A030078 -id:A030078 - cp's OEIS frontend

A133525 Sum of third powers of four consecutive primes.

503, 1826, 3996, 8784, 15300, 26136, 48328, 73206, 117000, 173754, 228872, 302904, 401128, 537586, 685060, 882000, 1091034, 1274672, 1540730, 1811754, 2158812, 2682468, 3219730, 3740670, 4260744, 4643100, 5055696, 6011352, 7034400

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=503 because 2^3+3^3+5^3+7^3=503.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A034963, A133524, A133526, A133527, A133528.

N:= 50: # for a(1)..a(N)
P3:= [0,seq(ithprime(i)^3,i=1..N+3)]:
S:= ListTools:-PartialSums(P3):
seq(S[i+4]-S[i],i=1..N); # Robert Israel, Jan 01 2024

a = 3; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a, {n, 1, 100}]
Total/@Partition[Prime[Range[40]]^3,4,1] (* Harvey P. Dale, Jan 06 2019 *)

a(n) = A133530(n) + A030078(n+3). - Michel Marcus, Nov 08 2013

A174905 Numbers with no pair (d,e) of divisors such that d < e < 2*d.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 76, 79, 81, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 98, 101, 103, 106

Offset: 1

Reinhard Zumkeller, Apr 01 2010

nonn

A174903(a(n)) = 0; complement of A005279;
sequences of powers of primes are subsequences;
a(n) = A129511(n) for n < 27, A129511(27) = 35 whereas a(27) = 37.
Also the union of A241008 and A241010 (see the link for a proof). - Hartmut F. W. Hoft, Jul 02 2015
In other words: numbers n with the property that all parts in the symmetric representation of sigma(n) have width 1. - Omar E. Pol, Dec 08 2016
Sequence A357581 shows the numbers organized in columns of a square array by the number of parts in their symmetric representation of sigma. - Hartmut F. W. Hoft, Oct 04 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Hartmut F. W. Hoft, Proof that this sequence equals union of A241008 and A241010

Cf. A000040, A000961, A001248, A005279, A030078, A030514, A129511, A174903, A237271, A237593, A241008, A241010.

Cf. A357581.

a174905 n = a174905_list !! (n-1)
a174905_list = filter ((== 0) . a174903) [1..]
-- Reinhard Zumkeller, Sep 29 2014

filter:= proc(n)
  local d,q;
   d:= numtheory:-divisors(n);
   min(seq(d[i+1]/d[i],i=1..nops(d)-1)) >= 2
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 08 2014

(* it suffices to test adjacent divisors *)
a174905[n_] := Module[{d = Divisors[n]}, ! Apply[Or, Map[2 #[[1]] > #[[2]] &, Transpose[{Drop[d, -1], Drop[d, 1]}]]]]
(* Hartmut F. W. Hoft, Aug 07 2014 *)
Select[Range[106], !MatchQ[Divisors[#], {_, d_, e_, _} /; e < 2d]& ] (* Jean-François Alcover, Jan 31 2018 *)

A120398 Sums of two distinct prime cubes.

Original entry on oeis.org

35, 133, 152, 351, 370, 468, 1339, 1358, 1456, 1674, 2205, 2224, 2322, 2540, 3528, 4921, 4940, 5038, 5256, 6244, 6867, 6886, 6984, 7110, 7202, 8190, 9056, 11772, 12175, 12194, 12292, 12510, 13498, 14364, 17080, 19026, 24397, 24416, 24514

Offset: 1

Tanya Khovanova, Jul 24 2007

If an element of this sequence is odd, it must be of the form a(n)=8+p^3, else it is a(n)=p^3+q^3 with two primes p>q>2. - M. F. Hasler, Apr 13 2008

			2^3+3^3=35=a(1), 2^3+5^3=133=a(2), 3^3+5^3=152=a(3), 2^3+7^3=351=a(4).

M. F. Hasler and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 284 terms from Hasler)
Index to sequences related to sums of cubes.

