A253784 -id:A253784 - cp's OEIS frontend

A342094 Number of integer partitions of n with no adjacent parts having quotient > 2.

1, 2, 3, 4, 5, 8, 9, 13, 16, 21, 27, 37, 44, 59, 75, 94, 117, 153, 186, 238, 296, 369, 458, 573, 701, 870, 1068, 1312, 1601, 1964, 2384, 2907, 3523, 4270, 5159, 6235, 7491, 9021, 10819, 12964, 15494, 18517, 22049, 26260, 31195, 37020, 43851, 51906, 61290

Offset: 1

Gus Wiseman, Mar 02 2021

nonn

The decapitation of such a partition (delete the greatest part) is term-wise greater than or equal to its negated first-differences.

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (211)   (221)    (42)      (322)      (53)
                    (1111)  (2111)   (222)     (421)      (332)
                            (11111)  (321)     (2221)     (422)
                                     (2211)    (3211)     (2222)
                                     (21111)   (22111)    (3221)
                                     (111111)  (211111)   (4211)
                                               (1111111)  (22211)
                                                          (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..250

The version with no adjacent parts having quotient < 2 is A000929.
The case of equality (all adjacent parts having quotient 2) is A154402.
A strong multiplicative version is A342083 or A342084.
The multiplicative version is A342085, with reciprocal version A337135.
The strict case is A342095.
The version with all adjacent parts having quotient < 2 is A342096, with strict case A342097.
The version with all adjacent parts having quotient > 2 is A342098.
The Heinz numbers of these partitions are listed by A342191.
A000009 counts strict partitions.
A003114 counts partitions with adjacent parts differing by more than 1.
A034296 counts partitions with adjacent parts differing by at most 1.
A038548 counts inferior (or superior) divisors, listed by A161906.
A161908 lists superior prime divisors.
Cf. A001055, A001227, A003242, A027193, A167606, A178470, A253784, A341591.

Table[Length[Select[IntegerPartitions[n],And@@Thread[Differences[-#]<=Rest[#]]&]],{n,30}]

A342191 Numbers with no adjacent prime indices having quotient < 1/2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 32, 35, 36, 37, 41, 42, 43, 45, 47, 48, 49, 53, 54, 55, 59, 60, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 84, 89, 90, 91, 96, 97, 101, 103, 105, 107, 108, 109

Offset: 1

Gus Wiseman, Mar 05 2021

nonn

Also Heinz numbers of integer partitions with no adjacent parts having quotient > 2 (counted by A342094). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}            18: {1,2,2}         42: {1,2,4}
      2: {1}           19: {8}             43: {14}
      3: {2}           21: {2,4}           45: {2,2,3}
      4: {1,1}         23: {9}             47: {15}
      5: {3}           24: {1,1,1,2}       48: {1,1,1,1,2}
      6: {1,2}         25: {3,3}           49: {4,4}
      7: {4}           27: {2,2,2}         53: {16}
      8: {1,1,1}       29: {10}            54: {1,2,2,2}
      9: {2,2}         30: {1,2,3}         55: {3,5}
     11: {5}           31: {11}            59: {17}
     12: {1,1,2}       32: {1,1,1,1,1}     60: {1,1,2,3}
     13: {6}           35: {3,4}           61: {18}
     15: {2,3}         36: {1,1,2,2}       63: {2,2,4}
     16: {1,1,1,1}     37: {12}            64: {1,1,1,1,1,1}
     17: {7}           41: {13}            65: {3,6}

The multiplicative version (squared instead of doubled) for prime factors is A253784.
These are the Heinz numbers of the partitions counted by A342094.
A003114 counts partitions with adjacent parts differing by more than 1.
A034296 counts partitions with adjacent parts differing by at most 1.
A038548 counts inferior or superior divisors, listed by A161906 or A161908.
Cf. A000929, A003242, A056239, A056924, A112798, A154402, A167606, A337135, A342085, A342096, A342098.

Select[Range[100],Min[Divide@@@Partition[PrimePi/@First/@FactorInteger[#],2,1]]>=1/2&]

A251726 Numbers n > 1 for which gpf(n) < lpf(n)^2, where lpf and gpf (least and greatest prime factor of n) are given by A020639(n) and A006530(n).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 55, 59, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 85, 89, 91, 95, 96, 97, 101, 103, 105, 107, 108, 109, 113, 115, 119, 121, 125, 127, 128, 131, 133, 135, 137, 139, 143, 144

Offset: 1

Antti Karttunen, Dec 17 2014

nonn

Numbers n > 1 for which there exists r <= gpf(n) such that r^k <= lpf(n) and gpf(n) < r^(k+1) for some k >= 0, where lpf and gpf (least and greatest prime factor of n) are given by A020639(n) and A006530(n) (the original, equivalent definition of the sequence).
Numbers n > 1 such that A252375(n) < 1 + A006530(n). Equally, one can substitute A251725 for A252375.
These are numbers n all of whose prime factors "fit between" two consecutive powers of some positive integer which itself is <= the largest prime factor of n.
Conjecture: If any n is in the sequence, then so is A003961(n).
Note: if Legendre's or Brocard's conjecture is true, then the above conjecture is true as well. See my comments at A251728. - Antti Karttunen, Jan 01 2015

			For 35 = 5*7, 7 is less than 5^2, thus 35 is included.
For 90 = 2*3*3*5, 5 is not less than 2^2, thus 90 is NOT included.
For 105 = 3*5*7, 7 is less than 3^2, thus 105 is included.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Ratio A251726(n)/A251727(n) (plotted with OEIS-server's Plot2-utility)

