A243055 -id:A243055 - cp's OEIS frontend

A307517 Numbers with at least two not necessarily distinct prime factors less than the largest prime factor.

12, 20, 24, 28, 30, 36, 40, 42, 44, 45, 48, 52, 56, 60, 63, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 130, 132, 135, 136, 138, 140, 144, 148, 150, 152, 153, 154, 156, 160, 164, 165, 168

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with at least two not necessarily distinct parts less than the largest part. The enumeration of these partitions by sum is given by A000094.

			The sequence of terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   30: {1,2,3}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   70: {1,3,4}
   72: {1,1,1,2,2}
   76: {1,1,8}

Cf. A000094, A000245, A000961, A001222, A052126, A056239, A061395, A064989, A105441, A112798, A243055, A256617, A307516, A325196.

q:= n-> (l-> add(l[i][2], i=1..nops(l)-1)>1)(sort(ifactors(n)[2])):
select(q, [$1..200])[];  # Alois P. Heinz, Apr 12 2019

Select[Range[100],PrimeOmega[#/Power@@FactorInteger[#][[-1]]]>1&]

A325198 Positive numbers whose maximum prime index minus minimum prime index is 2.

Original entry on oeis.org

10, 20, 21, 30, 40, 50, 55, 60, 63, 80, 90, 91, 100, 105, 120, 147, 150, 160, 180, 187, 189, 200, 240, 247, 250, 270, 275, 300, 315, 320, 360, 385, 391, 400, 441, 450, 480, 500, 525, 540, 551, 567, 600, 605, 637, 640, 713, 720, 735, 750, 800, 810, 900, 945

Offset: 1

Gus Wiseman, Apr 11 2019

nonn

Also Heinz numbers of integer partitions whose maximum minus minimum part is 2 (counted by A008805). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   10: {1,3}
   20: {1,1,3}
   21: {2,4}
   30: {1,2,3}
   40: {1,1,1,3}
   50: {1,3,3}
   55: {3,5}
   60: {1,1,2,3}
   63: {2,2,4}
   80: {1,1,1,1,3}
   90: {1,2,2,3}
   91: {4,6}
  100: {1,1,3,3}
  105: {2,3,4}
  120: {1,1,1,2,3}
  147: {2,4,4}
  150: {1,2,3,3}
  160: {1,1,1,1,1,3}
  180: {1,1,2,2,3}
  187: {5,7}

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 2's in A243055.
A061395(a(n)) - A055396((n)) = 2.
Cf. A000961, A008805, A046660, A056239, A093641, A112798, A118914, A174090, A195086, A256617, A325180, A325197.

N:= 1000: # for terms <= N
q:= 2: r:= 3:
Res:= NULL:
do
  p:= q; q:= r; r:= nextprime(r);
  if p*r > N then break fi;
  for i from 1 do
    pi:= p^i;
    if pi*r > N then break fi;
    for j from 0 do
      piqj:= pi*q^j;
      if piqj*r > N then break fi;
      Res:= Res, seq(piqj*r^k,k=1 .. floor(log[r](N/piqj)))
    od
  od
od:
sort([Res]); # Robert Israel, Apr 12 2019

Select[Range[100],PrimePi[FactorInteger[#][[-1,1]]]-PrimePi[FactorInteger[#][[1,1]]]==2&]

A325199 Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 2.

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 6, 3, 2, 9, 15, 12, 6, 12, 27, 38, 34, 22, 20, 43, 74, 94, 90, 67, 48, 69, 130, 194, 232, 230, 187, 132, 129, 218, 364, 497, 576, 578, 498, 367, 290, 378, 642, 977, 1264, 1435, 1448, 1290, 1000, 735, 728

Offset: 0

Gus Wiseman, Apr 11 2019

The Heinz numbers of these partitions are given by A325197.

			The a(3) = 2 through a(10) = 15 partitions (empty columns not shown):
  (3)    (41)    (33)    (43)    (521)    (333)    (433)
  (111)  (2111)  (42)    (2221)  (32111)  (441)    (442)
                 (222)   (4111)           (522)    (532)
                 (411)                    (531)    (541)
                 (2211)                   (3222)   (3322)
                 (3111)                   (5211)   (3331)
                                          (32211)  (4222)
                                          (33111)  (4411)
                                          (42111)  (5221)
                                                   (5311)
                                                   (32221)
                                                   (33211)
                                                   (42211)
                                                   (43111)
                                                   (52111)

Column k=2 of A325200.

