A344061
a(n) = Sum_{d|n} sigma(d)^(n/d).
Original entry on oeis.org
1, 4, 5, 17, 7, 56, 9, 146, 78, 298, 13, 1501, 15, 2276, 1265, 9219, 19, 25716, 21, 77519, 16929, 177328, 25, 739582, 7808, 1594562, 264382, 5611241, 31, 15699452, 33, 48863172, 4196081, 129140542, 312753, 447589422, 39, 1162261928, 67111665, 3771805472, 43, 10764897556, 45
Offset: 1
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a[n_] := DivisorSum[n, DivisorSigma[1 , #]^(n/#) &]; Array[a, 43] (* Amiram Eldar, May 08 2021 *)
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a(n) = sumdiv(n, d, sigma(d)^(n/d));
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my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)*x^k/(1-sigma(k)*x^k)))
A383309
Numbers whose prime indices are prime powers > 1 with a common sum of prime indices.
Original entry on oeis.org
1, 3, 5, 7, 9, 11, 17, 19, 23, 25, 27, 31, 35, 41, 49, 53, 59, 67, 81, 83, 97, 103, 109, 121, 125, 127, 131, 157, 175, 179, 191, 209, 211, 227, 241, 243, 245, 277, 283, 289, 311, 331, 343, 353, 361, 367, 391, 401, 419, 431, 461, 509, 529, 547, 563, 587, 599
Offset: 1
The systems with these MM-numbers begin:
1: {}
3: {{1}}
5: {{2}}
7: {{1,1}}
9: {{1},{1}}
11: {{3}}
17: {{4}}
19: {{1,1,1}}
23: {{2,2}}
25: {{2},{2}}
27: {{1},{1},{1}}
31: {{5}}
35: {{2},{1,1}}
41: {{6}}
49: {{1,1},{1,1}}
53: {{1,1,1,1}}
59: {{7}}
67: {{8}}
81: {{1},{1},{1},{1}}
83: {{9}}
97: {{3,3}}
Twice-partitions of this type are counted by
A279789.
For just a common sum we have
A326534.
For just constant blocks we have
A355743.
Numbers without a factorization of this type are listed by
A381871, counted by
A381993.
The multiplicative version is
A381995.
For strict instead of constant blocks we have
A382304.
A023894 counts partitions into prime-powers.
A034699 gives maximal prime-power divisor.
A050361 counts factorizations into distinct prime powers.
A355742 chooses a prime-power divisor of each prime index.
Cf.
A000688,
A000720,
A001222,
A006171,
A038041,
A279784,
A302242,
A302493,
A321455,
A326518,
A381719.
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prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SameQ@@Total/@prix/@prix[#]&&And@@PrimePowerQ/@prix[#]&]
A344195
a(n) = Sum_{k=1..n} tau(gcd(k,n))^(n/gcd(k,n)), where tau(n) is the number of divisors of n.
Original entry on oeis.org
1, 3, 4, 9, 6, 26, 8, 49, 25, 140, 12, 240, 14, 782, 156, 1215, 18, 3349, 20, 5130, 800, 20498, 24, 19558, 151, 98324, 3148, 111492, 30, 270624, 32, 551091, 20520, 2097176, 924, 1716189, 38, 9437210, 98348, 8630496, 42, 25362724, 44, 43714584, 266346, 184549406, 48, 137141048, 813, 671096867
Offset: 1
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a[n_] := DivisorSum[n, EulerPhi[n/#] * DivisorSigma[0, #]^(n/#) &]; Array[a, 50] (* Amiram Eldar, May 11 2021 *)
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a(n) = sum(k=1, n, numdiv(gcd(k, n))^(n/gcd(k, n)));
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a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d)^(n/d));
A381872
Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into blocks having a common sum.
Original entry on oeis.org
1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1
Offset: 1
The prime indices of 144 are {1,1,1,1,2,2}, with the following 4 multiset partitions having common block sum:
{{1,1,1,1,2,2}}
{{2,2},{1,1,1,1}}
{{1,1,2},{1,1,2}}
{{2},{2},{1,1},{1,1}}
with sums: 8, 4, 4, 2, of which 3 are distinct, so a(144) = 3.
The prime indices of 1296 are {1,1,1,1,2,2,2,2}, with the following 7 multiset partitions having common block sum:
{{1,1,1,1,2,2,2,2}}
{{2,2,2},{1,1,1,1,2}}
{{1,1,2,2},{1,1,2,2}}
{{2,2},{2,2},{1,1,1,1}}
{{2,2},{1,1,2},{1,1,2}}
{{1,2},{1,2},{1,2},{1,2}}
{{2},{2},{2},{2},{1,1},{1,1}}
with sums: 12, 6, 6, 4, 4, 3, 2, of which 5 are distinct, so a(1296) = 5.
With equal blocks instead of sums we have
A089723.
Positions of terms > 1 are
A321454.
With distinct instead of equal sums we have
A381637, before sums
A321469.
A265947 counts refinement-ordered pairs of integer partitions.
Other multiset partitions of prime indices:
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prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],SameQ@@Total/@#&]]],{n,100}]
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