A028983 -id:A028983 - cp's OEIS frontend

A011257 Numbers k such that the geometric mean of phi(k) and sigma(k) is an integer.

1, 14, 30, 51, 105, 170, 194, 248, 264, 364, 405, 418, 477, 595, 679, 714, 760, 780, 1023, 1455, 1463, 1485, 1496, 1512, 1524, 1674, 1715, 1731, 1796, 1804, 2058, 2080, 2651, 2754, 2945, 3080, 3135, 3192, 3410, 3534, 3567, 3596, 3828, 3956, 4064, 4381, 4420

Offset: 1

N. J. A. Sloane

nonn

For these terms the arithmetic mean is also an integer. It is conjectured that sigma(k) for these numbers is never odd. See also A065146, A028982, A028983. - Labos Elemer, Oct 18 2001
If p > 2 and 2^p - 1 is prime (a Mersenne prime) then m = 2^(p-2)*(2^p-1) is in the sequence because phi(m) = 2^(p-2)*(2^(p-1)-1); sigma(m) = (2^(p-1)-1)*2^p hence sqrt(phi(m)*sigma(m)) = 2^(p-1)*(2^(p-1)-1) is an integer. So for j > 1, 2^(A000043(j)-2)*2^(A000043(j)-1) is in the sequence. - Farideh Firoozbakht, Nov 27 2005
From a(2633) = 6931232 on, it is no longer true (as was once conjectured) that a(n) > n^2. - M. F. Hasler, Feb 07 2009
It follows from Theorems 1 and 2 in Broughan-Ford-Luca that a(n) << n^(24+e) for all e > 0. - Charles R Greathouse IV, May 09 2013

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 51, p. 19, Ellipses, Paris 2008.
Zhang Ming-Zhi (typescript submitted to Unsolved Problems section of Monthly, 96-01-10).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2000 terms from M. F. Hasler)
K. Broughan, K. Ford, and F. Luca, On square values of the product of the Euler totient function and sum of divisors function, Colloquium Mathematicum, (to appear).
Tristan Freiberg, Products of shifted primes simultaneously taking perfect power values, Journal of the Australian Mathematical Society 92:2 (2012), pp. 145-154. arXiv:1008.1978 [math.NT], 2010.
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
Luis Elesban Santos Cruz and Florian Luca, Power values of the product of the Euler function and the sum of divisors function, involve, Vol. 8 (2015), No. 5, 745-748.

Cf. A000043, A000668.

Cf. A293391 (sigma(m)/phi(m) is a perfect square), A327624 (this sequence \ A293391).

[k:k in [1..4500]| IsPower(EulerPhi(k)*DivisorSigma(1,k),2)]; // Marius A. Burtea, Sep 19 2019

Select[Range[8000], IntegerQ[Sqrt[DivisorSigma[1, #] EulerPhi[#]]] &] (* Carl Najafi, Aug 16 2011 *)

is(n)=issquare(eulerphi(n)*sigma(n)) \\ Charles R Greathouse IV, May 09 2013

A331590 Square array A(n,k) = A225546(A225546(n) * A225546(k)), n >= 1, k >= 1, read by descending antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 6, 6, 4, 5, 8, 5, 8, 5, 6, 10, 12, 12, 10, 6, 7, 5, 15, 9, 15, 5, 7, 8, 14, 10, 20, 20, 10, 14, 8, 9, 12, 21, 24, 7, 24, 21, 12, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 15, 27, 18, 35, 15, 35, 18, 27, 15, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 13, 24, 33, 40, 45, 20, 11, 20, 45, 40, 33, 24, 13

Offset: 1

Peter Munn, Jan 21 2020

As a binary operation, this sequence defines a commutative monoid over the positive integers that is isomorphic to multiplication. The self-inverse permutation A225546(.) provides an isomorphism. This monoid therefore has unique factorization. Its primes are the even terms of A050376: 2, 4, 16, 256, ..., which in standard integer multiplication are the powers of 2 with powers of 2 as exponents.
In this monoid, in contrast, the powers of 2 run through the squarefree numbers, the k-th power of 2 being A019565(k). 4 is irreducible and its powers are the squares of the squarefree numbers, the k-th power of 4 being A019565(k)^2 (where "^2" denotes standard integer squaring); and so on with powers of 16, 256, ...
In many cases the product of two numbers is the same here as in standard integer multiplication. See the formula section for details.

