A007916 -id:A007916 - cp's OEIS frontend

A298748 Heinz numbers of aperiodic (relatively prime multiplicities) integer partitions with relatively prime parts.

2, 6, 10, 12, 14, 15, 18, 20, 22, 24, 26, 28, 30, 33, 34, 35, 38, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105, 106, 108, 110, 112, 114

Offset: 1

Gus Wiseman, Mar 01 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of partitions begins: (1), (21), (31), (211), (41), (32), (221), (311), (51), (2111), (61), (411), (321), (52), (71), (43), (81), (3111), (421), (511), (322), (91), (21111), (331), (72), (611), (2221), (53), (4111).

Cf. A000740, A000837, A007916, A052410, A064989, A100953, A289508, A289509, A296302, A300272.

Select[Range[100],With[{t=Transpose[FactorInteger[#]]},And[GCD@@PrimePi/@t[[1]]===1,GCD@@t[[2]]===1]]&] (* Gus Wiseman, Apr 14 2018 *)

A336414 Number of divisors of n! with distinct prime multiplicities.

Original entry on oeis.org

1, 1, 2, 3, 7, 10, 20, 27, 48, 86, 147, 195, 311, 390, 595, 1031, 1459, 1791, 2637, 3134, 4747, 7312, 10766, 12633, 16785, 26377, 36142, 48931, 71144, 82591, 112308, 128023, 155523, 231049, 304326, 459203, 568095, 642446, 812245, 1137063, 1441067, 1612998, 2193307, 2429362

Offset: 0

Gus Wiseman, Jul 22 2020

nonn

A number has distinct prime multiplicities iff its prime signature is strict.

			The first and second columns below are the a(6) = 20 counted divisors of 6! together with their prime signatures. The third column shows the A000005(6!) - a(6) = 10 remaining divisors.
      1: ()      20: (2,1)    |    6: (1,1)
      2: (1)     24: (3,1)    |   10: (1,1)
      3: (1)     40: (3,1)    |   15: (1,1)
      4: (2)     45: (2,1)    |   30: (1,1,1)
      5: (1)     48: (4,1)    |   36: (2,2)
      8: (3)     72: (3,2)    |   60: (2,1,1)
      9: (2)     80: (4,1)    |   90: (1,2,1)
     12: (2,1)  144: (4,2)    |  120: (3,1,1)
     16: (4)    360: (3,2,1)  |  180: (2,2,1)
     18: (1,2)  720: (4,2,1)  |  240: (4,1,1)

Chai Wah Wu, Table of n, a(n) for n = 0..6245 (n = 0..94 from David A. Corneth)

Perfect-powers are A001597, with complement A007916.
Numbers with distinct prime multiplicities are A130091.
Divisors with distinct prime multiplicities are counted by A181796.
The maximum divisor with distinct prime multiplicities is A327498.
Divisors of n! with equal prime multiplicities are counted by A336415.
Cf. A000005, A098859, A118914, A124010, A327527, A336424, A336500, A336568.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416.

Table[Length[Select[Divisors[n!],UnsameQ@@Last/@FactorInteger[#]&]],{n,0,15}]

a(n) = sumdiv(n!, d, my(ex=factor(d)[,2]); #vecsort(ex,,8) == #ex); \\ Michel Marcus, Jul 24 2020

a(n) = A181796(n!).

a(21)-a(41) from Alois P. Heinz, Jul 24 2020

A376597 Inflection and undulation points in the sequence of prime-powers inclusive (A000961).

Original entry on oeis.org

1, 2, 3, 6, 8, 14, 15, 16, 27, 32, 50, 61, 67, 72, 85, 92, 93, 124, 129, 132, 136, 141, 185, 190, 211, 214, 221, 226, 268, 292, 301, 302, 322, 374, 394, 423, 456, 463, 502, 503, 547, 559, 560, 593, 604, 640, 646, 663, 671, 675, 710, 726, 727, 746, 754, 755

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

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

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 and shift left.

			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 zeros (A376597) at:
  1, 2, 3, 6, 8, 14, 15, 16, 27, 32, 50, 61, 67, 72, 85, 92, 93, 124, 129, 132, ...

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

The first differences were A057820, see also A053707, A376340.
These are the zeros of A376596 (sorted firsts A376653, exclusive A376654).
The complement is A376598.
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), A376596 (second differences), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A053707, A069623, A093555, A174965, A361102, A376653, A376654.

