A062770 -id:A062770 - cp's OEIS frontend

A325270 Numbers with 1 fewer distinct prime exponents than (not necessarily distinct) prime factors.

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 111, 115, 116, 117, 118, 119, 121, 122, 123, 124, 129, 133

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

Also Heinz numbers of integer partitions with 1 fewer distinct multiplicities than parts, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The enumeration of these partitions by sum is given by A117571.

Also numbers whose sorted prime signature is (1,1), (2), or (1,2). - Gus Wiseman, Jul 03 2019

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   25: {3,3}
   26: {1,6}
   28: {1,1,4}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   44: {1,1,5}

Cf. A001221, A001222, A000961, A005117, A060687, A062770, A071625, A072774, A090858, A117571, A118914, A130091, A325244, A325259.

Select[Range[100],PrimeOmega[#]==Length[Union[Last/@FactorInteger[#]]]+1&]

A323056 Numbers with exactly five distinct exponents in their prime factorization, or five distinct parts in their prime signature.

Original entry on oeis.org

174636000, 206388000, 244490400, 261954000, 269892000, 274428000, 288943200, 291060000, 301644000, 309582000, 343980000, 349272000, 365148000, 366735600, 377848800, 383292000, 404838000, 411642000, 412776000, 422301600, 433414800, 449820000, 452466000, 457380000

Offset: 1

Gus Wiseman, Jan 03 2019

nonn

The first term is A006939(5) = 174636000.
Positions of 5's in A071625.
Numbers k such that A001221(A181819(k)) = 5.

			174636000 = 2^5 * 3^4 * 5^3 * 7^2 * 11^1 has five distinct exponents so belongs to the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

One distinct exponent: A062770 or A072774.
Two distinct exponents: A323055.
Three distinct exponents: A323024.
Four distinct exponents: A323025.
Five distinct exponents: A323056.
Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024.

Select[Range[300000000],Length[Union[Last/@FactorInteger[#]]]==5&]

is(n) = #Set(factor(n)[, 2]) == 5 \\ David A. Corneth, Jan 12 2019

a(13)-a(24) from Daniel Suteu, Jan 12 2019

A325264 Numbers whose omega-sequence sums to 7.

Original entry on oeis.org

30, 36, 42, 64, 66, 70, 78, 100, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 196, 222, 225, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).

			The sequence of terms together with their prime indices and omega-sequences begins:
   30: {1,2,3} (3,3,1)
   36: {1,1,2,2} (4,2,1)
   42: {1,2,4} (3,3,1)
   64: {1,1,1,1,1,1} (6,1)
   66: {1,2,5} (3,3,1)
   70: {1,3,4} (3,3,1)
   78: {1,2,6} (3,3,1)
  100: {1,1,3,3} (4,2,1)
  102: {1,2,7} (3,3,1)
  105: {2,3,4} (3,3,1)
  110: {1,3,5} (3,3,1)
  114: {1,2,8} (3,3,1)
  130: {1,3,6} (3,3,1)
  138: {1,2,9} (3,3,1)
  154: {1,4,5} (3,3,1)
  165: {2,3,5} (3,3,1)
  170: {1,3,7} (3,3,1)
  174: {1,2,10} (3,3,1)
  182: {1,4,6} (3,3,1)
  186: {1,2,11} (3,3,1)
  190: {1,3,8} (3,3,1)
  195: {2,3,6} (3,3,1)
  196: {1,1,4,4} (4,2,1)

Positions of 7's in A325249.
Numbers with omega-sequence summing to m: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7).
Cf. A056239, A060687, A062770, A118914, A130091, A303555, A323023, A325238, A325265, A325277.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],Total[omseq[#]]==7&]

A384517 Nonsquarefree numbers that are squarefree numbers raised to an even power.

Original entry on oeis.org

4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 169, 196, 225, 256, 289, 361, 441, 484, 529, 625, 676, 729, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2401, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4096, 4225, 4356

Offset: 1

Amiram Eldar, Jun 01 2025

nonn

Differs from its subsequence A340674 by having the terms 64, 729, 1024, 4096, .... .

Numbers whose prime factorization exponents are equal and even.

Intersection of A000290 and A072777.
Equals A072777 \ A384518.
A340674 is a subsequence.
Cf. A005117, A062770, A072774, A231273, A231327.

Select[Range[2, 100], SameQ @@ FactorInteger[#][[;;, 2]] &]^2

isok(k) = {my(s, e = ispower(k, , &s)); !(e % 2) && issquarefree(s);}

from math import isqrt
from sympy import mobius, integer_nthroot
def A384517(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1		
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return n+x-sum(g(integer_nthroot(x,e)[0])-1 for e in range(2,x.bit_length(),2))
    return bisection(f,n,n) # Chai Wah Wu, Jun 01 2025

a(n) = A062770(n)^2 = A072774(n+1)^2.

Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2*k)/zeta(4*k)-1) = Sum{k>=1} (A231327(k)/(A231273(k)*Pi^(2*k)) - 1) = 0.62022193512079649421... .

A182856 a(0) = 1; for n > 0, a(n) = smallest positive integer whose prime signature contains, for k = 1 to n, exactly one positive number appearing exactly k times.

