A028983 -id:A028983 - cp's OEIS frontend

A347448 Number of integer partitions of n with alternating product > 1.

0, 0, 1, 2, 3, 5, 8, 12, 17, 25, 35, 49, 66, 90, 120, 161, 209, 275, 355, 460, 585, 750, 946, 1199, 1498, 1881, 2335, 2909, 3583, 4430, 5428, 6666, 8118, 9912, 12013, 14586, 17592, 21252, 25525, 30695, 36711, 43956, 52382, 62469, 74173, 88132, 104303, 123499

Offset: 0

Gus Wiseman, Sep 16 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The a(2) = 1 through a(7) = 12 partitions:
  (2)  (3)   (4)    (5)     (6)      (7)
       (21)  (31)   (32)    (42)     (43)
             (211)  (41)    (51)     (52)
                    (311)   (222)    (61)
                    (2111)  (321)    (322)
                            (411)    (421)
                            (3111)   (511)
                            (21111)  (2221)
                                     (3211)
                                     (4111)
                                     (31111)
                                     (211111)

The strict case is A000009, except that a(0) = a(1) = 0.
Allowing any alternating product >= 1 gives A000041, reverse A344607.
Ranked by A028983 (reverse A347465), which has complement A028982.
The complement is counted by A119620, reverse A347443.
The multiplicative version is A339890, weak A347456, reverse A347705.
The even-length case is A344608.
Allowing any integer reverse-alternating product gives A347445.
Allowing any integer alternating product gives A347446.
The reverse version is A347449, also the odd-length case.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A347461 counts possible alternating products of partitions.
Cf. A000070, A086543, A100824, A236913, A325534, A325535, A339846, A344654, A345196, A347440, A347444, A347462.

a:= n-> (p-> p(n)-p(iquo(n, 2)))(combinat[numbpart]):
seq(a(n), n=0..63);  # Alois P. Heinz, Oct 04 2021

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],altprod[#]>1&]],{n,0,30}]

a(n) = A000041(n) - A119620(n).

A347453 Heinz numbers of odd-length integer partitions with integer alternating (or reverse-alternating) product.

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 42, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 112, 113, 114, 116, 117, 124, 125, 127, 128, 130

Offset: 1

Gus Wiseman, Sep 24 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also numbers whose multiset of prime indices has odd length and integer alternating product, where a prime index of n is a number m such that prime(m) divides n.

			The terms and their prime indices begin:
      2: {1}         29: {10}            61: {18}
      3: {2}         31: {11}            63: {2,2,4}
      5: {3}         32: {1,1,1,1,1}     67: {19}
      7: {4}         37: {12}            68: {1,1,7}
      8: {1,1,1}     41: {13}            71: {20}
     11: {5}         42: {1,2,4}         72: {1,1,1,2,2}
     12: {1,1,2}     43: {14}            73: {21}
     13: {6}         44: {1,1,5}         75: {2,3,3}
     17: {7}         45: {2,2,3}         76: {1,1,8}
     18: {1,2,2}     47: {15}            78: {1,2,6}
     19: {8}         48: {1,1,1,1,2}     79: {22}
     20: {1,1,3}     50: {1,3,3}         80: {1,1,1,1,3}
     23: {9}         52: {1,1,6}         83: {23}
     27: {2,2,2}     53: {16}            89: {24}
     28: {1,1,4}     59: {17}            92: {1,1,9}

The reciprocal version is A000290.
Allowing any alternating product <= 1 gives A001105.
Allowing any alternating product gives A026424.
Factorizations of this type are counted by A347441.
These partitions are counted by A347444.
Allowing any length gives A347454.
Allowing any alternating product > 1 gives A347465.
A027193 counts odd-length partitions.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A344606 counts alternating permutations of prime indices.
A347446 counts partitions with integer alternating product.
A347457 ranks partitions with integer alt product, complement A347455.
A347461 counts possible alternating products of partitions.
A347462 counts possible reverse-alternating products of partitions.
Cf. A001222, A028260, A028982, A028983, A339890, A344617, A344653, A345958, A346703, A346704, A347437, A347443, A347450, A347451.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],OddQ[PrimeOmega[#]]&&IntegerQ[altprod[primeMS[#]]]&]