Join@@Position[Differences[Select[Range[1000],#==1||PrimePowerQ[#]&],2],0]

A144338 Squarefree numbers > 1.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113

Offset: 1

Ctibor O. Zizka, Sep 18 2008

Nontrivial products of distinct primes. Sequence A005117 without the initial 1.
Also numbers n for which the following equation holds : (2^r)-sigma_0(p(1)*...*p(r)) = 0. This sequence describes the way RMS numbers (A140480) are grouped. In general if n = p(1)^alpha(1) *...* p(s)^alpha(s), alpha(i)>=1, we have the equation [2^sum_i=1..s{alpha(i)}] - sigma_0(p(1)^alpha(1) *...* p(s)^alpha(s)) = T. In terms of OEIS sequences the equation is : 2^(A001055(n)) - (A000005(n)) = T. This sequence has T=0, n=p(1)*...*p(r). If T=(2^k)-(k+1) then n=p^k. T splits the set of integers into subsets according to the form of prime factorization of the number n.
These can be computed with a modified Sieve of Eratosthenes: [1] start at n=2, [2] if (n is crossed out an even number of times) then (append n to the sequence and cross out all multiples of n), [3] set n:=n+1 and go to step 2; compare with the sieve for the complement of perfect powers in A007916. - Reinhard Zumkeller, Mar 19 2009
Numbers such that the harmonic mean of Omega(n) (A001222) and omega(n) (A001221) is a positive integer. - Wesley Ivan Hurt, Oct 13 2013

Steven R. Finch, Kalmar's Composition Constant, CiteSeer (2003).
Steven R. Finch, Kalmar's composition constant, Jun 05 2003. [Cached copy, with permission of the author]
A. M. Legendre, Diviseurs de la formule t^2 - a*u^2, Essai sur la Théorie des Nombres An VI, Table III. See first column. [_Paul Curtz_, Apr 13 2019]
Eric Weisstein's World of Math, Ordered Factorization
Index entries for sequences generated by sieves

Cf. A001055, A140480, A000005.

Cf. A076259 (first differences, without the first 1).

A144338:= n->`if`(numtheory[issqrfree](n) = true,n,NULL); seq(A144338(k), k=2..113); # Wesley Ivan Hurt, Oct 13 2013

Select[Range[2,120],SquareFreeQ] (* Harvey P. Dale, May 07 2012 *)

is(n)=issquarefree(n) && n>1 \\ Charles R Greathouse IV, Nov 05 2017

from math import isqrt
from sympy import mobius
def A144338(n):
    def f(x): return int(n+x+1-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 14 2025

Corrected A-number typo. - R. J. Mathar, Feb 21 2009

Minor edits from Charles R Greathouse IV, Mar 18 2010

A075802 Characteristic function of perfect powers, A001597.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Oct 13 2002

nonn

Not multiplicative: for example, a(8)=a(9)=1, but a(72)=0. - Franklin T. Adams-Watters, Sep 09 2005

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Powers.
Index entries for characteristic functions

Cf. A001597, A007916, A052409, A057427, A072292, A112526.

a075802 1 = 1
a075802 n = signum $ a052409 n - 1  -- Reinhard Zumkeller, May 26 2012

a[n_] := Boole[GCD @@ FactorInteger[n][[All, 2]] > 1]; a[1] = 1; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Dec 12 2011 *)

from sympy import perfect_power
def A075802(n): return int(bool(perfect_power(n))) if n>1 else 1 # Chai Wah Wu, Mar 11 2025

a(n) = A057427(A052409(n) - 1);

a(A001597(n))=1 and a(A007916(n))=0.

A317089 Numbers whose prime factors span an initial interval of prime numbers and whose prime multiplicities span an initial interval of positive integers.

Original entry on oeis.org

2, 6, 12, 18, 30, 60, 90, 150, 180, 210, 300, 360, 420, 450, 540, 600, 630, 1050, 1260, 1350, 1470, 1500, 2100, 2250, 2310, 2520, 2940, 3150, 3780, 4200, 4410, 4620, 5880, 6300, 6930, 7350, 8820, 9450, 10500, 11550, 12600, 13230, 13860, 14700, 15750, 16170

Offset: 1

Gus Wiseman, Jul 21 2018

nonn

			The sequence of rows of A296150 indexed by the terms of this sequence begins: (1), (21), (211), (221), (321), (3211), (3221), (3321), (32211), (4321), (33211), (322111), (43211).

Andrew Howroyd, Table of n, a(n) for n = 1..300

Cf. A001222, A007916, A055932, A056239, A124010, A133808, A242414, A296150.

Cf. A317087, A317088, A317090, A317091, A317092.

normalQ[m_]:=Union[m]==Range[Max[m]];
Select[Range[10000],And[normalQ[PrimePi/@FactorInteger[#][[All,1]]],normalQ[FactorInteger[#][[All,2]]]]&]

ok(n)={my(f=factor(n), p=f[,1], e=vecsort(f[,2],,8)); n > 1 && #p==primepi(p[#p]) && #e==e[#e]} \\ Andrew Howroyd, Aug 26 2018

A294336 Number of ways to write n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 28 2017

nonn

Möbius-transform of A294337. - Antti Karttunen, Jun 12 2018

			The a(4096) = 7 ways are: 2^12, 4^6, 8^4, 8^(2^2), 16^3, 64^2, 4096.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000041, A001055, A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294337, A294338, A294339, A375598, A375599.