Complement: A251727. Subsequences: A251728, A000961 (after 1).
Characteristic function: A252372. Inverse function: A252373.
Gives the positions of zeros in A252459 (after its initial zero), cf. also A284261.
Cf. A252370 (gives the difference between the prime indices of gpf and lpf for each a(n)).
Sequence gives all n > 1 for which A284252(n) (equally: A284254) is 1, and A284256(n) (equally A284258) is 0, and also n > 1 such that A284260(n) = A006530(n).
Cf. A003961, A006530, A020639, A252375, A251725, A253784.
Related permutations: A252757-A252758.

pfQ[n_]:=Module[{f=FactorInteger[n]},f[[-1,1]]Harvey P. Dale, May 01 2015 *)

for(n=2, 150, if(vecmax(factor(n)[,1]) < vecmin(factor(n)[,1])^2, print1(n,", "))) \\ Indranil Ghosh, Mar 24 2017

from sympy import primefactors
print([n for n in range(2, 150) if max(primefactors(n))Indranil Ghosh, Mar 24 2017

Other identities. For all n >= 1:

A252373(a(n)) = n. [A252373 works as an inverse or ranking function for this sequence.]

A new simpler definition found Jan 01 2015 and the original definition moved to the Comments section.

A253567 Noncomposites together with such composites n = p_i * p_j * p_k * ... * p_u, p_i <= p_j <= p_k <= ... <= p_u, where there is at least one such pair of successive prime factors (when sorted into a nondecreasing order) that the square of the former is larger than the latter: (p_i)^2 > p_j or (p_j)^2 > p_k, etc.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 35, 36, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 55, 56, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 83, 84, 85, 88, 89, 90, 91, 92, 95, 96, 97, 98, 99, 100

Offset: 1

Antti Karttunen, Jan 16 2015

nonn

			4 = 2*2 is present as 2^2 > 2.
6 = 2*3 is present as 2^2 > 3.
70 = 2*5*7 is present as 5^2 > 7.

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

Complement: A253569.
Subsequences: A008578, A013929, A251728, A253784.
Cf. A001222.

A253785 Composite numbers n = prime(i_1) * ... * prime(i_k), prime(i_1) <= prime(i_2) <= ... <= prime(i_k), with at least one pair of successive prime factors (when sorted into monotonic order) where the latter prime factor is greater than the square of the former: prime(i_x)^2 < prime(i_{x+1}), for some x in 1 .. k-1, where k = A001222(n) and i_k = A061395(n).

Original entry on oeis.org

10, 14, 20, 22, 26, 28, 33, 34, 38, 39, 40, 44, 46, 50, 51, 52, 56, 57, 58, 62, 66, 68, 69, 70, 74, 76, 78, 80, 82, 86, 87, 88, 92, 93, 94, 98, 99, 100, 102, 104, 106, 110, 111, 112, 114, 116, 117, 118, 122, 123, 124, 129, 130, 132, 134, 136, 138, 140, 141, 142, 145, 146, 148, 152, 153, 154, 155, 156, 158, 159, 160, 164, 166, 170

Offset: 1

Antti Karttunen, Jan 16 2015

nonn

			10 = 2*5 is present as 2^2 < 5.
50 = 2*5*5 is present as 2^2 < 5.
51 = 3*17 is present as 3^2 < 17.
66 = 2*3*11 is present as 3^2 < 11.

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

Complement: A253784.
Subsequences: A138511, A253569.
Cf. A000290, A001222, A061395.
Differs from A245729 for the first time at n=14, where a(14) = 50, while A245729(14) = 51.

A379271 Composite numbers, k, whose prime factors, viewed on a log log scale, have a small standard deviation defined with respect to bigomega(k), as specified in the comments.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 88, 90, 91, 92, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 116, 117, 119, 120, 121, 124, 125, 126, 128

Offset: 1

Peter Munn, Feb 18 2025

nonn

Composite numbers k (written as a product of primes p_1 * p_2 * ... * p_m) such that s( {log(log(p_i)) : 1 <= i <= m} ) < s( {i : 1 <= i <= m} ), where s is standard deviation and m = bigomega(k).
Loosely described, these are numbers whose prime factors, including repetitions, are relatively close together. (Note we get the same criterion irrespective of whether s is sample standard deviation or population standard deviation.)
The author's intent is to divide the set of composite numbers into 2 parts whose asymptotic densities differ at most by a small factor. So his choice of criterion was guided by particular information relating to the statistics of prime factors of large numbers.
From Charles R Greathouse IV, May 19 2025: (Start)
For example, semiprimes p*q with p <= q are in this sequence if (and only if) q < p^e where e = 2.71... is the base of the natural logarithm.
For any m, there are finitely many primes p (perhaps none) such that p*m is in the sequence. (End)

Subsequences: A251728, the composites in A253784, A380438.

Select[Select[Range[128], CompositeQ], Less @@ Map[StandardDeviation, Transpose@ MapIndexed[{Log@ Log[#1], First[#2]} &, Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#] ] ] ] &] (* Michael De Vlieger, May 04 2025 *)

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A342094 Number of integer partitions of n with no adjacent parts having quotient > 2.

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A342191 Numbers with no adjacent prime indices having quotient < 1/2.

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A251726 Numbers n > 1 for which gpf(n) < lpf(n)^2, where lpf and gpf (least and greatest prime factor of n) are given by A020639(n) and A006530(n).

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A379271 Composite numbers, k, whose prime factors, viewed on a log log scale, have a small standard deviation defined with respect to bigomega(k), as specified in the comments.

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Mathematica