A347451 Numbers whose multiset of prime indices has integer reciprocal alternating product.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 14, 16, 18, 21, 22, 24, 25, 26, 32, 34, 36, 38, 39, 40, 46, 49, 50, 54, 56, 57, 58, 62, 64, 65, 72, 74, 81, 82, 84, 86, 87, 88, 90, 94, 96, 98, 100, 104, 106, 111, 115, 118, 121, 122, 126, 128, 129, 133, 134, 136, 142, 144, 146, 150, 152

Offset: 1

Gus Wiseman, Sep 24 2021

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.
We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
Also Heinz numbers integer partitions with integer reverse-reciprocal alternating product, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The terms and their prime indices begin:
      1: {}            32: {1,1,1,1,1}       65: {3,6}
      2: {1}           34: {1,7}             72: {1,1,1,2,2}
      4: {1,1}         36: {1,1,2,2}         74: {1,12}
      6: {1,2}         38: {1,8}             81: {2,2,2,2}
      8: {1,1,1}       39: {2,6}             82: {1,13}
      9: {2,2}         40: {1,1,1,3}         84: {1,1,2,4}
     10: {1,3}         46: {1,9}             86: {1,14}
     14: {1,4}         49: {4,4}             87: {2,10}
     16: {1,1,1,1}     50: {1,3,3}           88: {1,1,1,5}
     18: {1,2,2}       54: {1,2,2,2}         90: {1,2,2,3}
     21: {2,4}         56: {1,1,1,4}         94: {1,15}
     22: {1,5}         57: {2,8}             96: {1,1,1,1,1,2}
     24: {1,1,1,2}     58: {1,10}            98: {1,4,4}
     25: {3,3}         62: {1,11}           100: {1,1,3,3}
     26: {1,6}         64: {1,1,1,1,1,1}    104: {1,1,1,6}

The version for reversed prime indices is A028982, counted by A119620.
The additive version is A119899, strict A028260.
Allowing any alternating product >= 1 gives A344609.
Factorizations of this type are counted by A347439.
Allowing any alternating product <= 1 gives A347450.
The non-reciprocal version is A347454.
Allowing any alternating product > 1 gives A347465, reverse A028983.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A344606 counts alternating permutations of prime indices.
A347457 ranks partitions with integer alternating product.
Cf. A001222, A236913, A316523, A344617, A345958, A345959, A346703, A346704, A347437, A347446, A347455.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],IntegerQ[1/altprod[primeMS[#]]]&]

A248150 Numbers whose sum of divisors (A000203) is divisible by 4.

Original entry on oeis.org

3, 6, 7, 11, 12, 14, 15, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 35, 38, 39, 42, 43, 44, 46, 47, 48, 51, 54, 55, 56, 57, 59, 60, 62, 63, 65, 66, 67, 69, 70, 71, 75, 76, 77, 78, 79, 83, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 102, 103, 105, 107, 108, 110, 111, 112, 114, 115, 118

Offset: 1

M. F. Hasler, Oct 02 2014

nonn

A subsequence of A028983 (even sum of divisors) which contains all numbers but the squares and twice the squares, so no term of this sequence is of that form, either.

Any number having at least two odd prime factors to an odd power is in this sequence, therefore it has asymptotic density 1. - M. F. Hasler, Apr 26 2017

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

Cf. A000203, A028982, A028983, A248151.

First differs from A022544 by including 65.

Select[Range[200],Divisible[DivisorSigma[1,#],4]&] (* Harvey P. Dale, Feb 20 2015 *)

PARI
```
for(n=1,999,sigma(n)%4||print1(n","))
```

a(n) ~ n. - Charles R Greathouse IV, Sep 01 2015

A074627 Numbers n such that sigma(n) is divisible by 6.

