A024923 -id:A024923 - cp's OEIS frontend

A024918 Partial sums of the sequence of prime powers (A000961).

1, 3, 6, 10, 15, 22, 30, 39, 50, 63, 79, 96, 115, 138, 163, 190, 219, 250, 282, 319, 360, 403, 450, 499, 552, 611, 672, 736, 803, 874, 947, 1026, 1107, 1190, 1279, 1376, 1477, 1580, 1687, 1796, 1909, 2030, 2155, 2282, 2410, 2541, 2678, 2817, 2966, 3117, 3274, 3437

Offset: 1

Den Roussel (DenRoussel(AT)webtv.net)

nonn

The subsequence of prime partial sums of prime powers begins: 3, 79, 163, 499, 947, 1279, 5297. What is the smallest value which is a prime power p^k for k>1? [Jonathan Vos Post, Feb 11 2010]

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

Cf. A000961, A024923.

FoldList[Plus, 1, Select[Range[150], PrimePowerQ]] (* Amiram Eldar, Jun 22 2025 *)

ispp1(n) = isprimepower(n) || (n==1); \\ A000961
lista(nn) = {my(s=0); for (n=1, nn, if (ispp1(n), s+= n; print1(s, ", ")););} \\ Michel Marcus, Mar 26 2020

Offset 1 from Michel Marcus, Mar 26 2020

A385378 The maximum possible number of distinct factors in the factorization of n into prime powers (A246655).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2

Offset: 1

Amiram Eldar, Jun 27 2025

Differs from A376885 and A384422 at n = 32, 64, 96, 128, 160, 192, ... .
Differs from A086435 at n = 36, 100, 144, 180, 196, 225, ... .
Differs from A375272 at n = 128, 384, 640, 896, 1024, 1152, ...	.
a(n) depends only on the prime signature of n (A118914).
The indices of records in this sequence are the partial products of the sequence of powers of primes (A000961), i.e., the terms in A024923.
The least index n such that a(n) = k, for k = 0, 1, 2, ..., is A024923(k+1).

			      n | a(n) | factorization
  ------+------+--------------------------------
      2 |  1   | 2
      6 |  2   | 2 * 3
     24 |  3   | 2 * 3 * 2^2
    120 |  4   | 2 * 3 * 2^2 * 5
    840 |  5   | 2 * 3 * 2^2 * 5 * 7
   6720 |  6   | 2 * 3 * 2^2 * 5 * 7 * 2^3
  60480 |  7   | 2 * 3 * 2^2 * 5 * 7 * 2^3 * 3^2

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

Cf. A000430, A000961, A001221, A003056, A004709, A024923, A077761, A086435, A118914, A246655, A254578, A375272, A376885, A384422, A385379.

f[p_, e_] := Floor[(Sqrt[8*e + 1] - 1)/2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> (sqrtint(8*x+1)-1)\2 , factor(n)[, 2]));

Additive with a(p^e) = A003056(e).
a(n) >= A001221(n), with equality if and only if n is cubefree (A004709).
a(n) >= 1 for n >= 2, with equality if and only if n is a prime or a square of a prime (A000430).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{k>=2} P(k*(k+1)/2) = 0.19285739770001405035..., and P is the prime zeta function.

A334395 Partial products of A334393.

Original entry on oeis.org

1, 2, 6, 24, 120, 840, 6720, 60480, 665280, 8648640, 147026880, 2793510720, 64250746560, 1606268664000, 43369253928000, 1257708363912000, 38988959281272000, 1247646697000704000, 46162927789026048000, 1892680039350067968000, 81385241692052922624000

Offset: 1

Kevin Foote, Apr 26 2020

nonn

			a(6) = 1*2*3*4*5*7 = 840.

Amiram Eldar, Table of n, a(n) for n = 1..358

Cf. A008578, A024923, A334393.

