prime, its divisors 1, 2, and 4 sum to 7, and gcd(7, 4) = 1.
21 is in the sequences since it is not a prime, and its divisors 1, 3, 7, and 21 sum to 32, which is coprime to 21.

T. Koshy, Elementary number theory with applications, Academic Press, 2002, p. 141, exerc. 6,7,8 and 9.
L. Richard Duffy, The Duffinian numbers, Journal of Recreational Mathematics 12 (1979), pp. 112-115.
Peter Heichelheim, There exist five Duffinian consecutive integers but not six, Journal of Recreational Mathematics 14 (1981-1982), pp. 25-28.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 64.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Florian Luca, On the densities of some subsets of integers, Missouri Journal of Mathematical Sciences 19:3 (2007), pp. 167-170.
Rose Mary Zbiek, What can we say about the Duffinian numbers?, The Pentagon 42:2 (1983), pp. 99-109.

Cf. A000203, A002808, A014567, A025475.

a003624 n = a003624_list !! (n-1)
a003624_list = filter ((== 1) . a009194) a002808_list
-- Reinhard Zumkeller, Mar 23 2013

fQ[n_] := n != 1 && !PrimeQ[n] && GCD[n, DivisorSigma[1, n]] == 1; Select[ Range@ 280, fQ]

is(n)=gcd(n,sigma(n))==1&&!isprime(n) \\ Charles R Greathouse IV, Feb 13 2013

from math import gcd
from itertools import count, islice
from sympy import isprime, divisor_sigma
def A003624_gen(startvalue=2): # generator of terms
    return filter(lambda k:not isprime(k) and gcd(k,divisor_sigma(k))==1,count(max(startvalue,2)))
A003624_list = list(islice(A003624_gen(),30)) # Chai Wah Wu, Jul 06 2023

A009194(a(n)) * (1 - A010051(a(n))) = 1. - Reinhard Zumkeller, Mar 23 2013

a(n) >> n log log log n, see Luca. (Clearly excluding the primes only makes the n-th term larger.) - Charles R Greathouse IV, Feb 17 2014

A078137 Numbers which can be written as sum of squares>1.

Original entry on oeis.org

4, 8, 9, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 1

Reinhard Zumkeller, Nov 19 2002

A078134(a(n))>0.
Numbers which can be written as a sum of squares of primes. - Hieronymus Fischer, Nov 11 2007
Equivalently, numbers which can be written as a sum of squares of 2 and 3. Proof for numbers m>=24: if m=4*(k+6), k>=0, then m=(k+6)*2^2; if m=4*(k+6)+1 than m=(k+4)*2^2+3^2; if m=4*(k+6)+2 then m=(k+2)*2^2+2*3^2; if m=4*(k+6)+3 then m=k*2^2+3*3^2. Clearly, the numbers a(n)<24 can also be written as sums of squares of 2 and 3. Explicit representation as a sum of squares of 2 and 3 for numbers m>23: m=c*2^2+d*3^2, where c:=(floor(m/4) - 2*(m mod 4))>=0 and d:=m mod 4. - Hieronymus Fischer, Nov 11 2007

Index entries for sequences related to sums of squares
Eric Weisstein's World of Mathematics, Square Number.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Complement of A078135.

Cf. A000290, A078136, A078131, A001597, A025475, A078134, A078135, A078139, A090677, A134600, A134605, A134608, A134612, A134616, A134618, A134620.

Join[{4, 8, 9, 12, 13, 16, 17, 18, 20, 21, 22}, Range[24, 82]] (* Jean-François Alcover, Aug 01 2018 *)

a(n)=if(n>11,n+12,[4,8,9,12,13,16,17,18,20,21,22][n]) \\ Charles R Greathouse IV, Aug 21 2011

a(n)=n + 12 for n >= 12. - Hieronymus Fischer, Nov 11 2007

Edited by N. J. A. Sloane, Oct 17 2009 at the suggestion of R. J. Mathar.

A134617 Numbers such that the arithmetic mean of the squares of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

15, 20, 21, 28, 35, 39, 44, 48, 51, 52, 55, 65, 69, 85, 91, 92, 95, 108, 112, 115, 116, 129, 135, 141, 145, 159, 164, 172, 188, 189, 205, 208, 209, 215, 221, 225, 235, 236, 245, 249, 259, 268, 272, 295, 297, 299, 305, 309, 315, 316, 320, 325, 329, 339, 341, 365

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

			a(2)=20, since 20=2*2*5 and (2^2+2^2+5^2)/3=33/3=11.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134600, A134602, A134605, A134608, A134613, A134619, A134621.

amspQ[n_]:=PrimeQ[Mean[Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[ n]]^2]]; Select[Range[400],amspQ] (* Harvey P. Dale, Jan 21 2017 *)

Minor edits by the author, May 06 2013

A243056 If n is the i-th prime, p_i = A000040(i), then a(n) = i, otherwise the difference between the indices of the smallest and the largest prime dividing n: for n = p_i * ... * p_k, where p_i <= ... <= p_k, a(n) = (k-i); a(1) = 0 by convention.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 4, 0, 0, 2, 5, 1, 6, 3, 1, 0, 7, 1, 8, 2, 2, 4, 9, 1, 0, 5, 0, 3, 10, 2, 11, 0, 3, 6, 1, 1, 12, 7, 4, 2, 13, 3, 14, 4, 1, 8, 15, 1, 0, 2, 5, 5, 16, 1, 2, 3, 6, 9, 17, 2, 18, 10, 2, 0, 3, 4, 19, 6, 7, 3, 20, 1, 21, 11, 1, 7, 1, 5, 22, 2, 0, 12, 23, 3, 4, 13, 8, 4, 24, 2

Offset: 1

Antti Karttunen, May 31 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Useful when computing A243057 or A243059.
A025475 (prime powers that are not primes) gives the positions of zeros.
Differs from A241917 for the first time at n=18.
Cf. A243055, A241919, A242411, A049084, A027748, A055396, A061395.

(define (A243056 n) (if (prime? n) (A049084 n) (A243055 n)))

a(1) = 0, for n>1, if n = A000040(i), a(n) = i, otherwise a(n) = A061395(n) - A055396(n) = A243055(n).

cp's OEIS Frontend

A071139 Numbers k such that the sum of distinct primes dividing k is divisible by the largest prime dividing k.

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A134619 Numbers such that the arithmetic mean of the cubes of their prime factors (taken with multiplicity) is a prime.

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A376308 Run-compression of the sequence of first differences of prime-powers.

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A134611 Nonprime numbers such that the root mean cube of their prime factors is an integer (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

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A188585 Moebius inversion of sequence A000688, the number of factorizations of n into prime powers greater than 1.

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A376598 Points of nonzero curvature in the sequence of prime-powers inclusive (A000961).

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A003624 Duffinian numbers: composite numbers k relatively prime to sigma(k).

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