A024619 -id:A024619 - cp's OEIS frontend

A376599 Second differences of consecutive non-prime-powers inclusive (A024619). First differences of A375735.

-2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 1, -1, 1, -1, 0, 1, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0

Offset: 1

Gus Wiseman, Oct 02 2024

sign

Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, shift left once.

			The non-prime-powers inclusive (A024619) are:
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we had A375735, ones A375713(n) - 1.
Positions of zeros are A376600, complement A376601.
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers.
A057820 gives first differences of prime-powers inclusive, first appearances A376341, sorted A376340.
A321346/A321378 count integer partitions without prime-powers, factorizations A322452.
For non-prime-powers: A024619/A361102 (terms), A375735/A375708 (first differences), A376600 (inflections and undulations), A376601 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power).
Cf. A025475, A053707, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Differences[Select[Range[100],!(#==1||PrimePowerQ[#])&],2]

from sympy import primepi, integer_nthroot
def A376599(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return (a:=iterfun(f,n))-((b:=iterfun(lambda x:f(x)+1,a))<<1)+iterfun(lambda x:f(x)+2,b) # Chai Wah Wu, Oct 02 2024

A143731 Characteristic function of numbers with at least two distinct prime factors (A024619).

Original entry on oeis.org

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

Offset: 1

Reikku Kulon, Aug 30 2008

a(n) = 1 iff n is divisible by at least two distinct primes; otherwise 0.

Reikku Kulon, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions

Cf. A024619, A079275, A079872, A000961, A001222, A091304, A073093, A001221.

Cf. A215480.

Table[Boole[PrimeNu[n]>1],{n,1,200}] (* Enrique Pérez Herrero, Aug 13 2012 *)

a(n) = abs(A079872(n)) = A057427(A079275(n)).

For n>1: a(n) = 1 - floor(1/A001221(n)). - Enrique Pérez Herrero, Aug 13 2012

A373670 Numbers k such that the k-th run-length A110969(k) of the sequence of non-prime-powers (A024619) is different from all prior run-lengths.

Original entry on oeis.org

1, 5, 7, 12, 18, 28, 40, 53, 71, 109, 170, 190, 198, 207, 236, 303, 394, 457, 606, 774, 1069, 1100, 1225, 1881, 1930, 1952, 2247, 2281, 3140, 3368, 3451, 3493, 3713, 3862, 4595, 4685, 6625, 8063, 8121, 8783, 12359, 12650, 14471, 14979, 15901, 17129, 19155

Offset: 1

Gus Wiseman, Jun 15 2024

nonn

The unsorted version is A373669.

			The maximal runs of non-prime-powers begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60
So the a(n)-th runs begin:
   1
  14  15
  20  21  22
  33  34  35  36
  54  55  56  57  58

For nonsquarefree runs we have A373199 (if increasing), firsts of A053797.
For squarefree antiruns see A373200, unsorted A373128, firsts of A373127.
For composite runs we have A373400, unsorted A073051, firsts of A176246.
For prime antiruns we have A373402.
For runs of non-prime-powers:
- length A110969, firsts A373669, sorted A373670 (this sequence):
- min A373676
- max A373677
- sum A373678
For runs of prime-powers:
- length A174965
- min A373673
- max A373674
- sum A373675
A000961 lists the powers of primes (including 1).
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists the non-prime-powers, without 1 A024619.
Cf. A005381, A008864, A014963, A027833, A038664, A356068, A373401, A373574.

t=Length/@Split[Select[Range[10000],!PrimePowerQ[#]&],#1+1==#2&];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A376600 Inflection or undulation points in the sequence of non-prime-powers inclusive (A024619).

Original entry on oeis.org

2, 7, 9, 10, 11, 14, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 39, 41, 43, 44, 45, 47, 48, 50, 51, 52, 55, 56, 57, 58, 59, 62, 64, 66, 68, 70, 73, 74, 75, 76, 77, 80, 86, 87, 88, 90, 92, 93, 94, 95, 96, 97, 98, 100, 102, 103, 104, 107, 108, 109, 112, 114, 116

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376599) are zero.

Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, add 1 to all terms.

			The non-prime-powers inclusive are (A024619):
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...
with zeros at (A376600):
  2, 7, 9, 10, 11, 14, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 39, 41, 43, 44, ...

Gus Wiseman, Inflection and undulation points in the non-prime-powers inclusive.