Subsequence of A024670.

Select[Sort[ Flatten[Table[Prime[n]^3 + Prime[k]^3, {n, 15}, {k, n - 1}]]], # <= Prime[15^3] &]

isA030078(n)=n==round(sqrtn(n,3))^3 && isprime(round(sqrtn(n,3)))  \\ M. F. Hasler, Apr 13 2008

isA120398(n)={ n%2 & return(isA030078(n-8)); n<35 & return; forprime( p=ceil( sqrtn( n\2+1,3)),sqrtn(n-26.5,3), isA030078(n-p^3) & return(1))} \\ M. F. Hasler, Apr 13 2008

for( n=1,10^6, isA120398(n) & print1(n",")) \\ - M. F. Hasler, Apr 13 2008

list(lim)=my(v=List()); lim\=1; forprime(q=3,sqrtnint(lim-8,3), my(q3=q^3); forprime(p=2,min(sqrtnint(lim-q3,3),q-1), listput(v,p^3+q3))); Set(v) \\ Charles R Greathouse IV, Mar 31 2022

A120398 = (A030078 + A030078) - 2*A030078 = 8+(A030078\{8}) U { A030078(m)+A030078(n) ; 1M. F. Hasler, Apr 13 2008

A242378 Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 5, 5, 1, 0, 5, 9, 7, 7, 1, 0, 6, 7, 25, 11, 11, 1, 0, 7, 15, 11, 49, 13, 13, 1, 0, 8, 11, 35, 13, 121, 17, 17, 1, 0, 9, 27, 13, 77, 17, 169, 19, 19, 1, 0, 10, 25, 125, 17, 143, 19, 289, 23, 23, 1, 0, 11, 21, 49, 343, 19, 221, 23, 361, 29, 29, 1, 0

Offset: 0

Antti Karttunen, May 12 2014

Each row i is a multiplicative function, being in essence "the i-th power" of A003961, i.e., A(i,j) = A003961^i (j). Zeroth power gives an identity function, A001477, which occurs as the row zero.
The terms in the same column have the same prime signature.
The array is read by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

			The top-left corner of the array:
  0,   1,   2,   3,   4,   5,   6,   7,   8, ...
  0,   1,   3,   5,   9,   7,  15,  11,  27, ...
  0,   1,   5,   7,  25,  11,  35,  13, 125, ...
  0,   1,   7,  11,  49,  13,  77,  17, 343, ...
  0,   1,  11,  13, 121,  17, 143,  19,1331, ...
  0,   1,  13,  17, 169,  19, 221,  23,2197, ...
...
A(2,6) = A003961(A003961(6)) = p_{1+2} * p_{2+2} = p_3 * p_4 = 5 * 7 = 35, because 6 = 2*3 = p_1 * p_2.

Antti Karttunen, Table of n, a(n) for n = 0..10439; Antidiagonals n = 0..143, flattened

Taking every second column from column 2 onward gives array A246278 which is a permutation of natural numbers larger than 1.
Transpose: A242379.
Row 0: A001477, Row 1: A003961 (from 1 onward), Row 2: A357852 (from 1 onward), Row 3: A045968 (from 7 onward), Row 4: A045970 (from 11 onward).
Column 2: A000040 (primes), Column 3: A065091 (odd primes), Column 4: A001248 (squares of primes), Column 6: A006094 (products of two successive primes), Column 8: A030078 (cubes of primes).
Excluding column 0, a subtable of A297845.
Permutations whose formulas refer to this array: A122111, A241909, A242415, A242419, A246676, A246678, A246684.

A(0,j) = j, A(i,0) = 0, A(i > 0, j > 0) = A003961(A(i-1,j)).

For j > 0, A(i,j) = A297845(A000040(i+1),j) = A297845(j,A000040(i+1)). - Peter Munn, Sep 02 2025

A098999 Sum of cubes of the first n primes.