			From _Antti Karttunen_, Feb 02 2020: (Start)
The top left 16 X 16 corner of the array:
   1,  2,  3,  4,  5,  6,   7,   8,   9,  10,  11,  12,  13,  14,  15,  16, ...
   2,  3,  6,  8, 10,  5,  14,  12,  18,  15,  22,  24,  26,  21,  30,  32, ...
   3,  6,  5, 12, 15, 10,  21,  24,  27,  30,  33,  20,  39,  42,   7,  48, ...
   4,  8, 12,  9, 20, 24,  28,  18,  36,  40,  44,  27,  52,  56,  60,  64, ...
   5, 10, 15, 20,  7, 30,  35,  40,  45,  14,  55,  60,  65,  70,  21,  80, ...
   6,  5, 10, 24, 30, 15,  42,  20,  54,   7,  66,  40,  78,  35,  14,  96, ...
   7, 14, 21, 28, 35, 42,  11,  56,  63,  70,  77,  84,  91,  22, 105, 112, ...
   8, 12, 24, 18, 40, 20,  56,  27,  72,  60,  88,  54, 104,  84, 120, 128, ...
   9, 18, 27, 36, 45, 54,  63,  72,  25,  90,  99, 108, 117, 126, 135, 144, ...
  10, 15, 30, 40, 14,  7,  70,  60,  90,  21, 110, 120, 130, 105,  42, 160, ...
  11, 22, 33, 44, 55, 66,  77,  88,  99, 110,  13, 132, 143, 154, 165, 176, ...
  12, 24, 20, 27, 60, 40,  84,  54, 108, 120, 132,  45, 156, 168,  28, 192, ...
  13, 26, 39, 52, 65, 78,  91, 104, 117, 130, 143, 156,  17, 182, 195, 208, ...
  14, 21, 42, 56, 70, 35,  22,  84, 126, 105, 154, 168, 182,  33, 210, 224, ...
  15, 30,  7, 60, 21, 14, 105, 120, 135,  42, 165,  28, 195, 210,  35, 240, ...
  16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240,  81, ...
(End)

Antti Karttunen, Antidiagonals n = 1..144, flattened
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..80200; (antidiagonals n = 1..400)
Eric Weisstein's World of Mathematics, Monoid

Isomorphic to A003991 with A225546 as isomorphism.
Cf. A003961(main diagonal), A048675, A059895, A059896, A059897.
Rows/columns, sorted in ascending order: 2: A000037, 3: A028983, 4: A252849.
A019565 lists powers of 2 in order of increasing exponent.
Powers of k, sorted in ascending order: k=2: A005117, k=3: A056911, k=4: A062503, k=5: A276378, k=6: intersection of A325698 and A005117, k=7: intersection of A007775 and A005117, k=8: A062838.
Irreducibles are A001146 (even terms of A050376).

up_to = 1275;
A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331590sq(x,y) = if(1==x,y,if(1==y,x, my(fx=factor(x),fy=factor(y),u=max(#binary(vecmax(fx[, 2])),#binary(vecmax(fy[, 2]))),prodsx=vector(u,x,1),m=1); for(i=1,u,for(k=1,#fx~, if(bitand(fx[k,2],m),prodsx[i] *= fx[k,1])); for(k=1,#fy~, if(bitand(fy[k,2],m),prodsx[i] *= fy[k,1])); m<<=1); prod(i=1,u,A019565(A048675(prodsx[i]))^(1<<(i-1)))));
A331590list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A331590sq(col,(a-(col-1))))); (v); };
v331590 = A331590list(up_to);
A331590(n) = v331590[n]; \\ Antti Karttunen, Feb 02 2020