Original entry on oeis.org

5, 6, 10, 11, 14, 15, 17, 20, 22, 23, 24, 26, 29, 30, 33, 34, 35, 38, 40, 41, 42, 44, 45, 46, 47, 51, 53, 54, 55, 56, 58, 59, 60, 62, 65, 66, 68, 69, 70, 71, 74, 77, 78, 80, 82, 83, 85, 86, 87, 88, 89, 90, 92, 94, 95, 96, 99, 101, 102, 104, 105, 106, 107, 110, 113, 114, 115

Offset: 1

Labos Elemer, Aug 26 2002

n=10: sigma(10) = 1+2+5+10 = 18 = 3*6.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000203, A072862, A074384, A074628, A067051, A074630.

Select[Range@ 120, Divisible[DivisorSigma[1, #], 6] &] (* Michael De Vlieger, Feb 25 2017 *)

isok(n) = !(sigma(n) % 6); \\ Michel Marcus, Dec 17 2013

A000203(n) modulo 6 = 0.
{n: A084301(n) = 0 }. - R. J. Mathar, May 19 2020
A087943 INTERSECT A028983. - R. J. Mathar, May 19 2020

A347465 Numbers whose multiset of prime indices has alternating product > 1.

Original entry on oeis.org

3, 5, 7, 11, 12, 13, 17, 19, 20, 23, 27, 28, 29, 30, 31, 37, 41, 42, 43, 44, 45, 47, 48, 52, 53, 59, 61, 63, 66, 67, 68, 70, 71, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 112, 113, 114, 116, 117, 120, 124, 125, 127

Offset: 1

Gus Wiseman, Sep 27 2021

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.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
All terms have odd bigomega (A001222).
Also Heinz numbers integer partitions with reverse-alternating product > 1.

			The terms and their prime indices begin:
      3: {2}         37: {12}            68: {1,1,7}
      5: {3}         41: {13}            70: {1,3,4}
      7: {4}         42: {1,2,4}         71: {20}
     11: {5}         43: {14}            73: {21}
     12: {1,1,2}     44: {1,1,5}         75: {2,3,3}
     13: {6}         45: {2,2,3}         76: {1,1,8}
     17: {7}         47: {15}            78: {1,2,6}
     19: {8}         48: {1,1,1,1,2}     79: {22}
     20: {1,1,3}     52: {1,1,6}         80: {1,1,1,1,3}
     23: {9}         53: {16}            83: {23}
     27: {2,2,2}     59: {17}            89: {24}
     28: {1,1,4}     61: {18}            92: {1,1,9}
     29: {10}        63: {2,2,4}         97: {25}
     30: {1,2,3}     66: {1,2,5}         99: {2,2,5}
     31: {11}        67: {19}           101: {26}

The squarefree case is A030059 without 2.
The reverse version is A028983, counted by A119620.
The opposite version (< 1 instead of > 1) is A119899.
Factorizations of this type are counted by A339890, reverse A347705.
The weak version (>= 1 instead of > 1) is A344609.
Partitions of this type are counted by A347449, reverse A347448.
The complement is A347450, counted by A339846 or A347443.
Allowing any integer reverse-alternating product gives A347454.
Allowing any integer alternating product gives A347457.
A335433 ranks inseparable partitions, complement A335448.
A347446 counts partitions with integer alternating product, reverse A347445.
Cf. A008549, A344607, A344608, A344611, A347442, A347444, A347447, A347453, A347456, A347461, A347462.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],altprod[primeMS[#]]>1&]

A347705 Number of factorizations of n with reverse-alternating product > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 7, 1, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 7, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 1, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 8, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 3, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 12 2021

nonn

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

			The a(n) factorizations for n = 2, 6, 8, 12, 24, 30, 48, 60:
  2   6     8       12      24        30      48          60
      2*3   2*4     2*6     3*8       5*6     6*8         2*30
            2*2*2   3*4     4*6       2*15    2*24        3*20
                    2*2*3   2*12      3*10    3*16        4*15
                            2*2*6     2*3*5   4*12        5*12
                            2*3*4             2*3*8       6*10
                            2*2*2*3           2*4*6       2*5*6
                                              3*4*4       3*4*5
                                              2*2*12      2*2*15
                                              2*2*2*6     2*3*10
                                              2*2*3*4     2*2*3*5
                                              2*2*2*2*3