Cf. A046660, A065770, A071724, A243055, A325166, A325169, A325178, A325188, A325189, A325191, A325195, A325197, A325198.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otbmax[#]-otb[#]==2&]],{n,0,30}]

A359358 Let y be the integer partition with Heinz number n. Then a(n) is the size of the Young diagram of y after removing a rectangle of the same length as y and width equal to the smallest part of y.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 1, 0, 0, 2, 0, 2, 2, 4, 0, 1, 0, 5, 0, 3, 0, 3, 0, 0, 3, 6, 1, 2, 0, 7, 4, 2, 0, 4, 0, 4, 1, 8, 0, 1, 0, 4, 5, 5, 0, 3, 2, 3, 6, 9, 0, 3, 0, 10, 2, 0, 3, 5, 0, 6, 7, 5, 0, 2, 0, 11, 2, 7, 1, 6, 0, 2, 0, 12, 0, 4, 4, 13

Offset: 1

Gus Wiseman, Dec 27 2022

nonn

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

			The partition with Heinz number 7865 is (6,5,5,3), which has the following diagram. The 3 X 4 rectangle is shown in dots.
  . . . o o o
  . . . o o
  . . . o o
  . . .
The size of the complement is 7, so a(7865) = 7.

The opposite version is A326844.
Row sums of A356958 are a(n) + A001222(n) - 1, Heinz numbers A246277.
A055396 gives minimum prime index, maximum A061395.
A112798 list prime indices, sum A056239.
A243055 subtracts the least prime index from the greatest.
A326846 = size of the smallest rectangle containing the prime indices of n.
A358195 gives Heinz numbers of rows of A358172, even bisection A241916.
Cf. A124010, A243503, A268192, A316413, A325351, A325352, A326836, A326837, A355534, A359360.

Table[If[n==1,0,With[{q=Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Total[q]-q[[1]]*Length[q]]],{n,100}]

a(n) = A056239(n) - A001222(n) * A055396(n).

a(n) = A056239(n) - A359360(n).

A372430 Positive integers k such that the distinct prime indices of k are a subset of the binary indices of k.

Original entry on oeis.org

1, 3, 5, 15, 27, 39, 55, 63, 85, 121, 125, 135, 169, 171, 175, 209, 243, 247, 255, 299, 375, 399, 437, 459, 507, 539, 605, 637, 725, 735, 783, 841, 867, 891, 1085, 1215, 1323, 1331, 1375, 1519, 1767, 1815, 1863, 2079, 2125, 2187, 2223, 2295, 2299, 2331, 2405

Offset: 1

Gus Wiseman, May 02 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Conjecture: The only number whose binary indices are a subset of its prime indices is 4100, with binary indices {3,13} and prime indices {1,1,3,3,13}. Verified up to 10,000,000.

			The prime indices of 135 are {2,2,2,3}, and the binary indices are {1,2,3,8}. Since {2,3} is a subset of {1,2,3,8}, 135 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     3: {2}
     5: {3}
    15: {2,3}
    27: {2,2,2}
    39: {2,6}
    55: {3,5}
    63: {2,2,4}
    85: {3,7}
   121: {5,5}
   125: {3,3,3}
The terms together with their binary expansions and binary indices begin:
     1:              1 ~ {1}
     3:             11 ~ {1,2}
     5:            101 ~ {1,3}
    15:           1111 ~ {1,2,3,4}
    27:          11011 ~ {1,2,4,5}
    39:         100111 ~ {1,2,3,6}
    55:         110111 ~ {1,2,3,5,6}
    63:         111111 ~ {1,2,3,4,5,6}
    85:        1010101 ~ {1,3,5,7}
   121:        1111001 ~ {1,4,5,6,7}
   125:        1111101 ~ {1,3,4,5,6,7}

The version for equal lengths is A071814, zeros of A372441.
The version for equal sums is A372427, zeros of A372428.
For disjoint instead of subset we have A372431, complement A372432.
The version for equal maxima is A372436, zeros of A372442.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A001221, A230877, A243055, A304818, A355536, A358136, A372429.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],SubsetQ[bix[#],prix[#]]&]

Row k of A304038 is a subset of row k of A048793.