For first differences we had A375735, ones A375713(n)-1.
These are the zeros of A376599.
The complement is A376601.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A024619/A361102 list non-prime-powers inclusive.
A321346/A321378 count integer partitions into non-prime-powers, factorizations A322452.
For non-prime-powers: A375735/A375708 (first differences), A376599 (second differences), A376601 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power).
Cf. A025475, A053707, A057820, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Join@@Position[Differences[Select[Range[100], !(#==1||PrimePowerQ[#])&],2],0]

A085971 Union of primes and numbers that are not prime powers (A000040, A024619).

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Reinhard Zumkeller, Jul 06 2003

Complement of A025475;
A085972(n) = Max{k: a(k)<=n};
different from A007916 and A052485, as a(28)=36;
A085818(a(n)) = A000040(n).

With[{nn=100},Union[Join[Prime[Range[PrimePi[nn]]],DeleteCases[Range[2,80], ?(PrimePowerQ[#]&)]]]] (* _Harvey P. Dale, May 15 2019 *)

is(n)=isprimepower(n)<2 && n>1 \\ Charles R Greathouse IV, Oct 19 2015

from sympy import primepi, integer_nthroot
def A085971(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 20 2024

a(n) = n + o(sqrt n). - Charles R Greathouse IV, Oct 19 2015

A376601 Points of nonzero curvature in the sequence of non-prime-powers inclusive (A024619).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 12, 13, 16, 17, 19, 21, 23, 25, 27, 28, 32, 34, 35, 36, 37, 38, 40, 42, 46, 49, 53, 54, 60, 61, 63, 65, 67, 69, 71, 72, 78, 79, 81, 82, 83, 84, 85, 89, 91, 99, 101, 105, 106, 110, 111, 113, 115, 117, 118, 122, 124, 132, 134, 136, 138, 148

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376599) are nonzero.

Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, subtract 1 and shift left.

			The non-prime-powers inclusive (A024619) are:
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...
with nonzero terms (A376601) at:
  1, 3, 4, 5, 6, 8, 12, 13, 16, 17, 19, 21, 23, 25, 27, 28, 32, 34, 35, 36, 37, ...

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

For first differences we had A375735, ones A375713(n) - 1.
These are the nonzeros of A376599.
The complement is A376600.
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers.
A024619/A361102 list non-prime-powers inclusive.
A057820 gives first differences of prime-powers inclusive.
A321346/A321378 count integer partitions into non-prime-powers, factorizations A322452.
For non-prime-powers: A375735/A375708 (first differences), A376599 (second differences), A376600 (inflections and undulations).
For nonzero curvature: A333214 (prime), A376603 (composite), A376588 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376598 (prime-power).
Cf. A025475, A053707, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Join@@Position[Sign[Differences[Select[Range[100], !(#==1||PrimePowerQ[#])&],2]],1|-1]

A089992 Second prime divisor of numbers that are not powers of primes (A024619).

Original entry on oeis.org

3, 5, 3, 7, 5, 3, 5, 7, 11, 3, 13, 7, 3, 11, 17, 7, 3, 19, 13, 5, 3, 11, 5, 23, 3, 5, 17, 13, 3, 11, 7, 19, 29, 3, 31, 7, 13, 3, 17, 23, 5, 3, 37, 5, 19, 11, 3, 5, 41, 3, 17, 43, 29, 11, 3, 13, 23, 31, 47, 19, 3, 7, 11, 5, 3, 13, 5, 53, 3, 5, 37, 7, 3, 23, 29, 13, 59, 17, 3, 61, 41, 31, 3, 43

Offset: 1

Cino Hilliard, Jan 14 2004

nonn

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

Cf. A024619, A119288, A089993.

Select[Table[If[Length[(f = FactorInteger[n])] > 1, f[[2, 1]], 1], {n, 1, 150}], # > 1 &]

f(n) = a=factor(n);v=a[,1];ln=length(v);if(ln>1,return(v[2]));
g(m) = for(x=2,m,if(f(x)>0,print1(f(x)",")));

a(n) = A119288(A024619(n)). - Amiram Eldar, Apr 12 2021

Offset corrected by Amiram Eldar, Apr 12 2021

A335268 Numbers that are not powers of primes (A024619) whose harmonic mean of their unitary divisors that are larger than 1 is an integer.