Alternative definition: A(n,1) = n; A(n,k) = A(A059897(n,k), A003961(A059895(n,k))).
Main derived identities: (Start)
A(n,k) = A(k,n).
A(1,n) = n.
A(n, A(m,k)) = A(A(n,m), k).
A(m,m) = A003961(m).
A(n^2, k^2) = A(n,k)^2.
A(A003961(n), A003961(k)) = A003961(A(n,k)).
A(A019565(n), A019565(k)) = A019565(n+k).
(End)
Characterization of conditions for A(n,k) = n * k: (Start)
The following 4 conditions are equivalent:
(1) A(n,k) = n * k;
(2) A(n,k) = A059897(n,k);
(3) A(n,k) = A059896(n,k);
(4) A059895(n,k) = 1.
If gcd(n,k) = 1, A(n,k) = n * k.
If gcd(n,k) = 1, A(A225546(n), A225546(k)) = A225546(n) * A225546(k).
The previous formula implies A(n,k) = n * k in the following cases:
(1) for n = A005117(m), k = j^2;
(2) more generally for n = A005117(m_1)^(2^i_1), k = A005117(m_2)^(2^i_2), with A004198(i_1, i_2) = 0.
(End)

A347457 Heinz numbers of integer partitions with integer alternating product.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 71, 72, 73, 74, 75, 76, 78

Offset: 1

Gus Wiseman, Sep 26 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 integer reverse-alternating product.

			The prime indices of 525 are {2,3,3,4}, with reverse-alternating product 2, so 525 is in the sequence
The prime indices of 135 are {2,2,2,3}, with reverse-alternating product 3/2, so 135 is not in the sequence.

The reciprocal version is A028982.
Allowing any alternating product > 1 gives A028983, reverse A347465.
Factorizations of this type are counted by A347437.
These partitions are counted by A347446.
The reverse reciprocal version A347451.
The odd-length case is A347453.
The reverse version is A347454.
The complement is A347455.
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.
A347461 counts possible alternating products of partitions, reverse A347462.
Cf. A001105, A001222, A028260, A119620, A119899, A316523, A344606, A344617, A346703, A346704, A347448, A347450.

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[altprod[Reverse[primeMS[#]]]]&]

A145204 Numbers whose representation in base 3 (A007089) ends in an odd number of zeros.

Original entry on oeis.org

0, 3, 6, 12, 15, 21, 24, 27, 30, 33, 39, 42, 48, 51, 54, 57, 60, 66, 69, 75, 78, 84, 87, 93, 96, 102, 105, 108, 111, 114, 120, 123, 129, 132, 135, 138, 141, 147, 150, 156, 159, 165, 168, 174, 177, 183, 186, 189, 192, 195, 201, 204, 210, 213, 216, 219, 222, 228, 231

Offset: 1

Reinhard Zumkeller, Oct 04 2008

nonn

Previous name: Complement of A007417.
Also numbers having infinitary divisor 3, or the same, having factor 3 in their Fermi-Dirac representation as product of distinct terms of A050376. - Vladimir Shevelev, Mar 18 2013
For n > 1: where even terms occur in A051064. - Reinhard Zumkeller, May 23 2013
If we exclude a(1) = 0, these are numbers whose squarefree part is divisible by 3, which can be partitioned into numbers whose squarefree part is congruent to 3 mod 9 (A055041) and 6 mod 9 (A055040) respectively. - Peter Munn, Jul 14 2020
The inclusion of 0 as a term might be viewed as a cultural preference: if we habitually wrote numbers enclosed in brackets and then used a null string of digits for zero, the natural number sequence in ternary would be [], [1], [2], [10], [11], [12], [20], ... . - Peter Munn, Aug 02 2020
The asymptotic density of this sequence is 1/4. - Amiram Eldar, Sep 20 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.
Index entries for 3-automatic sequences.

Subsequence of A008585, A028983.
Subsequences: A016051, A055040, A055041, A329575.
Cf. A007089, A007417 (complement), A050376, A182581 (characteristic function).
Positions of 0s in A014578.
Excluding 0: the positions of odd numbers in A007949; equivalently, of even numbers in A051064; symmetric difference of A003159 and A036668.
Related to A042964 via A052330.
Related to A036554 via A064614.