Original entry on oeis.org

8, 35, 160, 503, 1834, 4031, 8944, 15803, 27970, 52359, 82150, 132803, 201724, 281231, 385054, 533931, 739310, 966291, 1267054, 1624965, 2013982, 2507021, 3078808, 3783777, 4696450, 5726751, 6819478, 8044521, 9339550, 10782447

Offset: 1

Suzanne O' Regan (s.m.oregan(AT)student.ucc.ie), Nov 06 2004

nonn

Robert Price, Table of n, a(n) for n = 1..1000
V. Shevelev, Asymptotics of sum of the first n primes with a remainder term
OEIS Wiki, Sums of powers of primes divisibility sequences

Partial sums of A030078.

P3[n_]:=Sum[Prime[i]^3, {i, 1, n}];Table[P3[n], {n, 1, 60}]

a(n) = sum(i=1, n, prime(i)^3); \\ Michel Marcus, Jan 20 2014

a(n) = 0.25*n^4*log(n)^3 + O(n^4*log(n)^2*log(log(n))). The proof is similar to proof for A007504(n) (see link of Shevelev). - Vladimir Shevelev, Aug 02 2013

A095904 Triangular array of natural numbers (greater than 1) arranged by prime signature.

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 10, 8, 11, 49, 14, 27, 12, 13, 121, 15, 125, 18, 16, 17, 169, 21, 343, 20, 81, 24, 19, 289, 22, 1331, 28, 625, 40, 30, 23, 361, 26, 2197, 44, 2401, 54, 42, 32, 29, 529, 33, 4913, 45, 14641, 56, 66, 243, 36, 31, 841, 34, 6859, 50, 28561, 88, 70

Offset: 0

Alford Arnold, Jul 10 2004

The unit, 1, has the empty prime signature { } (thus not in triangle).
Downwards diagonals:
* Rightmost diagonal: smallest numbers of a given prime signature in increasing order (A025487). This defines the order of signatures used.
  This special ordering of prime signatures (by increasing smallest numbers of a given prime signature, A181087) is unrelated to any of the 8 variants of graded lexicographic or colexicographic orderings (based on the exponents only) since it depends on the magnitudes of the prime numbers. It is not even graded by Omega(n).
* Second rightmost diagonal: second smallest numbers of a given prime signature (A077560). (They are not increasing anymore.)
Upwards diagonals:
* Leftmost diagonal: primes.                           {1}     (A000040)
* 2nd leftmost diagonal: squares of primes.            {2}     (A001248)
* 3rd leftmost diagonal: squarefree biprimes.          {1,1}   (A006881)
* 4th leftmost diagonal: cubes of primes.              {3}     (A030078)
* 5th leftmost diagonal: signature (Achilles numbers)  {1,2}   (A054753)
* 6th leftmost diagonal: fourth powers of primes.      {4}     (A030514)
* 7th leftmost diagonal: signature (Achilles numbers)  {1,3}   (A065036)
* 8th leftmost diagonal: squarefree triprimes.         {1,1,1} (A007304)
  The Achilles numbers are nonsquarefree while not perfect powers.
Prime signatures are often expressed in increasing order of exponents. The decreasing order of exponents (as on the Wiki page, see links) has the advantage of listing the exponents in the same order (with the canonical factorization convention) as the smallest number of a given prime signature.

			343 is in the 4th left- and 4th rightmost diagonal, because it is the 4th value with the 4th prime signature {3}.
First 8 rows of triangular array (Cf. table link for this sequence):
                                   2
                              3         4
                         5         9         6
                    7        25        10         8
               11       49        14        27        12
          13      121        15       125        18        16
     17       169       21       343        20        81        24
19       289       22       1331       28       625        40        30

OEIS Wiki, Prime signature.
OEIS Wiki, Orderings.

Cf. A083140, A064839, A181087.

