A032741 -id:A032741 - cp's OEIS frontend

A294926 Number of proper divisors of n that are deficient (A005100).

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 4, 1, 3, 3, 4, 1, 4, 1, 5, 3, 3, 1, 5, 2, 3, 3, 5, 1, 6, 1, 5, 3, 3, 3, 5, 1, 3, 3, 6, 1, 6, 1, 5, 5, 3, 1, 6, 2, 5, 3, 5, 1, 5, 3, 6, 3, 3, 1, 7, 1, 3, 5, 6, 3, 6, 1, 5, 3, 7, 1, 6, 1, 3, 5, 5, 3, 6, 1, 7, 4, 3, 1, 7, 3, 3, 3, 7, 1, 8, 3, 5, 3, 3, 3, 7, 1, 5, 5, 7, 1, 6, 1, 7, 7

Offset: 1

Antti Karttunen, Nov 14 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A032741, A080226, A294886, A294927, A294928, A294929, A294934.

a[n_] := DivisorSum[n, 1 &, # < n && DivisorSigma[1, #] < 2*# &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)

A294926(n) = sumdiv(n, d, (dAntti Karttunen, Nov 14 2017

a(n) = Sum_{d|n, dA294934(d).
a(n) = A080226(n) - A294934(n).
a(n) + A294927(n) = A032741(n).

A304104 a(n) = Product_{d|n, d>1} prime(A304101(d)-1).

Original entry on oeis.org

1, 2, 2, 6, 2, 20, 3, 12, 10, 20, 3, 420, 2, 30, 20, 60, 11, 300, 11, 420, 12, 30, 5, 4200, 22, 20, 130, 990, 3, 11000, 11, 420, 102, 44, 30, 31500, 5, 242, 20, 10920, 11, 3000, 13, 1170, 1100, 190, 3, 231000, 33, 2420, 506, 420, 19, 66300, 12, 9900, 110, 30, 11, 8085000, 13, 242, 300, 5460, 52, 56100, 19, 660, 130, 19500, 13, 9135000, 11, 290, 4180, 2178, 99

Offset: 1

Antti Karttunen, May 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4181

Cf. A304101, A304102, A304105 (restricted growth sequence transform of this sequence).

\\ Needs also code from A304101:
A304104(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(A304101(d)-1))); (m); };

a(n) = Product_{d|n, d>1} A000040(A304101(d)-1).
a(n) = (1/2) * A304102(n) * A000040(A304101(n)-1).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A001511(a(n)) = A005086(n).
A007949(a(n)) = A304096(n).

A309307 Number of unitary divisors of n (excluding 1).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 7, 1, 1, 3, 3, 3, 3, 1, 3, 3, 3, 1, 7, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 7, 1, 3, 3, 1, 3, 7, 1, 3, 3, 7, 1, 3, 1, 3, 3, 3, 3, 7, 1, 3, 1, 3, 1, 7, 3, 3, 3, 3, 1, 7, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3

Offset: 1

Ilya Gutkovskiy, Jul 21 2019

nonn

Also the number of squarefree divisors > 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor

Cf. A000225, A001221, A001222, A032741, A034444, A070824, A077610, A087893.

nmax = 100; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x] // Rest
Table[2^PrimeNu[n] - 1, {n, 1, 100}]

G.f.: Sum_{k>=2} mu(k)^2*x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)*(zeta(s)/zeta(2*s) - 1).
a(n) = 2^omega(n) - 1.
a(n) = A000225(A001221(n)) = A034444(n) - 1.
Sum_{k=1..n} a(k) ~ 6*n*(log(n) + 2*gamma - 1 - Pi^2/6 - 12*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 16 2019
a(n) = -1 + Sum_{d|n} mu(d)^2. - Wesley Ivan Hurt, Feb 04 2022

A324202 a(n) = A046523(A332461(n)), where A332461(n) = Product_{d|n, d>1} prime(1+A297167(d)).

Original entry on oeis.org

1, 2, 2, 6, 2, 12, 2, 30, 6, 12, 2, 120, 2, 12, 12, 210, 2, 180, 2, 420, 12, 12, 2, 2520, 6, 12, 30, 420, 2, 720, 2, 2310, 12, 12, 12, 7560, 2, 12, 12, 9240, 2, 720, 2, 420, 120, 12, 2, 138600, 6, 180, 12, 420, 2, 6300, 12, 60060, 12, 12, 2, 151200, 2, 12, 420, 30030, 12, 720, 2, 420, 12, 720, 2, 831600, 2, 12, 180, 420, 12, 720, 2, 360360

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences computed from indices in prime factorization

Cf. A032741, A061395, A297113, A297167, A324190, A324203 (rgs-transform), A324537, A332461.

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324202(n) = A046523(factorback(apply(x -> prime(1+x),apply(A297167, select(d -> d>1,divisors(n))))));

a(n) = A046523(A332461(n)).
A001221(a(n)) = A324190(n).
A001222(a(n)) = A032741(n).