a145204 n = a145204_list !! (n-1)
a145204_list = 0 : map (+ 1) (findIndices even a051064_list)
-- Reinhard Zumkeller, May 23 2013

isA145204 := proc(n) local d, c;
if n = 0 then return true fi;
d := A007089(n); c := 0;
while irem(d, 10) = 0 do c := c+1; d := iquo(d, 10) od;
type(c, odd) end:
select(isA145204, [$(0..231)]); # Peter Luschny, Aug 05 2020

Select[ Range[0, 235], (# // IntegerDigits[#, 3]& // Split // Last // Count[#, 0]& // OddQ)&] (* Jean-François Alcover, Mar 18 2013 *)
Join[{0}, Select[Range[235], OddQ @ IntegerExponent[#, 3] &]] (* Amiram Eldar, Sep 20 2020 *)

import numpy as np
def isA145204(n):
    if n == 0: return True
    c = 0
    d = int(np.base_repr(n, base = 3))
    while d % 10 == 0:
        c += 1
        d //= 10
    return c % 2 == 1
print([n for n in range(231) if isA145204(n)]) # Peter Luschny, Aug 05 2020

from sympy import integer_log
def A145204(n):
    if n == 1: return 0
    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): return n-1+sum(((m:=x//9**i)-2)//3+(m-1)//3+2 for i in range(integer_log(x,9)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 15 2025

a(n) = 3 * A007417(n-1) for n > 1.
A014578(a(n)) = 0.
For n > 1, A007949(a(n)) mod 2 = 1. [Edited by Peter Munn, Aug 02 2020]
{a(n) : n >= 2} = {A052330(A042964(k)) : k >= 1} = {A064614(A036554(k)) : k >= 1}. - Peter Munn, Aug 31 2019 and Dec 06 2020

New name using a comment of Vladimir Shevelev by Peter Luschny, Aug 05 2020

A062699 Numbers n such that sigma(n) = 2*phi(n).

Original entry on oeis.org

3, 35, 1045, 24871, 29029, 50065, 58435, 64285, 87685, 137885, 140335, 1390753, 1529983, 1739507, 2011009, 2086903, 3189625, 3281663, 3501605, 3722875, 3830827, 3852155, 6605945, 7711405, 8409305, 9815195, 11413205, 11569805, 13321295, 13932919, 16540205

Offset: 1

Jason Earls, Jul 11 2001

nonn

3 is the only prime term of this sequence. There is no term of the form p^k where p is a prime and k>1. All terms are odd because if n is even then 2*phi(n)=phi(2n)<=nFarideh Firoozbakht, Apr 01 2005, Feb 24 2007

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..50 from Harry J. Smith, terms 51..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.

Cf. A068390, A104900, A104901.

Subsequence of A028983 (sigma(k) is even).

Select[Range[10^6], DivisorSigma[1, #] == 2 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)

for(n=1,500000, if(sigma(n)==eulerphi(n)*2,print(n)))

n=0; for (m=1, 10^9, if(sigma(m)==2*eulerphi(m), write("b062699.txt", n++, " ", m); if (n==50, break)) ) \\ Harry J. Smith, Aug 09 2009

is(n)=my(f=factor(n)); sigma(f)==2*eulerphi(f) \\ Charles R Greathouse IV, Aug 13 2015

More terms from Labos Elemer, Nov 23 2001

A087943 Numbers n such that 3 divides sigma(n).

Original entry on oeis.org

2, 5, 6, 8, 10, 11, 14, 15, 17, 18, 20, 22, 23, 24, 26, 29, 30, 32, 33, 34, 35, 38, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 56, 58, 59, 60, 62, 65, 66, 68, 69, 70, 71, 72, 74, 77, 78, 80, 82, 83, 85, 86, 87, 88, 89, 90, 92, 94, 95, 96, 98, 99, 101, 102, 104, 105, 106

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 27 2003

nonn

Numbers n such that in the prime factorization n = Product_i p_i^e_i, there is some p_i == 1 (mod 3) with e_i == 2 (mod 3) or some p_i == 2 (mod 3) with e_i odd. - Robert Israel, Nov 09 2016

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Cf. A000203, A059269, A066498, A034020, A028983, A074216, A329963 (complement).

select(n -> numtheory:-sigma(n) mod 3 = 0, [$1..1000]); # Robert Israel, Nov 09 2016