A286471 If n is noncomposite, then a(n) = 0, otherwise 1 + difference between indices of the two smallest (not necessarily distinct) prime factors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 1, 1, 3, 0, 1, 0, 4, 2, 1, 0, 2, 0, 1, 3, 5, 0, 1, 1, 6, 1, 1, 0, 2, 0, 1, 4, 7, 2, 1, 0, 8, 5, 1, 0, 2, 0, 1, 1, 9, 0, 1, 1, 3, 6, 1, 0, 2, 3, 1, 7, 10, 0, 1, 0, 11, 1, 1, 4, 2, 0, 1, 8, 3, 0, 1, 0, 12, 2, 1, 2, 2, 0, 1, 1, 13, 0, 1, 5, 14, 9, 1, 0, 2, 3, 1, 10, 15, 6, 1, 0, 4, 1, 1, 0, 2, 0, 1, 2, 16, 0, 1, 0, 3, 11, 1, 0, 2, 7, 1, 1, 17

Offset: 1

Antti Karttunen, May 11 2017

nonn

			For n = 1, 2 and 3, which are all noncomposite numbers, a(n) = 0.
For n = 4 = 2*2 = prime(1)*prime(1), the difference 1-1 = 0, plus one is 1, thus a(4) = 1.
For n = 6 = 2*3 = prime(1)*prime(2), the difference 2-1 = 1, plus one is 2, thus a(6) = 2.

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

Cf. A001222, A032742, A055396.

Cf. also A241917, A241919, A242411, A243055, A243056, A286472.

Table[If[Length@ # < 2, 0, First@ Differences@ PrimePi@ Take[#, 2] + 1] &@ Flatten[FactorInteger[n] /. {p_, e_} /; p > 0 :> ConstantArray[p, e]], {n, 118}] (* Michael De Vlieger, May 12 2017 *)

from sympy import primepi, isprime, primefactors, divisors
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a(n): return 0 if n==1 or isprime(n) else 1 + a055396(divisors(n)[-2]) - a055396(n) # Indranil Ghosh, May 12 2017

(define (A286471 n) (if (or (= 1 n) (= 1 (A001222 n))) 0 (+ 1 (- (A055396 (A032742 n)) (A055396 n)))))

If n is noncomposite, then a(n) = 0, otherwise a(n) = 1 + A055396(A032742(n)) - A055396(n).

A358169 Row n lists the first differences plus one of the prime indices of n with 1 prepended.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 4, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 2, 6, 1, 4, 2, 2, 1, 1, 1, 1, 7, 1, 2, 1, 8, 1, 1, 3, 2, 3, 1, 5, 9, 1, 1, 1, 2, 3, 1, 1, 6, 2, 1, 1, 1, 1, 4, 10, 1, 2, 2, 11, 1, 1, 1, 1, 1, 2, 4, 1, 7, 3, 2, 1, 1, 2, 1, 12, 1, 8, 2, 5, 1, 1, 1, 3

Offset: 2

Gus Wiseman, Nov 01 2022

Every nonempty composition appears as a row exactly once.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Here this multiset is regarded as a sequence in weakly increasing order.
Also the reversed augmented differences of the integer partition with Heinz number n, where the augmented differences aug(q) of a sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k, and the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The non-reversed version is A355534.

			Triangle begins:
   2: 1
   3: 2
   4: 1 1
   5: 3
   6: 1 2
   7: 4
   8: 1 1 1
   9: 2 1
  10: 1 3
  11: 5
  12: 1 1 2
  13: 6
  14: 1 4
  15: 2 2
  16: 1 1 1 1
  17: 7
  18: 1 2 1
  19: 8
  20: 1 1 3

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Row-lengths are A001222.
The first term of each row is A055396.
Row-sums are A252464.
The rows appear to be ranked by A253566.
Another variation is A287352.
Constant rows have indices A307824.
The Heinz numbers of the rows are A325351.
Strict rows have indices A325366.
Row-minima are A355531, also A355524 and A355525.
Row-maxima are A355532, non-augmented A286470, also A355526.
Reversing rows gives A355534.
The non-augmented version A355536, also A355533.
A112798 lists prime indices, sum A056239.
Cf. A124010, A243055, A243056, A325352, A325390, A355523, A355528.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Differences[Prepend[primeMS[n],1]]+1,{n,30}]

A372431 Positive integers k such that the prime indices of k are disjoint from the binary indices of k.