A328960 Number of integer partitions of n whose number of nontrivial submultisets is greater than their number of distinct parts times their number of parts minus 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 6, 10, 18, 28, 45, 63, 93, 129, 178, 238, 321, 419, 551, 708, 911, 1158, 1472, 1845, 2316, 2883, 3583, 4421, 5453, 6680, 8180, 9964, 12122, 14687, 17771, 21418, 25788, 30949, 37092, 44324, 52906, 62980, 74885, 88832, 105243, 124429

Offset: 0

Gus Wiseman, Nov 02 2019

nonn

These partitions are conjectured to be precisely those that have a pair of multiset partitions such that no part of one is a submultiset of any part of the other (see A320632). For example, such a pair of partitions of {1,1,2,2} is ({{1,1},{2,2}}, {{1,2},{1,2}}).

			The a(6) = 1 through a(10) = 18 partitions:
  (2211)  (3211)   (3221)    (3321)     (3322)
          (22111)  (3311)    (4221)     (4321)
                   (4211)    (4311)     (4411)
                   (22211)   (5211)     (5221)
                   (32111)   (32211)    (5311)
                   (221111)  (33111)    (6211)
                             (42111)    (32221)
                             (222111)   (33211)
                             (321111)   (42211)
                             (2211111)  (43111)
                                        (52111)
                                        (222211)
                                        (322111)
                                        (331111)
                                        (421111)
                                        (2221111)
                                        (3211111)
                                        (22111111)
For example, the partition (4,2,2,1,1) has 16 nontrivial submultisets: {(1), (2), (4), (11), (21), ..., (2211), (4211), (4221)}, and 5 parts, 3 of which are distinct. Since 16 > (5 - 1) * 3 = 12, the partition (42211) is counted under a(10)

The Heinz numbers of these partitions are conjectured to be A320632.
A307409(n) is (omega(n) - 1) * nu(n).
A328958(n) is sigma_0(n) - omega(n) * nu(n).
A328959(n) is sigma_0(n) - 2 - (omega(n) - 1) * nu(n).
Cf. A008284, A032741, A116608, A328956, A328961, A328963.

Table[Length[Select[IntegerPartitions[n],0

A358103 Quotient of the n-th divisible pair, where pairs are ordered by Heinz number. Quotient of prime indices of A318990(n).

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 5, 1, 6, 7, 8, 3, 9, 1, 4, 10, 11, 2, 12, 13, 14, 5, 15, 16, 6, 3, 17, 1, 18, 7, 2, 19, 20, 21, 22, 8, 23, 1, 24, 9, 4, 25, 26, 27, 10, 28, 29, 30, 5, 11, 31, 3, 32, 12, 33, 34, 1, 35, 36, 13, 6, 37, 2, 14, 38, 39, 15, 40, 41, 1, 42, 7, 4, 43

Offset: 1

Gus Wiseman, Nov 02 2022

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.

			The 12th divisible pair is (2,6) so a(12) = 3.

The divisible pairs are ranked by A318990, proper A339005.
Quotient of A358104 and A358105.
A different ordering is A358106.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A001358 lists the semiprimes, squarefree A006881.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
A358192/A358193 gives quotients of semiprime indices.
Cf. A000720, A027751, A032741, A215366, A289508, A289509, A296150, A300912, A318991, A318992.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Join@@Table[Cases[primeMS[n],{x_,y_}/;Divisible[y,x]:>y/x,{0}],{n,100}]

a(n) = A358104(n)/A358105(n).

A358104 Unreduced numerator of the n-th divisible pair, where pairs are ordered by Heinz number. Greater prime index of A318990(n).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 3, 6, 7, 8, 6, 9, 4, 8, 10, 11, 6, 12, 13, 14, 10, 15, 16, 12, 9, 17, 5, 18, 14, 8, 19, 20, 21, 22, 16, 23, 6, 24, 18, 12, 25, 26, 27, 20, 28, 29, 30, 15, 22, 31, 12, 32, 24, 33, 34, 7, 35, 36, 26, 18, 37, 10, 28, 38, 39, 30, 40, 41, 8, 42

Offset: 1

Gus Wiseman, Nov 02 2022

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.

			The 12th divisible pair is (2,6) so a(12) = 6.

The divisible pairs are ranked by A318990, proper A339005.
For all semiprimes we have A338913.
The quotient of the pair is A358103.
The denominator is A358105.
The reduced version for all semiprimes is A358192, denominator A358193.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A001358 lists the semiprimes, squarefree A006881.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
A318991 ranks divisor-chains.
Cf. A000720, A032741, A128301, A215366, A289508, A289509, A296150, A300912, A318992, A358106.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Join@@Table[Cases[primeMS[n],{x_,y_}/;Divisible[y,x]:>y,{0}],{n,1000}]

A358103(n) = a(n)/A358105(n).

A358106 Quotient of the n-th divisible pair, where pairs are ordered first by sum and then by denominator.