Select[Range[1000],Mod[DivisorSigma[1,#],3]==0&] (* Enrique Pérez Herrero, Sep 03 2013 *)

is(n)=sigma(n)%3==0 \\ Charles R Greathouse IV, Sep 04 2013

is(n)=forprime(p=2,997,my(e=valuation(n,p)); if(e && Mod(p,3*p-3)^(e+1)==1, return(1), n/=p^e)); sigma(n)%3==0 \\ Charles R Greathouse IV, Sep 04 2013

a(n) << n^k for any k > 1, where << is the Vinogradov symbol. - Charles R Greathouse IV, Sep 04 2013
a(n) ~ n as n -> infinity: since Sum_{primes p == 2 (mod 3)} 1/p diverges, asymptotically almost every number is divisible by some prime p == 2 (mod 3) but not by p^2. - Robert Israel, Nov 09 2016
Because sigma(n) and sigma(3n)=A144613(n) differ by a multiple of 3, these are also the numbers n such that n divides sigma(3n). - R. J. Mathar, May 19 2020

More terms from Benoit Cloitre and Ray Chandler, Oct 27 2003

A347461 Number of distinct possible alternating products of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 10, 12, 16, 19, 23, 27, 34, 41, 49, 57, 67, 78, 91, 106, 125, 147, 166, 187, 215, 245, 277, 317, 357, 405, 460, 524, 592, 666, 740, 829, 928, 1032, 1147, 1273, 1399, 1555, 1713, 1892, 2087, 2298, 2523, 2783, 3070, 3383, 3724, 4104, 4504

Offset: 0

Gus Wiseman, Oct 06 2021

nonn

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

			Partitions representing each of the a(7) = 10 alternating products are:
     (7) -> 7
    (61) -> 6
    (52) -> 5/2
   (511) -> 5
    (43) -> 4/3
   (421) -> 2
  (4111) -> 4
   (331) -> 1
   (322) -> 3
  (3211) -> 3/2

The version for alternating sum is A004526.
Counting only integers gives A028310, reverse A347707.
The version for factorizations is A347460, reverse A038548.
The reverse version is A347462.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A122768 counts distinct submultisets of partitions.
A126796 counts complete partitions.
A293627 counts knapsack factorizations by sum.
A301957 counts distinct subset-products of prime indices.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
A304793 counts distinct positive subset-sums of prime indices.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A000070, A001055, A002033, A002219, A028983, A119620, A325768, A345926, A347443, A347444, A347445, A347446.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@IntegerPartitions[n]]],{n,0,30}]

A347450 Numbers whose multiset of prime indices has alternating product <= 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 14, 15, 16, 18, 21, 22, 24, 25, 26, 32, 33, 34, 35, 36, 38, 39, 40, 46, 49, 50, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 69, 72, 74, 77, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 98, 100, 104, 106, 111, 115, 118, 119, 121, 122

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 alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also Heinz numbers integer partitions with reverse-alternating product <= 1, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also numbers whose multiset of prime indices has alternating sum <= 1.

			The initial terms and their prime indices:
      1: {}            26: {1,6}           56: {1,1,1,4}
      2: {1}           32: {1,1,1,1,1}     57: {2,8}
      4: {1,1}         33: {2,5}           58: {1,10}
      6: {1,2}         34: {1,7}           60: {1,1,2,3}
      8: {1,1,1}       35: {3,4}           62: {1,11}
      9: {2,2}         36: {1,1,2,2}       64: {1,1,1,1,1,1}
     10: {1,3}         38: {1,8}           65: {3,6}
     14: {1,4}         39: {2,6}           69: {2,9}
     15: {2,3}         40: {1,1,1,3}       72: {1,1,1,2,2}
     16: {1,1,1,1}     46: {1,9}           74: {1,12}
     18: {1,2,2}       49: {4,4}           77: {4,5}
     21: {2,4}         50: {1,3,3}         81: {2,2,2,2}
     22: {1,5}         51: {2,7}           82: {1,13}
     24: {1,1,1,2}     54: {1,2,2,2}       84: {1,1,2,4}
     25: {3,3}         55: {3,5}           85: {3,7}

The additive version (alternating sum <= 0) is A028260.
The reverse version is A028982, counted by A119620.
Allowing any alternating product < 1 gives A119899.
Factorizations of this type are counted by A339846, complement A339890.
Allowing any alternating product >= 1 gives A344609, multiplicative A347456.
Partitions of this type are counted by A347443.
Allowing any integer alternating product gives A347454, reciprocal A347451.
The complement is A347465, reverse A028983, counted by A347448.
A056239 adds up prime indices, row sums of A112798.
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
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 lists Heinz numbers of partitions with integer alternating product.
Cf. A001105, A001222, A027193, A344617, A345958, A346703, A346704, A347449, 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&]

Union of A028982 and A119899.