Original entry on oeis.org

1, 2, 4, 7, 8, 9, 10, 11, 12, 13, 16, 17, 19, 21, 23, 24, 25, 26, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 53, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 72, 73, 74, 76, 79, 80, 81, 82, 83, 84, 86, 89, 92, 93, 94, 96, 97, 98, 101

Offset: 1

Gus Wiseman, May 03 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The binary indices of 65 are {1,7}, and the prime indices are {3,6}, so 65 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     7: {4}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    11: {5}
    12: {1,1,2}
    13: {6}
    16: {1,1,1,1}
The terms together with their binary expansions and binary indices begin:
   1:       1 ~ {1}
   2:      10 ~ {2}
   4:     100 ~ {3}
   7:     111 ~ {1,2,3}
   8:    1000 ~ {4}
   9:    1001 ~ {1,4}
  10:    1010 ~ {2,4}
  11:    1011 ~ {1,2,4}
  12:    1100 ~ {3,4}
  13:    1101 ~ {1,3,4}
  16:   10000 ~ {5}

For subset instead of disjoint we have A372430.
The complement is A372432.
Equal lengths: A071814, zeros of A372441.
Equal sums: A372427, zeros of A372428.
Equal maxima: A372436, zeros of A372442.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A001221, A059893, A096111, A230877, A243055, A304818, A355536, A358136, A372429.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[bix[#],prix[#]]=={}&]

A297173 Smallest difference between indices of prime divisors of n, or 0 if n is a prime power.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 1, 0, 0, 1, 0, 2, 2, 4, 0, 1, 0, 5, 0, 3, 0, 1, 0, 0, 3, 6, 1, 1, 0, 7, 4, 2, 0, 1, 0, 4, 1, 8, 0, 1, 0, 2, 5, 5, 0, 1, 2, 3, 6, 9, 0, 1, 0, 10, 2, 0, 3, 1, 0, 6, 7, 1, 0, 1, 0, 11, 1, 7, 1, 1, 0, 2, 0, 12, 0, 1, 4, 13, 8, 4, 0, 1, 2, 8, 9, 14, 5, 1, 0, 3, 3, 2, 0, 1, 0, 5, 1

Offset: 1

Antti Karttunen, Mar 03 2018

nonn

			For n = 130 = 2*5*13 = prime(1)*prime(3)*prime(6), the smallest difference between indices is 3-1 = 2, thus a(130) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A073490, A073491.

Cf. also A100573, A243055, A242411, A261079, A286470, A286471.

A297173(n) = if(omega(n)<=1,0,my(ps=factor(n)[,1]); vecmin(vector((#ps)-1,i,primepi(ps[i+1])-primepi(ps[i]))));

a(A073491(n)) <= 1.

A307516 Numbers whose maximum prime index and minimum prime index differ by more than 1.

Original entry on oeis.org

10, 14, 20, 21, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114, 115, 116, 117, 118

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose maximum and minimum parts differ by more than 1. The enumeration of these partitions by sum is given by A000094.

Differs from A069900 first at n = 43.

			The sequence of terms together with their prime indices begins:
   10: {1,3}
   14: {1,4}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   38: {1,8}
   39: {2,6}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   46: {1,9}
   50: {1,3,3}
   51: {2,7}
   52: {1,1,6}
   55: {3,5}

Positions of numbers > 1 in A243055. Complement of A000961 and A256617.

Cf. A000094, A000245, A001222, A052126, A056239, A061395, A064989, A069900, A105441, A112798, A307517, A325196.

with(numtheory):
q:= n-> (l-> pi(l[-1])-pi(l[1])>1)(sort([factorset(n)[]])):
select(q, [$2..200])[];  # Alois P. Heinz, Apr 12 2019

Select[Range[100],PrimePi[FactorInteger[#][[-1,1]]]-PrimePi[FactorInteger[#][[1,1]]]>1&]

cp's OEIS Frontend

A307517 Numbers with at least two not necessarily distinct prime factors less than the largest prime factor.

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A325198 Positive numbers whose maximum prime index minus minimum prime index is 2.

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A325199 Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 2.

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A359358 Let y be the integer partition with Heinz number n. Then a(n) is the size of the Young diagram of y after removing a rectangle of the same length as y and width equal to the smallest part of y.

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A372430 Positive integers k such that the distinct prime indices of k are a subset of the binary indices of k.

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A286471 If n is noncomposite, then a(n) = 0, otherwise 1 + difference between indices of the two smallest (not necessarily distinct) prime factors of n.

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A358169 Row n lists the first differences plus one of the prime indices of n with 1 prepended.

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A372431 Positive integers k such that the prime indices of k are disjoint from the binary indices of k.

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