A050997 -id:A050997 - cp's OEIS frontend

A182944 Square array A(i,j), i >= 1, j >= 1, of prime powers prime(i)^j, by descending antidiagonals.

2, 4, 3, 8, 9, 5, 16, 27, 25, 7, 32, 81, 125, 49, 11, 64, 243, 625, 343, 121, 13, 128, 729, 3125, 2401, 1331, 169, 17, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23

Offset: 1

Clark Kimberling, Dec 14 2010

We alternatively refer to this sequence as a triangle T(.,.), with T(n,k) = A(k,n-k+1) = prime(k)^(n-k+1).
The monotonic ordering of this sequence, prefixed by 1, is A000961.
The joint-rank array of this sequence is A182869.
Main diagonal gives A062457. - Omar E. Pol, Sep 11 2018

			Square array A(i,j) begins:
  i \ j: 1      2      3      4      5  ...
  ---\-------------------------------------
  1:     2,     4,     8,    16,    32, ...
  2:     3,     9,    27,    81,   243, ...
  3:     5,    25,   125,   625,  3125, ...
  4:     7,    49,   343,  2401, 16807, ...
  ...
The triangle T(n,k) begins:
  n\k:  1     2     3     4     5     6  ...
  1:    2
  2:    4     3
  3:    8     9     5
  4:   16    27    25     7
  5:   32    81   125    49    11
  6:   64   243   625   343   121    13
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened)
Michael De Vlieger, Diagram showing row n of triangle in a semicircle as noted, with a color function associated with the magnitude of T(n,k) compared to 2^n in light blue, where prime(n) is the smallest and the prime power indicated in red the largest in the row.

Cf. A000961, A006939 (row products of triangle), A062457, A182945, A332979 (row maxima of triangle).
Columns: A000040 (1), A001248 (2), A030078 (3), A030514 (4), A050997 (5), A030516 (6), A092759 (7), A179645 (8), A179665 (9), A030629 (10).
A319075 extends the array with 0th powers.
Subtable of A242378, A284457, A329332.

TableForm[Table[Prime[n]^j,{n,1,14},{j,1,8}]]

From Peter Munn, Dec 29 2019: (Start)
A(i,j) = A182945(j,i) = A319075(j,i).
A(i,j) = A242378(i-1,2^j) = A329332(2^(i-1),j).
A(i,i) = A062457(i).
(End)

Clarified in respect of alternate reading as a triangle by Peter Munn, Aug 28 2022

A189975 Numbers with prime factorization pqr^3 for distinct p, q, r.

Original entry on oeis.org

120, 168, 264, 270, 280, 312, 378, 408, 440, 456, 520, 552, 594, 616, 680, 696, 702, 728, 744, 750, 760, 888, 918, 920, 945, 952, 984, 1026, 1032, 1064, 1128, 1144, 1160, 1240, 1242, 1272, 1288, 1416, 1464, 1480, 1485, 1496, 1566, 1608, 1624, 1640, 1672

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 03 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of prime signatures
Index to sequences related to prime signature

Cf. A030634, A050997, A178739.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,3}; Select[Range[2000],f]

list(lim)=my(v=List(),t);forprime(p=2,(lim\6)^(1/3),forprime(q=2,sqrt(lim\p^3),if(p==q,next);t=p^3*q;forprime(r=q+1,lim\t,if(p==r,next);listput(v,t*r))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 19 2011

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A189975(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((t:=primepi(s:=isqrt(y:=x//r**3)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,3)[0]+1))+sum(primepi(x//p**4) for p in primerange(integer_nthroot(x,4)[0]+1))-primepi(integer_nthroot(x,5)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

A131992 a(n) = 1 + prime(n) + prime(n)^2 + prime(n)^3 + prime(n)^4.