Union of A028260 and A001105.

A347454 Numbers whose multiset of prime indices has integer alternating product.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 36, 37, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 108, 109, 112, 113

Offset: 1

Gus Wiseman, Sep 26 2021

nonn

First differs from A265640 in having 42.
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)).
Also Heinz numbers of partitions with integer reverse-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: {}            20: {1,1,3}         47: {15}
      2: {1}           23: {9}             48: {1,1,1,1,2}
      3: {2}           25: {3,3}           49: {4,4}
      4: {1,1}         27: {2,2,2}         50: {1,3,3}
      5: {3}           28: {1,1,4}         52: {1,1,6}
      7: {4}           29: {10}            53: {16}
      8: {1,1,1}       31: {11}            59: {17}
      9: {2,2}         32: {1,1,1,1,1}     61: {18}
     11: {5}           36: {1,1,2,2}       63: {2,2,4}
     12: {1,1,2}       37: {12}            64: {1,1,1,1,1,1}
     13: {6}           41: {13}            67: {19}
     16: {1,1,1,1}     42: {1,2,4}         68: {1,1,7}
     17: {7}           43: {14}            71: {20}
     18: {1,2,2}       44: {1,1,5}         72: {1,1,1,2,2}
     19: {8}           45: {2,2,3}         73: {21}

The even-length case is A000290.
The additive version is A026424.
Allowing any alternating product < 1 gives A119899, strict A028260.
Allowing any alternating product >= 1 gives A344609, multiplicative A347456.
Factorizations of this type are counted by A347437.
These partitions are counted by A347445, reverse A347446.
Allowing any alternating product <= 1 gives A347450.
The reciprocal version is A347451.
The odd-length case is A347453.
The version for reversed prime indices is A347457, complement A347455.
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.
A347461 counts possible alternating products of partitions.
A347462 counts possible reverse-alternating products of partitions.
Cf. A001105, A001222, A028982, A119620, A236913, A316523, A344653, A346703, A346704, A347443, A347439.

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[altprod[primeMS[#]]]&]

A347462 Number of distinct possible reverse-alternating products of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 13, 17, 22, 28, 33, 42, 51, 59, 69, 84, 100, 117, 137, 163, 191, 222, 256, 290, 332, 378, 429, 489, 564, 643, 729, 819, 929, 1040, 1167, 1313, 1473, 1647, 1845, 2045, 2272, 2521, 2785, 3076, 3398, 3744, 4115, 4548, 5010, 5524, 6086

Offset: 0

Gus Wiseman, Oct 06 2021

nonn

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.

			Partitions representing each of the a(7) = 11 reverse-alternating products:
     (7) -> 7
    (61) -> 1/6
    (52) -> 2/5
   (511) -> 5
    (43) -> 3/4
   (421) -> 2
  (4111) -> 1/4
   (331) -> 1
   (322) -> 3
  (3211) -> 2/3
  (2221) -> 1/2

The version for non-reverse alternating sum instead of product is A004526.
Counting only integers gives A028310, non-reverse A347707.
The version for factorizations is A038548, non-reverse A347460.
The non-reverse version is A347461.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A122768 counts distinct submultisets of partitions.
A126796 counts complete partitions.
A293627 counts knapsack factorizations by sum.
A301957 counts distinct subset-products of prime indices.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
A304793 counts distinct positive subset-sums of prime indices.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A000070, A001055, A002033, A002219, A028983, A119620, A325768, A345926, A347443, A347444, A347445, A347446.

revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[revaltprod/@IntegerPartitions[n]]],{n,0,30}]

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