Array[1+Sum[#0[g],{g,Rest[Divisors[GCD@@FactorInteger[#1][[All,2]]]]}]&,200]

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A294336(n) = if(1==n,n,sumdiv(A052409(n),d,A294336(d))); \\ Antti Karttunen, Jun 12 2018, after Mathematica-code.

a(1) = 1; for n > 1, a(n) = Sum_{d|A052409(n)} a(d). - Antti Karttunen, Jun 12 2018, after Mathematica-code.

a(n) = A294337(A052409(n)) for n >= 2. - Pontus von Brömssen, Aug 20 2024

More terms from Antti Karttunen, Jun 12 2018

A377466 Numbers k such that there is more than one perfect power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

4, 9, 11, 30, 327, 445, 3512, 7789, 9361, 26519413

Offset: 1

Gus Wiseman, Nov 02 2024

Perfect powers (A001597) are numbers with a proper integer root, the complement of A007916.
Is this sequence finite?
The Redmond-Sun conjecture (see A308658) implies that this sequence is finite. - Pontus von Brömssen, Nov 05 2024

			Primes 9 and 10 are 23 and 29, and the interval (24,25,26,27,28) contains two perfect powers (25,27), so 9 is in the sequence.

For powers of 2 see A013597, A014210, A014234, A188951, A244508, A377467.
For no prime-powers we have A377286, ones in A080101.
For a unique prime-power we have A377287.
For squarefree numbers see A377430, A061398, A377431, A068360, A224363.
These are the positions of terms > 1 in A377432.
For a unique perfect power we have A377434.
For no perfect powers we have A377436.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A046933 counts the interval from A008864(n) to A006093(n+1).
A081676 gives the greatest perfect power <= n.
A131605 lists perfect powers that are not prime-powers.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053607, A304521.
A377468 gives the least perfect power > n.
Cf. A000720, A023055, A031218, A045542, A052410, A053706, A069623, A116086, A116455, A216765, A308658, A336416, A345531, A375740, A376560, A376561, A377057.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[100],Count[Range[Prime[#]+1, Prime[#+1]-1],_?perpowQ]>1&]

from itertools import islice
from sympy import prime
from gmpy2 import is_power, next_prime
def A377466_gen(startvalue=1): # generator of terms >= startvalue
    k = max(startvalue,1)
    p = prime(k)
    while (q:=next_prime(p)):
        c = 0
        for i in range(p+1,q):
            if is_power(i):
                c += 1
                if c>1:
                    yield k
                    break
        k += 1
        p = q
A377466_list = list(islice(A377466_gen(),9)) # Chai Wah Wu, Nov 04 2024

a(n) = A000720(A116086(n)) = A000720(A116455(n)) for n <= 10. This would hold for all n if there do not exist more than two perfect powers between any two consecutive primes, which is implied by the Redmond-Sun conjecture. - Pontus von Brömssen, Nov 05 2024

a(10) from Pontus von Brömssen, Nov 04 2024

A288636 Height of power-tower factorization of n. Row lengths of A278028.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 12 2017

nonn

After a(1)=0 this sequence has many terms equal to A089723. It first differs at a(64)=2, A089723(64)=4.

First positions of a(n) = {0, 1, 2, 3, 4} are n = {1, 2, 4, 16, 65536}. - Michael De Vlieger, Nov 24 2017

Michael De Vlieger, Table of n, a(n) for n = 1..65536

Cf. A007916, A089723, A164337, A277562, A277564, A278028.

a[n_] := If[n == 1, 0, 1+a[GCD @@ FactorInteger[n][[All, 2]]]];
Array[a,100]

A316888 Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is 1.

Original entry on oeis.org

2, 195, 3185, 6475, 10527, 16401, 20445, 20535, 21045, 25365, 46155, 164255, 171941, 218855, 228085, 267883, 312785, 333925, 333935, 335405, 343735, 355355, 414295, 442975, 474513, 527425, 549575, 607475, 633777, 691041, 711321, 722425, 753865, 804837, 822783

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A partition is aperiodic if its multiplicities are relatively prime.
Does not contain 29888089, which belongs to A316890 and is the Heinz number of a periodic partition.

			The partition (6,4,4,3) with Heinz number 3185 is aperiodic, has relatively prime parts, and 1/6 + 1/4 + 1/4 + 1/3 = 1, so 3185 belongs to the sequence.
The sequence of partitions whose Heinz numbers belong to the sequence begins: (1), (6,3,2), (6,4,4,3), (12,4,3,3), (10,5,5,2), (20,5,4,2), (15,10,3,2), (12,12,3,2), (18,9,3,2), (24,8,3,2), (42,7,3,2).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A007916, A051908, A100953, A289509, A296150, A316855, A316856, A316857, A316888-A316904.

Select[Range[2,100000],And[GCD@@FactorInteger[#][[All,2]]==1,GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]==1]&]

cp's OEIS Frontend

A298748 Heinz numbers of aperiodic (relatively prime multiplicities) integer partitions with relatively prime parts.

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A336414 Number of divisors of n! with distinct prime multiplicities.

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A376597 Inflection and undulation points in the sequence of prime-powers inclusive (A000961).

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A144338 Squarefree numbers > 1.

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A075802 Characteristic function of perfect powers, A001597.

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A317089 Numbers whose prime factors span an initial interval of prime numbers and whose prime multiplicities span an initial interval of positive integers.

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A294336 Number of ways to write n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

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