A200768 -id:A200768 - cp's OEIS frontend

A373459 Expansion of Sum_{p prime} x^p/(1 - (p*x)^p).

0, 1, 1, 4, 1, 43, 1, 64, 729, 3381, 1, 20707, 1, 827639, 10297066, 16384, 1, 14414443, 1, 30517840269, 678610493338, 285312719187, 1, 10464547507, 95367431640625, 302875123369469, 282429536481, 558545864150392871, 1, 298030849742873568, 1, 1073741824

Offset: 1

Seiichi Manyama, Jun 06 2024

Cf. A200768, A373458.

a[n_]:=Sum[Boole[PrimeQ[d]]d^(n-d),{d,Divisors[n]}]; Array[a,32] (* Stefano Spezia, Mar 30 2025 *)

a(n) = sumdiv(n, d, isprime(d)*d^(n-d));

a(n) = Sum_{p|n prime} p^(n - p).

If p is prime, a(p) = 1.

A322080 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{p|n, p prime} p^k.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 4, 3, 1, 0, 8, 9, 2, 1, 0, 16, 27, 4, 5, 2, 0, 32, 81, 8, 25, 5, 1, 0, 64, 243, 16, 125, 13, 7, 1, 0, 128, 729, 32, 625, 35, 49, 2, 1, 0, 256, 2187, 64, 3125, 97, 343, 4, 3, 2, 0, 512, 6561, 128, 15625, 275, 2401, 8, 9, 7, 1, 0, 1024, 19683, 256, 78125, 793, 16807, 16, 27, 29, 11, 2

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
  0,  0,   0,    0,    0,     0,  ...
  1,  2,   4,    8,   16,    32,  ...
  1,  3,   9,   27,   81,   243,  ...
  1,  2,   4,    8,   16,    32,  ...
  1,  5,  25,  125,  625,  3125,  ...
  2,  5,  13,   35,   97,   275,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Index entries for sequences related to sums of divisors

Columns k=0..4 give A001221, A008472, A005063, A005064, A005065.

Cf. A109974, A200768 (diagonal), A285425, A286880, A321258.

Table[Function[k, Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[Prime[j]^k x^Prime[j]/(1 - x^Prime[j]), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

T(n,k)={vecsum([p^k | p<-factor(n)[,1]])}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

G.f. of column k: Sum_{j>=1} prime(j)^k*x^prime(j)/(1 - x^prime(j)).

A318969 Expansion of exp(Sum_{k>=1} ( Sum_{p|k, p prime} p^k ) * x^k/k).

Original entry on oeis.org

1, 0, 2, 9, 6, 643, 182, 118953, 6019, 242630, 2243190, 25938251679, 78106516, 23349992199606, 288964822371, 46755212195033, 226472341461312, 48661337027901364945, 18066374340919781, 104224677113940850317679, 440728415311733637734, 208546898802899685866735, 972477473959172989443327

Offset: 0

Ilya Gutkovskiy, Sep 06 2018

nonn

Cf. A000607, A023881, A200768, A219224, A220427, A318913, A318968.

nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 22; CoefficientList[Series[Product[1/(1 - Prime[k]^Prime[k] x^Prime[k])^(1/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 22}]

G.f.: Product_{k>=1} 1/(1 - prime(k)^prime(k)*x^prime(k))^(1/prime(k)).

cp's OEIS Frontend

A373459 Expansion of Sum_{p prime} x^p/(1 - (p*x)^p).

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A322080 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{p|n, p prime} p^k.

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A318969 Expansion of exp(Sum_{k>=1} ( Sum_{p|k, p prime} p^k ) * x^k/k).

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