A007445 -id:A007445 - cp's OEIS frontend

A061152 Expansion of Product_{n>=1} (1+x^n)^prime(n).

1, 2, 4, 11, 23, 51, 107, 216, 430, 839, 1614, 3046, 5684, 10465, 19046, 34321, 61225, 108245, 189779, 330093, 569916, 977139, 1664304, 2817039, 4740000, 7930740, 13198108, 21851556, 36001483, 59035979, 96373100, 156644241, 253550911

Offset: 0

Vladeta Jovovic, Apr 16 2001

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A007441, A007445, A030009, A061150, A061151.

a(n) = (1/n) * Sum_{k=1..n} a(n-k)*b(k), k>0, a(0)=1, b(k)=Sum_{d|k} (-1)^(k/d+1)*d*prime(d).

A275700 a(n) = Product_{d|n} prime(d).

Original entry on oeis.org

2, 6, 10, 42, 22, 390, 34, 798, 230, 1914, 62, 101010, 82, 4386, 5170, 42294, 118, 547170, 134, 951258, 12410, 14694, 166, 170807910, 2134, 24846, 23690, 3285114, 218, 660741510, 254, 5540514, 42470, 49206, 55726, 21399271530, 314, 65526, 68470, 3126785046, 358

Offset: 1

Jaroslav Krizek, Aug 05 2016

nonn

a(n) mod n = 0 for n: 1, 2, 6, 30, 78, 330, 390, 870, 1410, 3198, ...

			a(4) = 42 because the divisors of 4 are: 1, 2 and 4; and prime(1) * prime(2) * prime(4) = 2 * 3 * 7 = 42.

Cf. A007445 (Sum_{d|n} prime(d)).
A version for binary indices is A034729.
Partitions of this type are counted by A054973, strict case of A371284.
The sorted version is A371283, squarefree case of A371288.
These numbers have products A371286, unsorted version A371285.
A000005 counts divisors, row-lengths of A027750.
A027746 lists prime factors, indices A112798, length A001222.
Cf. A000720, A001221, A005179, A007416.

[(&*[NthPrime(d): d in Divisors(n)]): n in [1..100]]

Table[Times@@(Prime[#]&/@Divisors[n]),{n,50}] (* Harvey P. Dale, Jun 16 2017 *)

a(n) = my(d=divisors(n)); prod(i=1, #d, prime(d[i])) \\ Felix Fröhlich, Aug 05 2016

use ntheory ":all"; sub a275700 { vecprod(map { nth_prime($) } divisors($[0])); } # Dana Jacobsen, Aug 09 2016

A061150 a(n) = Sum_{d|n} d*prime(d).

Original entry on oeis.org

2, 8, 17, 36, 57, 101, 121, 188, 224, 353, 343, 573, 535, 729, 777, 1036, 1005, 1406, 1275, 1801, 1669, 2087, 1911, 2861, 2482, 3167, 3005, 3753, 3163, 4541, 3939, 5228, 4879, 5737, 5391, 7314, 5811, 7475, 7063, 8873, 7341, 9957, 8215, 10607, 9849

Offset: 1

Vladeta Jovovic, Apr 16 2001

			a(4)=36 because the divisors of 4 are 1,2,4 and 1*p(1) + 2*p(2) + 4*p(4) = 1*2 + 2*3 + 4*7 = 36.

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

Cf. A007441, A007445, A030009, A061151-A061152.

Cf. A127093.

with(numtheory): a:=proc(n) local div: div:=divisors(n): sum(div[j]*ithprime(div[j]),j=1..tau(n)) end: seq(a(n),n=1..55); # Emeric Deutsch, Jan 20 2007

a(n) = sumdiv(n, d, d*prime(d)); \\ Michel Marcus, Jun 24 2018

Equals M * V, where M = A127093 as an infinite lower triangular matrix and V = A000040, the sequence of primes as a vector. E.g., a(4) = 36 = 1*2 + 2*3 + 4*7, where (1, 2, 0, 4) = row 4 of A127093 and 2, 3 and 7 are p(1), p(2), p(4). - Gary W. Adamson, Jan 11 2007

L.g.f.: log(Product_{k>=1} 1/(1 - x^k)^prime(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 10 2017

Edited by N. J. A. Sloane, May 04 2007

A061151 1 + Sum_{n>=1} a_n x^n = 1/Product_{n>=1} (1+x^n)^prime(n).

Original entry on oeis.org

1, -2, 0, -3, 5, -3, 14, -4, 25, -32, 16, -88, 18, -155, 108, -153, 393, -88, 855, -160, 1255, -974, 1122, -3172, 370, -6794, 383, -10017, 5004, -9460, 19380, -2635, 45790, 5008, 76263, -7353, 87597, -77967, 48886, -244397, -45016, -500016, -115318, -734277, 56213, -710603, 810177, -161662, 2432173, 910752, 4767086

Offset: 0

Vladeta Jovovic, Apr 16 2001

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A007441, A007445, A030009, A061150, A061152.

a(n) = (1/n) * Sum_{k=1..n} a(n-k)*b(k), k>0, a(0)=1, b(k)=Sum_{d|k} (-1)^(k/d)*d*prime(d).