Original entry on oeis.org

1, 2, 60, 1801800, 11657093261814000, 7167827541370578634694420017740000, 291943326350524088652207164949980988754136887856059678357800000

Offset: 0

Matthew Vandermast, Jan 05 2011

a(n) = smallest number m such that A181819(m) = A006939(n).  a(n) belongs to A182855 iff n > 1.
Next term has 105 digits.
Smallest number k with A323022(k) = n, where A323022(m) is the number of distinct multiplicities in the prime signature of m. - Gus Wiseman, Jan 03 2019

			The canonical prime factorization of a(3) = 1801800 is 2^3*3^2*5^2*7*11*13. The prime signature of 1801800 is therefore (3,2,2,1,1,1). Note that (3,2,2,1,1,1) contains exactly one number that appears once (3), one number that appears twice (2), and one number that appears three times (1).

David A. Corneth, Table of n, a(n) for n = 0..12
Dario Alpern, Factorization using the Elliptic Curve Method

Cf. A006939, A181819, A182855.

Cf. A056239, A059404, A062770, A071625, A118914, A181821, A182857, A323022.

Table[Product[Times@@Prime[i*(i-1)/2+Ceiling[Range[i*(n-i)]/(n-i)]],{i,n-1}],{n,6}] (* Gus Wiseman, Jan 03 2019 *)

a(n) = if(n == 0, return(1)); my(f = matrix(binomial(n+1,2), 2)); f[, 1] = primes(#f~ )~; f[, 2] = Vecrev(concat(vector(n, i, vector(n+1-i, j, i))))~; factorback(f) \\ David A. Corneth, Jan 03 2019

Partial products of A113511.
log a(n) ~ (1/3) n^3 log n. [Charles R Greathouse IV, Jan 13 2012]
A001222(a(n)) = A000292(n). - Gus Wiseman, Jan 03 2019
a(0) = 1; a(n + 1) = A002110(binomial(n + 2, 2)) * a(n). - David A. Corneth, Jan 03 2019

A304449 Numbers that are either squarefree or a perfect power.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89

Offset: 1

Gus Wiseman, May 12 2018

nonn

First differs from A072774 at a(105) = 144, A072774(105) = 145.

Apparently the 1 and the members of A062770. - R. J. Mathar, May 22 2018

Cf. A000009, A000720, A000837, A000837, A001222, A001597, A005117, A007916, A013929, A047966, A072774, A073576, A126706, A303554, A303946.

Select[Range[150],SquareFreeQ[#]||GCD@@FactorInteger[#][[All,2]]>1&]

isok(n) = issquarefree(n) || ispower(n); \\ Michel Marcus, May 13 2018

from math import isqrt
from sympy import mobius, integer_nthroot
def A304449(n):
    def f(x): return int(n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 19 2024

Union of A005117 and A001597. Complement of A303946.

A325243 Number of integer partitions of n with exactly two distinct multiplicities.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 12, 19, 26, 39, 47, 70, 89, 115, 148, 189, 235, 294, 362, 450, 558, 669, 817, 980, 1197, 1421, 1709, 2012, 2429, 2836, 3380, 3961, 4699, 5433, 6457, 7433, 8770, 10109, 11818, 13547, 15912, 18109, 21105, 24121, 27959, 31736, 36840, 41670

Offset: 0

Gus Wiseman, Apr 15 2019

nonn

For example, (32211) has two distinct multiplicities (1 and 2) so is counted under a(9).

The Heinz numbers of these partitions are given by A323055.

			The a(4) = 1 through a(9) = 19 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (3221)     (3222)
                          (3211)    (4211)     (3321)
                          (4111)    (5111)     (4221)
                          (22111)   (22211)    (4311)
                          (31111)   (32111)    (5211)
                          (211111)  (41111)    (6111)
                                    (221111)   (22221)
                                    (311111)   (32211)
                                    (2111111)  (33111)
                                               (42111)
                                               (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

Column k = 2 of A325242. Dominated by A325267.

Cf. A008284, A062770, A071625, A098859, A116608, A244515, A323055, A325244.

Table[Length[Select[IntegerPartitions[n],Length[Union[Length/@Split[#]]]==2&]],{n,0,30}]

A325265 Numbers with sum of omega-sequence > 4.

Original entry on oeis.org

6, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).