Rest @ FoldList[Times, 1, Select[Range[43], Length[(f = FactorInteger[#])] == 1 && ((e = f[[1, 2]]) == 1 || PrimeQ[e]) &]] (* Amiram Eldar, May 11 2020 *)

isok(n) = if (n==1, return (1)); my(k=isprimepower(n)); (k==1) || isprime(k); \\ A334393
lista(nn) = {my(v = select(x->isok(x), [1..nn]), p=1); for (n=1, #v, p *= v[n]; print1(p, ", "););} \\ Michel Marcus, May 11 2020

a(n) = Product_{i=1..n} A334393(i).

A385380 Partial products of the sequence nonprime powers of primes (A025475).

Original entry on oeis.org

1, 4, 32, 288, 4608, 115200, 3110400, 99532800, 4877107200, 312134860800, 25282923724800, 3059233770700800, 382404221337600000, 48947740331212800000, 8272168115974963200000, 2010136852181916057600000, 514595034158570510745600000, 148717964871826877605478400000

Offset: 1

Amiram Eldar, Jun 27 2025

Indices of records in A385379.

a(n) is the least index k such that A385379(k) = n-1.

Amiram Eldar, Table of n, a(n) for n = 1..217
Index entries for sequences related to powerful numbers.

Cf. A024923, A025475, A385379.

Subsequence of A001694 and A025487 (i.e., of A181800).

FoldList[Times, 1, Select[Range[250], !PrimeQ[#] && PrimePowerQ[#] &]] (* Amiram Eldar, Jun 27 2025 *)

list(lim) = {my(p = 1); print1(p, ", "); for(k = 2, lim, if(isprimepower(k) > 1, p *= k; print1(p, ", ")));}

a(n) = Product_{k=1..n} A025475(k).

A308819 Product of prime powers <= n.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 120, 840, 6720, 60480, 60480, 665280, 665280, 8648640, 8648640, 8648640, 138378240, 2352430080, 2352430080, 44696171520, 44696171520, 44696171520, 44696171520, 1028011944960, 1028011944960, 25700298624000, 25700298624000, 693908062848000

Offset: 0

Ilya Gutkovskiy, Jun 26 2019

nonn

a(n) is the smallest number that's divisible by all numbers less than or equal to n. - Keith F. Lynch, Apr 24 2025

			a(9) = 60480 because 2, 3, 4, 5, 7, 8, 9 are the prime powers less than or equal to 9 and 2 * 3 * 4 * 5 * 7 * 8 * 9 = 60480.

Cf. A000961, A024923, A034386, A065515, A074793, A100994, A179215, A246655.

a[n_] := Product[If[PrimePowerQ[k], k, 1], {k, 1, n}]; Table[a[n], {n, 0, 27}]
Module[{nn=50,ppwr},ppwr=Select[Range[nn],PrimePowerQ[#]&];Table[Times@@ Select[ ppwr,#<= n&],{n,0,nn}]] (* Harvey P. Dale, Apr 03 2024 *)

a(n) = prod(k=1, n, if (isprime(k) || isprimepower(k), k, 1)); \\ Michel Marcus, Jun 27 2019

A380318 Product of the first n perfect powers (A001597).

Original entry on oeis.org

1, 1, 4, 32, 288, 4608, 115200, 3110400, 99532800, 3583180800, 175575859200, 11236854988800, 910185254092800, 91018525409280000, 11013241574522880000, 1376655196815360000000, 176211865192366080000000, 25374508587700715520000000, 4288291951321420922880000000, 840505222458998500884480000000, 181549128051143676191047680000000

Offset: 0

Ilya Gutkovskiy, Jan 20 2025

nonn

Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A000442, A001044, A001597, A002110, A024923, A036691, A076408, A111059.

Join[{1},FoldList[Times,Join[{1},Select[Range[250],GCD@@FactorInteger[#][[All,2]]>1&]]]] (* Harvey P. Dale, May 03 2025 *)

cp's OEIS Frontend

A024918 Partial sums of the sequence of prime powers (A000961).

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A385378 The maximum possible number of distinct factors in the factorization of n into prime powers (A246655).

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A334395 Partial products of A334393.

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A385380 Partial products of the sequence nonprime powers of primes (A025475).

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A308819 Product of prime powers <= n.

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A380318 Product of the first n perfect powers (A001597).

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