A127639 A051731 * A127640, where A127640 = infinite lower triangular matrix with the sequence of primes in the main diagonal and the rest zeros.

Original entry on oeis.org

2, 2, 3, 2, 0, 5, 2, 3, 0, 7, 2, 0, 0, 0, 11, 2, 3, 5, 0, 0, 13, 2, 0, 0, 0, 0, 0, 17, 2, 3, 0, 7, 0, 0, 0, 19, 2, 0, 5, 0, 0, 0, 0, 0, 23, 2, 3, 0, 0, 11, 0, 0, 0, 0, 29, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 2, 3, 5, 7, 0, 13, 0, 0, 0, 0, 0, 37, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 2, 3, 0, 0, 0, 0, 17, 0, 0

Offset: 1

Gary W. Adamson, Jan 21 2007

Row sums = A007445, inverse Mobius transform of the primes: (2, 5, 7, 12, 13, 23, ...)

			First few rows of the triangle are:
2;
2, 3;
2, 0, 5;
2, 3, 0, 7;
2, 0, 0, 0, 11;
2, 3, 5, 0, 0, 13;
...

Cf. A051731, A007445, A127639.

A051731 := proc(n,k) if n mod k = 0 then 1 ; else 0 ; fi ; end: A127639 := proc(n,k) A051731(n,k)*ithprime(k) ; end: for n from 1 to 16 do for k from 1 to n do printf("%d,", A127639(n,k)) ; od ; od ; # R. J. Mathar, Mar 14 2007

More terms from R. J. Mathar, Mar 14 2007

A062774 Inverse Moebius transform of PrimePi function.

Original entry on oeis.org

0, 1, 2, 3, 3, 6, 4, 7, 6, 8, 5, 13, 6, 11, 11, 13, 7, 17, 8, 18, 14, 14, 9, 26, 12, 16, 15, 22, 10, 29, 11, 24, 18, 19, 18, 35, 12, 21, 20, 34, 13, 37, 14, 30, 29, 24, 15, 47, 19, 32, 24, 33, 16, 42, 24, 42, 26, 27, 17, 61, 18, 30, 36, 42, 27, 48, 19, 40, 30, 48, 20, 68, 21, 34

Offset: 1

Labos Elemer, Jul 18 2001

nonn

			n = 12: divisors = D(12) = {1,2,3,4,6,12}, pi(D(12)) = {0,1,2,2,3,5} of which the sum is 0+1+2+2+3+5 = 13 so a(12) = 13; a(p(n)) = 0+n = n, for n-th prime p(n).

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000720, A007444, A007445.

{ for (n=1, 1000, d=divisors(n); write("b062774.txt", n, " ", sum(k=1, length(d), primepi(d[k]))) ) } \\ Harry J. Smith, Aug 10 2009

a(n) = Sum_{d|n} pi(d).
G.f.: Sum_{k>=1} pi(k)*x^k/(1 - x^k), where pi(k) is the number of primes <= k (A000720). - Ilya Gutkovskiy, Jan 16 2017
a(n) = Sum_{d|n} omega(d!). - Wesley Ivan Hurt, May 23 2021

Offset changed from 0 to 1 by Harry J. Smith, Aug 10 2009

A130069 Primes by columns, interspersed with (k-1) zeros.

Original entry on oeis.org

2, 3, 2, 5, 0, 2, 7, 3, 0, 2, 11, 0, 0, 0, 2, 13, 5, 3, 0, 0, 2, 17, 0, 0, 0, 0, 0, 2, 19, 7, 0, 3, 0, 0, 0, 23, 0, 5, 0, 0, 0, 0, 0, 2, 29, 11, 0, 0, 3, 0, 0, 0, 0, 2

Offset: 1

Gary W. Adamson, May 05 2007

Left column = the primes, A000040; row sums = A007445, (2, 5, 7, 12, 13, 23, ...), the inverse Moebius transform of the primes. A130070 = A130069 with nonzero terms reversed and mapped on A051731.

			First few rows of the triangle:
  2;
  3,  2;
  5,  0, 2;
  7,  3, 0, 2;
 11,  0, 0, 0, 2;
 13,  5, 3, 0, 0, 2;
 17,  0, 0, 0, 0, 0, 2;
 19,  7, 0, 3, 0, 0, 0, 2;
 23,  0, 5, 0, 0, 0, 0, 0, 2;
 29, 11, 0, 0, 3, 0, 0, 0, 0, 2;
 ...

Cf. A000040, A051731, A130070.

Primes by columns, interspersed with (k-1) zeros. Map the primes on the "1" terms of A051731, by columns; and rest zeros.

A361706 Inverse Moebius transform applied twice to primes.