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 2, 1, 6, 7, 3, 1, 8, 2, 9, 4, 1, 10, 11, 5, 3, 2, 1, 12, 13, 6, 1, 14, 4, 2, 15, 7, 3, 1, 16, 17, 8, 5, 2, 1, 18, 19, 9, 4, 3, 1, 20, 6, 2, 21, 10, 1, 22, 23, 11, 7, 5, 3, 2, 1, 24, 4, 25, 12, 1, 26, 8, 2, 27, 13, 6, 3, 1, 28, 29, 14, 9, 5, 4, 2, 1

Offset: 2

Gus Wiseman, Nov 03 2022

			Grouping by sum gives:
   2:  1
   3:  2
   4:  3 1
   5:  4
   6:  5 2 1
   7:  6
   8:  7 3 1
   9:  8 2
  10:  9 4 1
  11: 10
  12: 11 5 3 2 1
  13: 12
  14: 13 6 1
  15: 14 4 2
  16: 15 7 3 1
  17: 16
  18: 17 8 5 2 1

Row-lengths are A032741.
This is A208460/A027751.
A ranking of divisible pairs is A318990, proper A339005.
A different ordering is A358103 = A358104 / A358105.
A000041 counts partitions, strict A000009.
A001358 lists semiprimes, squarefree A006881.
A318991 ranks divisor-chains.
A358192/A358193 gives quotients of semiprime indices.
Cf. A000837, A003238, A122934.

Table[Divide@@@Select[IntegerPartitions[n,{2}],Divisible@@#&],{n,2,30}]

a(n) = A208460(n)/A027751(n).

A358193 Denominator of the quotient of the prime indices of the n-th semiprime.

Original entry on oeis.org

1, 2, 1, 3, 4, 3, 2, 5, 1, 6, 5, 7, 4, 8, 3, 9, 1, 7, 5, 4, 10, 11, 2, 9, 12, 5, 13, 7, 14, 5, 3, 11, 15, 8, 16, 6, 3, 17, 7, 1, 18, 13, 7, 2, 19, 15, 20, 6, 10, 21, 11, 22, 8, 9, 23, 1, 17, 24, 9, 4, 7, 25, 19, 26, 5, 13, 27, 8, 10, 28, 14, 11, 29, 21, 7, 30

Offset: 1

Gus Wiseman, Nov 03 2022

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 31-st semiprime has prime indices (4,6), so the quotient is 4/6 = 2/3; hence a(31) = 3.

The divisible pairs are ranked by A318990, proper A339005.
The unreduced pair is (A338912, A338913).
The quotients of divisible pairs are A358103.
The restriction to divisible pairs is A358105, numerator A358104.
The numerator is A358192.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A001358 lists the semiprimes, squarefree A006881.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
Cf. A000720, A027751, A032741, A128301, A215366, A289508, A289509, A296150, A300912, A318991, A358106.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Denominator/@Divide@@@primeMS/@Select[Range[100],PrimeOmega[#]==2&]

A363605 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^5.

Original entry on oeis.org

0, 1, 5, 16, 35, 76, 126, 226, 335, 531, 715, 1092, 1365, 1947, 2420, 3286, 3876, 5251, 5985, 7861, 8986, 11342, 12650, 16252, 17585, 21841, 24086, 29367, 31465, 38946, 40920, 49662, 53080, 62782, 66206, 80082, 82251, 97376, 102640, 120001, 123410, 146628

Offset: 1

Seiichi Manyama, Jun 11 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A013663, A032741, A065608, A069153, A363604, A363606.

Cf. A000203, A001157, A001158, A001159.

a[n_] := DivisorSum[n, Binomial[# + 2, 4] &]; Array[a, 40] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^5)))

a(n) = my(f = factor(n)); (sigma(f, 4) + 2*sigma(f, 3) - sigma(f, 2) - 2*sigma(f)) / 24; \\ Amiram Eldar, Dec 30 2024

G.f.: Sum_{k>0} binomial(k+2,4) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d+2,4).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_4(n) + 2*sigma_3(n) - sigma_2(n) - 2*sigma_1(n)) / 24.
Dirichlet g.f.: zeta(s) * (zeta(s-4) + 2*zeta(s-3) - zeta(s-2) - 2*zeta(s-1)) / 24.
Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)

cp's OEIS Frontend

A294926 Number of proper divisors of n that are deficient (A005100).

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A304104 a(n) = Product_{d|n, d>1} prime(A304101(d)-1).

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A309307 Number of unitary divisors of n (excluding 1).

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A324202 a(n) = A046523(A332461(n)), where A332461(n) = Product_{d|n, d>1} prime(1+A297167(d)).

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A328960 Number of integer partitions of n whose number of nontrivial submultisets is greater than their number of distinct parts times their number of parts minus 1.

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A358103 Quotient of the n-th divisible pair, where pairs are ordered by Heinz number. Quotient of prime indices of A318990(n).

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A358104 Unreduced numerator of the n-th divisible pair, where pairs are ordered by Heinz number. Greater prime index of A318990(n).

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A358106 Quotient of the n-th divisible pair, where pairs are ordered first by sum and then by denominator.

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A358193 Denominator of the quotient of the prime indices of the n-th semiprime.

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