Original entry on oeis.org

31, 121, 781, 2801, 16105, 30941, 88741, 137561, 292561, 732541, 954305, 1926221, 2896405, 3500201, 4985761, 8042221, 12326281, 14076605, 20456441, 25774705, 28792661, 39449441, 48037081, 63455221, 89451461, 105101005, 113654321

Offset: 1

Reinhard Zumkeller, Aug 06 2007

Thébault shows that a(2) = 121 is the only square in this sequence. - Charles R Greathouse IV, Jul 23 2013

Giovanni Resta has found that 28792661 is the first Sophie Germain prime of this form (and actually of the form p = (n^m-1)/(n-1) for any p-1 > n, m > 1). - M. F. Hasler, Mar 03 2020

			a(1) = 31 because prime(1) = 2 and 1 + 2 + 2^2 + 2^3 + 2^4 = 1 + 2 + 4 + 8 + 16 = 31.

Victor Thébault, Curiosités arithmétiques, Mathesis 62 (1953), pp. 120-129.

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A030514, A008864, A060800, A131993.

Equals A053699 restricted to prime indices. Subsequence of primes is A190527.

Table[Sum[Prime[n]^k, {k, 0, 4}], {n, 30}] (* Alonso del Arte, May 24 2015 *)

a(n)=sigma(prime(n)^4)  \\ Charles R Greathouse IV, Jul 23 2013

a(n) = 1 + A131991(n)*A000040(n).
a(n) = (A050997(n) - 1)/A006093(n).
a(n) = A000203(prime(n)^4). - R. J. Mathar, Mar 15 2018
a(n) = (prime(n)^5 - 1)/(prime(n) - 1) = A053699(prime(n)). (This is also meant by the 2nd formula.) - M. F. Hasler, Mar 03 2020

A133536 Sum of fifth powers of two consecutive primes.

Original entry on oeis.org

275, 3368, 19932, 177858, 532344, 1791150, 3895956, 8912442, 26947492, 49140300, 97973108, 185200158, 262864644, 376353450, 647540500, 1133119792, 1559520600, 2194721408, 3154354458, 3877300944, 5150127992, 7016097042, 9523100092

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=2^5+3^5=275.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133537, A133538.

a = 5; Table[Prime[n]^a + Prime[n + 1]^a, {n, 1, 100}]

a(n) = A050997(n) + A050997(n+1). - Michel Marcus, Nov 09 2013

A138404 a(n) = prime(n)^5 - prime(n).

Original entry on oeis.org

30, 240, 3120, 16800, 161040, 371280, 1419840, 2476080, 6436320, 20511120, 28629120, 69343920, 115856160, 147008400, 229344960, 418195440, 714924240, 844596240, 1350125040, 1804229280, 2073071520, 3077056320, 3939040560

Offset: 1

Artur Jasinski, Mar 19 2008

Subsequence of A061167. - Bernard Schott, Feb 06 2023

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index to sequences related to prime powers

Cf. A000040, A006093, A008864, A050997, A061167, A066872, A138430.

[NthPrime((n))^5 - NthPrime((n)): n in [1..30] ]; // Vincenzo Librandi, Jun 17 2011

a = {}; Do[p = Prime[n]; AppendTo[a, p^5 - p], {n, 1, 50}]; a
#^5-#&/@Prime[Range[30]] (* Harvey P. Dale, Dec 25 2022 *)

forprime(p=2,1e3,print1(p^5-p", ")) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = A050997(n) - A000040(n). - Elmo R. Oliveira, Jan 27 2023
From Bernard Schott, Feb 09 2023: (Start)
a(n) = A061167(A000040(n)).
a(n) = 30 * A138430(n).
a(n) = A000040(n) * A006093(n) * A008864(n) * A066872(n). (End)

A115975 Numbers of the form p^k, where p is a prime and k is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Giovanni Teofilatto, Mar 15 2006; corrected Apr 23 2006

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

Subsequence of A000961 (powers of primes).
Cf. A117245 (partial sums).
Subsequences: A000040, A001248, A030078, A050997, A138031, A179645.