Extended by Ray Chandler, Jul 31 2004
Corrected (minor) by Daniel Forgues, Jan 21 2011
Example, comments by Daniel Forgues, Jan 21 2011
Edited by Alois P. Heinz, Jan 23 2011
Edited by Daniel Forgues, Jan 23 2011

A229125 Numbers of the form p * m^2, where p is prime and m > 0: union of A228056 and A000040.

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 112, 113, 116, 117, 124, 125, 127, 128, 131, 137, 139, 147, 148, 149

Offset: 1

Chris Boyd, Sep 14 2013

nonn

No term is the product of two other terms.
Squares of terms and pairwise products of distinct terms form a subsequence of A028260.
Numbers n such that A162642(n) = 1. - Jason Kimberley, Oct 10 2016
Numbers k such that A007913(k) is a prime number. - Amiram Eldar, Jul 27 2020

Chris Boyd, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes, IX. On the set of integers representable as a product of a prime and square, Acta Arithmetica, Vol. 7 (1962), pp. 417-420.

Subsequence of A026424.
Cf. A028260, A162642, A229153.
Cf. A007913, A013661.
Subsequences: A000040, A030078, A050997, A054753, A092759, A179643, A179665, A246551.

With[{nn=70},Take[Union[Flatten[Table[p*m^2,{p,Prime[Range[nn]]},{m,nn}]]], nn]] (* Harvey P. Dale, Dec 02 2014 *)

test(n)=isprime(core(n))
for(n=1,200,if(test(n), print1(n",")))

from math import isqrt
from sympy import primepi
def A229125(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//y**2) for y in range(1,isqrt(x)+1))
    return bisection(f,n,n) # Chai Wah Wu, Jan 30 2025

The number of terms not exceeding x is (Pi^2/6) * x/log(x) + O(x/(log(x))^2) (Cohen, 1962). - Amiram Eldar, Jul 27 2020

A157289 Decimal expansion of Zeta(3)/Zeta(6).

Original entry on oeis.org

1, 1, 8, 1, 5, 6, 4, 9, 4, 9, 0, 1, 0, 2, 5, 6, 9, 1, 2, 5, 6, 9, 3, 9, 9, 7, 3, 4, 1, 6, 0, 4, 5, 4, 2, 6, 0, 5, 4, 7, 0, 2, 3, 2, 6, 0, 7, 6, 8, 6, 8, 2, 6, 1, 0, 2, 8, 3, 0, 4, 3, 1, 4, 8, 8, 7, 7, 2, 0, 5, 4, 2, 1, 1, 1, 0, 3, 1, 8, 8, 3, 9, 9, 0, 0, 2, 9, 9, 4, 8, 7, 1, 1, 8, 4, 4, 4, 9, 2, 7, 0, 1, 1, 4, 8

Offset: 1

R. J. Mathar, Feb 26 2009

The Product_{p = primes = A000040} (1+1/p^3), the cubic analog to A082020.

			1.181564949010256912569399734... = (1+1/2^3)*(1+1/3^3)*(1+1/5^3)*(1+1/7^3)*...

Cf. A002117, A005117, A013664, A030078, A062838, A082020.

Maple
```
evalf(Zeta(3)/Zeta(6)) ;
```

RealDigits[Zeta[3]/Zeta[6],10,120][[1]] (* Harvey P. Dale, Jul 23 2016 *)

Equals A002117/A013664 = Product_{i} (1+1/A030078(i)).

Equals Sum_{k>=1} 1/A062838(k) = Sum_{k>=1} 1/A005117(k)^3. - Amiram Eldar, May 22 2020

A195086 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 2.