Positions of 1's are A000430.
The weak version (>= instead of >) is A001055, non-reverse A347456.
The non-reverse version is A339890, strict A347447.
The version for reverse-alternating product 1 is A347438.
Allowing any integer reciprocal alternating product gives A347439.
The even-length case is A347440, also the opposite reverse version.
Allowing any integer rev-alt product gives A347442, non-reverse A347437.
The version for partitions is A347449, non-reverse A347448.
A001055 counts factorizations (strict A045778, ordered A074206).
A038548 counts possible rev-alt products of factorizations, integer A046951.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A292886 counts knapsack factorizations, by sum A293627.
A347707 counts possible integer reverse-alternating products of partitions.
Cf. A028983, A119620, A339846, A347441, A347443, A347450, A347463, A347466.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
revaltprod[q_]:=Product[q[[-i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],revaltprod[#]>1&]],{n,100}]

a(n) = A001055(n) - A347438(n).

A071189 Smallest prime factor of sum of divisors of n.

Original entry on oeis.org

1, 3, 2, 7, 2, 2, 2, 3, 13, 2, 2, 2, 2, 2, 2, 31, 2, 3, 2, 2, 2, 2, 2, 2, 31, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 127, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, May 15 2002

nonn

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

Cf. A000203, A020639, A028982, A028983, A071187, A071190.

Table[FactorInteger[DivisorSigma[1,n]][[1,1]],{n,90}] (* Harvey P. Dale, May 15 2011 *)

A071189(n) = if(1==n, n, my(f = factor(sigma(n))); vecmin(f[, 1])); \\ Antti Karttunen, Jul 24 2017

first(n) = {my(v = vector(n, i, 2), sq = List()); for(i=1, sqrtint(n), listput(sq, i^2); listput(sq, 2*i^2)); listsort(sq); v[1]=1; for(i=2, #sq, if(sq[i]>n,break); v[sq[i]] = factor(sigma(sq[i]))[, 1]~[1]);v} \\ David A. Corneth, Jul 24 2017

a(n) = A020639(A000203(n)).

a(n) = 2 if and only if n is in A028983. - Amiram Eldar, Mar 24 2024

A329575 Numbers whose smallest Fermi-Dirac factor is 3.

Original entry on oeis.org

3, 12, 15, 21, 27, 33, 39, 48, 51, 57, 60, 69, 75, 84, 87, 93, 105, 108, 111, 123, 129, 132, 135, 141, 147, 156, 159, 165, 177, 183, 189, 192, 195, 201, 204, 213, 219, 228, 231, 237, 240, 243, 249, 255, 267, 273, 276, 285, 291, 297, 300, 303, 309, 321, 327, 336, 339, 345

Offset: 1

Peter Munn, Apr 27 2020

nonn

Every positive integer is the product of a unique subset of the terms of A050376 (sometimes called Fermi-Dirac primes). This sequence lists the numbers where the relevant subset includes 3 but not 2.
Numbers whose squarefree part is divisible by 3 but not 2.
Positive multiples of 3 that survive sieving by the rule: if m appears then 2m, 3m and 6m do not. Asymptotic density is 1/6.

			6 is the product of the following terms of A050376: 2, 3. These terms include 2, so 6 is not in the sequence.
12 is the product of the following terms of A050376: 3, 4. These terms include 3, but not 2, so 12 is in the sequence.
20 is the product of the following terms of A050376: 4, 5. These terms do not include 3, so 20 is not in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Cf. A007913, A036668, A050376, A059897, A223490, A307150, A339690.
Intersection of any 2 of A003159, A145204 and A325424; also subsequence of A028983.
Ordered 3rd quadrisection of A052330.