With[{nn=60},Take[Join[{1},Union[First[#]^Last[#]&/@Union[Flatten[ Outer[List,Prime[Range[nn]],Fibonacci[Range[nn/6]]],1]]]],70]] (* Harvey P. Dale, Jun 05 2012 *)
fib[lim_] := Module[{s = {}, f = 1, k = 2}, While[f <= lim, AppendTo[s, f]; k++; f = Fibonacci[k]]; s]; seq[max_] := Module[{s = {1}, p = 2, e = 1, f = {}}, While[e > 0, e = Floor[Log[p, max]]; If[f == {}, f = fib[e], f = Select[f, # <= e &]]; s = Join[s, p^f]; p = NextPrime[p]]; Sort[s]]; seq[250] (* Amiram Eldar, Aug 09 2024 *)

{m=240;v=Set([]);forprime(p=2,m,i=0;while((s=p^fibonacci(i))

Edited and corrected by Klaus Brockhaus, Apr 25 2006

A122103 Sum of the fifth powers of the first n primes.

Original entry on oeis.org

32, 275, 3400, 20207, 181258, 552551, 1972408, 4448507, 10884850, 31395999, 60025150, 129369107, 245225308, 392233751, 621578758, 1039774251, 1754698550, 2599294851, 3949419958, 5753649309, 7826720902, 10903777301, 14842817944

Offset: 1

Alexander Adamchuk, Aug 20 2006

a(n) is prime for n = {66, 148, 150, 164, 174, 214, 238, 264, 312, 328, 354, 440, 516, 536, 616, 624, 724, 744, 774, 836, 940, ...} = A122125. Primes of this form are listed in A122126 = {32353461605953, 9874820441996857, 10821208357045699, ...}.

			a(2) = 275 because the first two primes are 2 and 3, the fifth powers of which are 32 and 243, and 32 + 243 = 275.
a(3) = 3400, because the third prime is 5, its fifth power if 3125 and 275 + 3125 = 3400.

OEIS Wiki, Sums of powers of primes divisibility sequences
V. Shevelev, Asymptotics of sum of the first n primes with a remainder term

Cf. A050997, A007504, A024450, A098999, A122102, A122125, A122126.

Table[Sum[Prime[k]^5, {k, n}], {n, 100}]

a(n)=sum(i=1,n,prime(i)) \\ Charles R Greathouse IV, Nov 30 2013

a(n) = sum(k = 1 .. n, prime(k)^5).

a(n) = 1/6*n^6*log(n)^5 + O(n^6*log(n)^4*log(log(n))). The proof is similar to proof for A007504(n) (see link of Shevelev). For a generalization, see comment in A122102. - Vladimir Shevelev, Aug 14 2013

A131993 1 + prime(n) + prime(n)^2 + prime(n)^3 + prime(n)^4 + prime(n)^5.

Original entry on oeis.org

63, 364, 3906, 19608, 177156, 402234, 1508598, 2613660, 6728904, 21243690, 29583456, 71270178, 118752606, 150508644, 234330768, 426237714, 727250580, 858672906, 1370581548, 1830004056, 2101864254, 3116505840, 3987077724

Offset: 1

Reinhard Zumkeller, Aug 06 2007

nonn

a(n) = 1 + A131992(n)*A000040(n).

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A050997, A008864, A060800, A131991.

[1+(&+[NthPrime(n)^(k): k in [1..5]]): n in [1..100]]; // Berselli - Librandi, Apr 20 2011

Total[#^Range[0,5]]&/@Prime[Range[30]]  (* Harvey P. Dale, Apr 20 2011 *)

a(n) = (A030516(n) - 1)/A006093(n).

A255231 The number of factorizations n = Product_i b_i^e_i, where all bases b_i are distinct, and all exponents e_i are distinct >=1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 6, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3

Offset: 1

Saverio Picozzi, Feb 18 2015

nonn

Not multiplicative: a(48) = a(2^4*3) = 5 <> a(2^4)*a(3) = 4*1 = 4. - R. J. Mathar, Nov 05 2016