Original entry on oeis.org

2, 7, 9, 19, 15, 37, 21, 50, 39, 65, 35, 116, 45, 91, 87, 134, 63, 174, 71, 200, 125, 155, 87, 322, 125, 197, 172, 282, 113, 383, 131, 349, 217, 271, 213, 555, 161, 311, 267, 546, 183, 555, 195, 482, 402, 379, 215, 857, 267, 546, 369, 602, 245, 768, 349, 774, 421, 503, 281, 1204, 287, 561, 582, 875, 425

Offset: 1

Ilya Gutkovskiy, Mar 21 2023

nonn

Dirichlet convolution of primes with the number of divisors function.

Winston de Greef, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A000005, A000040, A007429, A007445, A361707.

a:= (proc(p) proc(n) uses numtheory;
       add(p(d), d=divisors(n))
     end end@@2)(ithprime):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 23 2023

Table[Sum[DivisorSigma[0, n/d] Prime[d], {d, Divisors[n]}], {n, 1, 65}]

a(n) = sumdiv(n, d, numdiv(n/d)*prime(d)); \\ Michel Marcus, Mar 23 2023

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

a(n) = Sum_{d|n} A000005(n/d) * prime(d).

A361708 Inverse Moebius transform of nonprimes.

Original entry on oeis.org

1, 5, 7, 13, 10, 21, 13, 27, 22, 30, 19, 49, 22, 39, 40, 52, 27, 63, 29, 68, 51, 56, 35, 98, 46, 64, 61, 87, 43, 114, 46, 98, 73, 80, 72, 142, 53, 87, 83, 138, 58, 145, 61, 126, 118, 103, 66, 189, 81, 135, 103, 144, 75, 177, 104, 178, 113, 127, 82, 254, 85, 135, 152, 185, 119

Offset: 1

Ilya Gutkovskiy, Mar 21 2023

Paolo Xausa, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A007445, A018252, A361709.

Nonprime:= remove(isprime, [$1..1000]): N:= nops(Nonprime):
seq(add(Nonprime[d],d=numtheory:-divisors(n)),n=1..N); # Robert Israel, Mar 23 2023

With[{np = Select[Range[100],!PrimeQ[#] &]}, Table[DivisorSum[n, np[[#]] &], {n, Length[np]}]] (* Paolo Xausa, Aug 21 2025 *)

a(n) = Sum_{d|n} A018252(d).

A062778 Values of Moebius-transform of PrimePi function.

Original entry on oeis.org

0, 1, 2, 1, 3, 0, 4, 2, 2, 0, 5, 1, 6, 1, 1, 2, 7, 2, 8, 3, 2, 2, 9, 2, 6, 2, 5, 2, 10, 3, 11, 5, 4, 3, 4, 2, 12, 3, 4, 2, 13, 3, 14, 5, 6, 4, 15, 4, 11, 5, 6, 5, 16, 4, 8, 5, 6, 5, 17, 2, 18, 6, 8, 7, 9, 4, 19, 7, 8, 6, 20, 5, 21, 8, 9, 8, 12, 6, 22, 8, 13, 8, 23, 6, 13, 8, 11, 7, 24, 4, 14, 9, 11, 8

Offset: 1

Labos Elemer, Jul 18 2001

nonn

			n=12, divisors = D(12) = {1,2,3,4,6,12}, pi(12/divisors) = {5,3,2,2,1,0}, mu(divisors) = {1,-1,-1,0,1,0}, Sum = 5*1 - 3*1 - 2*1 + 0 + 1*1 + 0 = 1, thus a(12)=1; for p=prime(n), pi(p/divisor) = {n,0}, mu({1,p})={1,-1}, Sum = 1*n + 0 = n, so a(prime(n)) = n.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A007444, A007445.

f[n_] := Block[{d = Divisors@n}, Plus @@ (MoebiusMu /@ (n/d)*PrimePi /@ d)]; Array[f, 94] (* Robert G. Wilson v, Dec 07 2005 *)

{ for (n=1, 1000, d=divisors(n); write("b062778.txt", n, " ", sum(k=1, length(d), primepi(n/d[k]) * moebius(d[k]))) ) } \\ Harry J. Smith, Aug 10 2009

a(n) = sumdiv(n, d, primepi(d)*moebius(n/d)); \\ Michel Marcus, Nov 05 2018

a(n) = Sum_{d|n} pi(n/d)*mu(d).

cp's OEIS Frontend

A061152 Expansion of Product_{n>=1} (1+x^n)^prime(n).

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A275700 a(n) = Product_{d|n} prime(d).

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A061150 a(n) = Sum_{d|n} d*prime(d).

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A061151 1 + Sum_{n>=1} a_n x^n = 1/Product_{n>=1} (1+x^n)^prime(n).

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A127639 A051731 * A127640, where A127640 = infinite lower triangular matrix with the sequence of primes in the main diagonal and the rest zeros.

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A062774 Inverse Moebius transform of PrimePi function.

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PARI

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A130069 Primes by columns, interspersed with (k-1) zeros.

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A361706 Inverse Moebius transform applied twice to primes.

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Maple

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