Original entry on oeis.org

8, 24, 27, 36, 40, 54, 56, 88, 100, 104, 120, 125, 135, 136, 152, 168, 180, 184, 189, 196, 225, 232, 248, 250, 252, 264, 270, 280, 296, 297, 300, 312, 328, 343, 344, 351, 375, 376, 378, 396, 408, 424, 440, 441, 450, 456, 459, 468, 472, 484, 488

Offset: 1

Harvey P. Dale, Sep 08 2011

nonn

From Amiram Eldar, Nov 07 2020: (Start)
Numbers whose powerful part (A057521) is either a cube of a prime (A030078) or a square of a squarefree semiprime (A085986).
The asymptotic density of this sequence is (6/Pi^2) * (Sum_{p prime} 1/(p^2*(p+1)) + Sum_{p=4} (-1)^(k+1)*(k-1)*P(k) + (Sum_{k>=2} (-1)^k*P(k))^2)/2 = 0.0963023158..., where P is the prime zeta function. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Alfred Rényi, On the density of certain sequences of integers, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.
Index entries for sequences computed from exponents in factorization of n

Subsequence of A048108.
Subsequences: A030078 and A085986.
Cf. A001221, A001222, A025487, A057521, A060687, A195069, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A046660, A257851, A261256, A264959.

a195086 n = a195086_list !! (n-1)
a195086_list = filter ((== 2) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[500],PrimeOmega[#]-PrimeNu[#]==2&]

is(n)=bigomega(n)-omega(n)==2 \\ Charles R Greathouse IV, Sep 14 2015

is(n)=my(f=factor(n)[,2]); vecsum(f)==#f+2 \\ Charles R Greathouse IV, Aug 01 2016

A001222(a(n)) - A001221(a(n)) = 2.

A046660(a(n)) = 2. - Reinhard Zumkeller, Nov 29 2015

A083686 a(0) = 8; for n>0, a(n) = 2*a(n-1) - 1.

Original entry on oeis.org

8, 15, 29, 57, 113, 225, 449, 897, 1793, 3585, 7169, 14337, 28673, 57345, 114689, 229377, 458753, 917505, 1835009, 3670017, 7340033, 14680065, 29360129, 58720257, 117440513, 234881025, 469762049, 939524097, 1879048193, 3758096385, 7516192769, 15032385537

Offset: 0

N. J. A. Sloane, Jun 15 2003

An Engel expansion of 2/7 to the base 2 as defined in A181565, with the associated series expansion 2/7 = 2/8 + 2^2/(8*15) + 2^3/(8*15*29) + 2^4/(8*15*29*57) + ... . - Peter Bala, Oct 29 2013

The initial 8 is the only cube in this sequence. - Antti Karttunen, Sep 24 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A020737, A083575, A083683, A083705, A168596, A181565, A195744.

Cf. A005940, A030078.

[7*2^n+1 : n in [0..30]]; // Vincenzo Librandi, Nov 03 2011

7*2^Range[0, 50] + 1 (* Paolo Xausa, Apr 02 2024 *)

Vec((8-9*x)/((1-x)*(1-2*x)) + O(x^40)) \\ Colin Barker, Sep 20 2016

a(n)=7<Charles R Greathouse IV, Sep 20 2016

a(n) = 7*2^n + 1. - David Brotherton (dbroth01(AT)aol.com), Jul 29 2003
a(n) = 3*a(n-1) - 2*a(n-2), n>1. - Vincenzo Librandi, Nov 03 2011
G.f.: (8-9*x) / ((1-x)*(1-2*x)). - Colin Barker, Sep 20 2016
E.g.f.: exp(x)*(1 + 7*exp(x)). - Stefano Spezia, Oct 08 2022
For n >= 0, A005940(a(n)) = A030078(1+n). - Antti Karttunen, Sep 24 2023

cp's OEIS Frontend

A133525 Sum of third powers of four consecutive primes.

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A174905 Numbers with no pair (d,e) of divisors such that d < e < 2*d.

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A120398 Sums of two distinct prime cubes.

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A242378 Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).

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A098999 Sum of cubes of the first n primes.

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Mathematica

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A095904 Triangular array of natural numbers (greater than 1) arranged by prime signature.

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Extensions

A229125 Numbers of the form p * m^2, where p is prime and m > 0: union of A228056 and A000040.

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Mathematica

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A157289 Decimal expansion of Zeta(3)/Zeta(6).

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