			The sequence of terms together with their omega-sequences begins:
   6: 2 2 1       46: 2 2 1         80: 5 2 2 1       112: 5 2 2 1
  10: 2 2 1       48: 5 2 2 1       81: 4 1           114: 3 3 1
  12: 3 2 2 1     50: 3 2 2 1       82: 2 2 1         115: 2 2 1
  14: 2 2 1       51: 2 2 1         84: 4 3 2 2 1     116: 3 2 2 1
  15: 2 2 1       52: 3 2 2 1       85: 2 2 1         117: 3 2 2 1
  16: 4 1         54: 4 2 2 1       86: 2 2 1         118: 2 2 1
  18: 3 2 2 1     55: 2 2 1         87: 2 2 1         119: 2 2 1
  20: 3 2 2 1     56: 4 2 2 1       88: 4 2 2 1       120: 5 3 2 2 1
  21: 2 2 1       57: 2 2 1         90: 4 3 2 2 1     122: 2 2 1
  22: 2 2 1       58: 2 2 1         91: 2 2 1         123: 2 2 1
  24: 4 2 2 1     60: 4 3 2 2 1     92: 3 2 2 1       124: 3 2 2 1
  26: 2 2 1       62: 2 2 1         93: 2 2 1         126: 4 3 2 2 1
  28: 3 2 2 1     63: 3 2 2 1       94: 2 2 1         128: 7 1
  30: 3 3 1       64: 6 1           95: 2 2 1         129: 2 2 1
  32: 5 1         65: 2 2 1         96: 6 2 2 1       130: 3 3 1
  33: 2 2 1       66: 3 3 1         98: 3 2 2 1       132: 4 3 2 2 1
  34: 2 2 1       68: 3 2 2 1       99: 3 2 2 1       133: 2 2 1
  35: 2 2 1       69: 2 2 1        100: 4 2 1         134: 2 2 1
  36: 4 2 1       70: 3 3 1        102: 3 3 1         135: 4 2 2 1
  38: 2 2 1       72: 5 2 2 1      104: 4 2 2 1       136: 4 2 2 1
  39: 2 2 1       74: 2 2 1        105: 3 3 1         138: 3 3 1
  40: 4 2 2 1     75: 3 2 2 1      106: 2 2 1         140: 4 3 2 2 1
  42: 3 3 1       76: 3 2 2 1      108: 5 2 2 1       141: 2 2 1
  44: 3 2 2 1     77: 2 2 1        110: 3 3 1         142: 2 2 1
  45: 3 2 2 1     78: 3 3 1        111: 2 2 1         143: 2 2 1

Positions of terms > 4 in A325249.
Numbers with omega-sequence summing to m: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7).
Cf. A056239, A060687, A062770, A118914, A130091, A303555, A323023, A325238, A325277.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],Total[omseq[#]]>4&]

A325279 Number of integer partitions of n whose maximum multiplicity is one greater than their minimum multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 5, 6, 9, 10, 18, 18, 31, 34, 48, 57, 80, 86, 122, 138, 183, 211, 275, 311, 402, 461, 576, 663, 825, 942, 1163, 1334, 1621, 1865, 2248, 2566, 3084, 3532, 4193, 4794, 5674, 6472, 7617, 8685, 10153, 11576, 13483, 15320, 17790, 20200, 23342

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The Heinz numbers of these partitions are given by A325241.

For example, the partition (44111) has two multiplicities (2 and 3) which differ by 1, so is counted under a(11).

			The a(4) = 1 through a(11) = 18 partitions:
  (211)  (221)  (411)  (322)    (332)    (441)    (433)      (443)
         (311)         (331)    (422)    (522)    (442)      (533)
                       (511)    (611)    (711)    (622)      (551)
                       (3211)   (3221)   (3321)   (811)      (722)
                       (22111)  (4211)   (4221)   (5221)     (911)
                                (22211)  (4311)   (5311)     (4322)
                                         (5211)   (6211)     (4331)
                                         (32211)  (33211)    (4421)
                                         (33111)  (42211)    (5411)
                                                  (2221111)  (6221)
                                                             (6311)
                                                             (7211)
                                                             (33221)
                                                             (33311)
                                                             (43211)
                                                             (44111)
                                                             (52211)
                                                             (2222111)

Cf. A047966, A062770, A071625, A098859, A117571, A127002, A325242, A325244, A325245, A325280.

Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]-Min@@Length/@Split[#]==1&]],{n,0,30}]

A383106 Numbers k such that A382883(k) != 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100

Offset: 1

Peter Luschny, Apr 16 2025

nonn

See the comments in A382883.

Cf. A382883, A382943 (complement), A383016, A383017.

Cf. A005117 and similars: A062770, A072774, A085924, A242414, A304449.

V[n_, e_] := If[e == 1, 1, IntegerExponent[n, e]]; f[n_] := f[n] = -DivisorSum[n, V[n, #] * f[#] &, # < n &]; f[1] = 1; Select[Range[100], f[#] != 0 &] (* Amiram Eldar, Apr 29 2025 *)

def A383106List(upto): return [n for n in srange(1, upto) if A382883(n) != 0]

cp's OEIS Frontend

A325270 Numbers with 1 fewer distinct prime exponents than (not necessarily distinct) prime factors.

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A323056 Numbers with exactly five distinct exponents in their prime factorization, or five distinct parts in their prime signature.

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A325264 Numbers whose omega-sequence sums to 7.

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A384517 Nonsquarefree numbers that are squarefree numbers raised to an even power.

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A182856 a(0) = 1; for n > 0, a(n) = smallest positive integer whose prime signature contains, for k = 1 to n, exactly one positive number appearing exactly k times.

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A304449 Numbers that are either squarefree or a perfect power.

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A325243 Number of integer partitions of n with exactly two distinct multiplicities.

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A325265 Numbers with sum of omega-sequence > 4.

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