f[p_, e_] := p^(2^IntegerExponent[e, 2]); fdmin[n_] := Min @@ f @@@ FactorInteger[n]; Select[Range[350], fdmin[#] == 3 &] (* Amiram Eldar, Nov 27 2020 *)

isok(m) = core(m) % 6 == 3; \\ Michel Marcus, May 01 2020

from itertools import count
from sympy import integer_log
def A329575(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 f(x):
        c = n+x
        for i in range(integer_log(x,9)[0]+1):
            i2 = 9**i
            for j in count(0,2):
                k = i2<x:
                    break
                m = x//k
                c -= (m-1)//6+(m-5)//6+2
        return c
    return 3*bisection(f,n,n) # Chai Wah Wu, Apr 10 2025

A223490(a(n)) = 3.
A007913(a(n)) == 3 (mod 6).
A059897(2, a(n)) = 2 * a(n).
A059897(3, a(n)) * 3 = a(n).
{a(n) : n >= 1} = {k : 3 * A307150(k) = 2 * k}.
A003159 = {a(n) / 3 : n >= 1} U {a(n) : n >= 1}.
A036668 = {a(n) / 3 : n >= 1} U {a(n) * 2 : n >= 1}.
A145204 \ {0} = {a(n) : n >= 1} U {a(n) * 2 : n >= 1}.
a(n) = 3*A339690(n). - Chai Wah Wu, Apr 10 2025

A055008 Numbers k such that gcd(phi(k), sigma(k)) = 1 with phi = A000010, sigma = A000203.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 25, 32, 36, 50, 64, 81, 100, 121, 128, 144, 225, 242, 256, 289, 324, 400, 484, 512, 529, 576, 578, 625, 729, 800, 841, 900, 1024, 1058, 1089, 1156, 1250, 1296, 1600, 1681, 1682, 1936, 2025, 2048, 2116, 2209, 2304, 2312, 2401, 2500, 2601

Offset: 1

Labos Elemer, May 31 2000

nonn

The asymptotic density of this sequence is 0 (Dressler, 1974). - Amiram Eldar, Jul 23 2020
Conjecture: Every term is a square or twice a square. - Jason Yuen, May 16 2024
The conjecture is true: If k is neither a square nor twice a square (i.e., in A028983), then sigma(k) is even. Since gcd(phi(k), sigma(k)) = 1, then phi(k) must be odd, but phi(k) is odd only for k = 1 and 2. - Amiram Eldar, May 19 2024

			For n = 484, phi(484) = 220 = 2*2*5*11, sigma(484) = 931 = 7*7*19, and gcd(220,931) = 1.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Robert E. Dressler, On a theorem of Niven, Canadian Mathematical Bulletin, Vol. 17, No. 1 (1974), pp. 109-110.

Cf. A000010, A000203, A009223, A028983.

Select[Range@ 2700, CoprimeQ[EulerPhi@ #, DivisorSigma[1, #]] &] (* Michael De Vlieger, Feb 05 2017 *)
Select[With[{max = 51}, Union[Array[#^2 &, max], Array[2*#^2 &, Floor[max / Sqrt[2]]]]], CoprimeQ[EulerPhi[#], DivisorSigma[1, #]] &] (* Amiram Eldar, May 19 2024 *)

is(n)=gcd(sigma(n),eulerphi(n))==1 \\ Charles R Greathouse IV, Feb 19 2013

Incorrect comment removed by Charles R Greathouse IV, Feb 19 2013

cp's OEIS Frontend

A347448 Number of integer partitions of n with alternating product > 1.

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A347453 Heinz numbers of odd-length integer partitions with integer alternating (or reverse-alternating) product.

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A347451 Numbers whose multiset of prime indices has integer reciprocal alternating product.

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A248150 Numbers whose sum of divisors (A000203) is divisible by 4.

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PARI

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A074627 Numbers n such that sigma(n) is divisible by 6.

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PARI

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A347465 Numbers whose multiset of prime indices has alternating product > 1.

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A347705 Number of factorizations of n with reverse-alternating product > 1.

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A071189 Smallest prime factor of sum of divisors of n.

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PARI