Original entry on oeis.org

6, 15, 20, 24, 28, 30, 45, 60, 72, 90, 91, 96, 100, 112, 153, 216, 220, 240, 264, 272, 325, 352, 360, 364, 378, 496, 703, 765, 780, 816, 832, 1056, 1125, 1170, 1225, 1360, 1431, 1512, 1656, 1760, 1891, 1900, 1984, 2275, 2448, 2520, 2701, 2912, 3024, 3168, 3321

Offset: 1

Amiram Eldar, May 29 2020

nonn

Since the unitary divisors of a power of prime (A000961), p^e, are {1, p^e}, they are trivial terms and hence they are excluded from this sequence.
The corresponding harmonic means are 3, 5, 6, 6, 7, 5, 9, 7, 12, 7, 13, 8, 10, 14, 17, ...
Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-m) | m * (2^omega(m)-1), or A063919(m) | (m * A309307(m)), where usigma is the sum of unitary divisors (A034448), and 2^omega(m) = A034444(m) is the number of the unitary divisors of m.
The squarefree terms of A335267 are also terms of this sequence.
The terms with 2 distinct prime divisors are of the form p^e * (2*p^e - 1), when the second factor is also a prime power. The least term which both of its 2 prime divisors are nonunitary (with multiplicity larger than 1) is 1225 = 5^2 * 7^2 = 5^2 * (2 * 5^2 - 1).
The unitary perfect numbers (A002827) are terms of this sequence: if m is a unitary perfect number then usigma(m)-m = m.

			6 is a term since its unitary divisors other than 1 are 2, 3 and 6, and their harmonic mean, 3/(1/2 + 1/3 + 1/6) = 3, is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..1000

The unitary version of A335267.
A002827 is subsequence.
Cf. A006086, A000961, A024619, A034444, A034448, A063919, A077610, A309307, A335269, A335270.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[3000], (omega = PrimeNu[#]) > 1 && Divisible[# * (2^omega-1), usigma[#] - #] &]

A056835 a(n) = lcm(s(1),...,s(n)) where {s(n)} = A024619 and a(0) = 1.

Original entry on oeis.org

1, 6, 30, 60, 420, 420, 1260, 1260, 1260, 13860, 27720, 360360, 360360, 360360, 360360, 6126120, 6126120, 6126120, 116396280, 116396280, 116396280, 116396280, 116396280, 116396280, 2677114440, 5354228880, 26771144400

Offset: 0

Labos Elemer, Aug 30 2000

nonn

			Overlaps with but is different from A003418 and A051451. E.g., 840 or 720720 are not here, but 1260 and 13860 are here, not there.

Robert Israel, Table of n, a(n) for n = 0..3930

Cf. A024619, A003418, A051451. Range of values is A056836. All members belong to A025487.

N:= 100: # to get a(0) to a(N)
A[0]:= 1:
m:= 0:
for s from 2 while m < N do
if nops(numtheory:-factorset(s)) > 1 then
  m:= m+1;
  A[m]:= ilcm(A[m-1],s);
fi
od:
seq(A[i],i=0..N); # Robert Israel, Dec 20 2015

s = Select[Range@ 50, ! PrimePowerQ@ # &]; Table[LCM @@ Take[s, n], {n, Length@ s}] (* Michael De Vlieger, Dec 20 2015 *)

Link typos corrected, cross-reference added by Matthew Vandermast, Nov 13 2010

A056836 Distinct values of sequence obtained when LCM is applied to initial segments of sequence A024619 union {1}.

Original entry on oeis.org

1, 6, 30, 60, 420, 1260, 13860, 27720, 360360, 6126120, 116396280, 2677114440, 5354228880, 26771144400, 80313433200, 2329089562800, 72201776446800, 2671465728531600, 109530094869795600, 4709794079401210800

Offset: 0

Labos Elemer, Aug 30 2000

nonn

A003418, A051451, A000961, A024619.

Distinct values of A056835. Subsequence of A025487.

Link typo corrected, cross-references added by Matthew Vandermast, Nov 15 2010

cp's OEIS Frontend

A376599 Second differences of consecutive non-prime-powers inclusive (A024619). First differences of A375735.

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A143731 Characteristic function of numbers with at least two distinct prime factors (A024619).

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A373670 Numbers k such that the k-th run-length A110969(k) of the sequence of non-prime-powers (A024619) is different from all prior run-lengths.

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A376600 Inflection or undulation points in the sequence of non-prime-powers inclusive (A024619).

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A085971 Union of primes and numbers that are not prime powers (A000040, A024619).

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A376601 Points of nonzero curvature in the sequence of non-prime-powers inclusive (A024619).

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A089992 Second prime divisor of numbers that are not powers of primes (A024619).

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A335268 Numbers that are not powers of primes (A024619) whose harmonic mean of their unitary divisors that are larger than 1 is an integer.

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