			From _R. J. Mathar_, Nov 05 2016: (Start)
a(4)=2: 4^1 = 2^2.
a(8)=2: 8^1 = 2^3.
a(9)=2: 9^1 = 3^2.
a(12)=2: 12^1 = 2^2*3^1.
a(16)=4: 16^1 = 4^2 = 2^2*4^1 = 2^4.
a(18)=2: 18^1 = 2*3^2.
a(20)=2: 20^1 = 2^2*5^1.
a(24)=3: 24^1 = 2^2*6^1 = 2^3*3^1.
a(32)=5: 32^1 = 2^1*4^2 = 2^2*8^1 = 2^3*4^1 = 2^5.
a(36)=4: 36^1 = 6^2 = 3^2*4^1 = 2^2*9^1.
a(48)=5: 48^1 = 3^1*4^2 = 2^2*12^1 = 2^3*6^1 = 2^4*3^1.
a(60)=2 : 60^1 = 2^2*15^1.
a(64)=7: 64^1 = 8^2 = 4^3 = 2^2*16^1 = 2^3*8^1 = 2^4*4^1 = 2^6.
a(72)=6 : 72^1 = 3^2*8^1 = 2^1*6^2 = 2^2*18^1 = 2^3*9^1 = 2^3*3^2.
(End)

R. J. Mathar, Table of n, a(n) for n = 1..419
R. J. Mathar, Factorizations of integers into factors with distinct bases and exponents
Index to sequences related to prime signature

Cf. A000688 (b_i not necessarily distinct).

Cf. A001248, A005117, A030078, A030514, A054753, A065036, A085986, A085987, A143610, A178739.

# Count solutions for products if n = dvs_i^exps(i) where i=1..pividx are fixed
Apiv := proc(n,dvs,exps,pividx)
    local dvscnt, expscopy,i,a,expsrt,e ;
    dvscnt := nops(dvs) ;
    a := 0 ;
    if pividx > dvscnt then
        # have exhausted the exponent list: leave of the recursion
        # check that dvs_i^exps(i) is a representation
        if n = mul( op(i,dvs)^op(i,exps),i=1..dvscnt) then
            # construct list of non-0 exponents
            expsrt := [];
            for i from 1 to dvscnt do
                if op(i,exps) > 0 then
                    expsrt := [op(expsrt),op(i,exps)] ;
                end if;
            end do;
            # check that list is duplicate-free
            if nops(expsrt) = nops( convert(expsrt,set)) then
                return 1;
            else
                return 0;
            end if;
        else
            return 0 ;
        end if;
    end if;
    # need a local copy of the list to modify it
    expscopy := [] ;
    for i from 1 to nops(exps) do
        expscopy := [op(expscopy),op(i,exps)] ;
    end do:
    # loop over all exponents assigned to the next base in the list.
    for e from 0 do
        candf := op(pividx,dvs)^e ;
        if modp(n,candf) <> 0 then
            break;
        end if;
        # assign e to the local copy of exponents
        expscopy := subsop(pividx=e,expscopy) ;
        a := a+procname(n,dvs,expscopy,pividx+1) ;
    end do:
    return a;
end proc:
A255231 := proc(n)
    local dvs,dvscnt,exps ;
    if n = 1 then
        return 1;
    end if;
    # candidates for the bases are all divisors except 1
    dvs := convert(numtheory[divisors](n) minus {1},list) ;
    dvscnt := nops(dvs) ;
    # list of exponents starts at all-0 and is
    # increased recursively
    exps := [seq(0,e=1..dvscnt)] ;
    # take any subset of dvs for the bases, i.e. exponents 0 upwards
    Apiv(n,dvs,exps,1) ;
end proc:
seq(A255231(n),n=1..120) ; # R. J. Mathar, Nov 05 2016

a(n)=1 for all n in A005117. a(n)=2 for all n in A001248 and for all n in A054753 and for all n in A085987 and for all n in A030078. a(n)=3 for all n in A065036. a(n)=4 for all n in A085986 and for all n in A030514. a(n)=5 for all n in A178739, all n in A179644 and for all n in A050997. a(n)=6 for all n in A143610, all n in A162142 and all n in A178740. a(n)=7 for all n in A030516. a(n)=9 for all n in A189988 and all n in A189987. a(n)=10 for all n in A092759. a(n) = 11 for all n in A179664. a(n)=12 for all n in A179646. - R. J. Mathar, Nov 05 2016, May 20 2017

Values corrected. Incorrect comments removed. - R. J. Mathar, Nov 05 2016

A133521 Smallest k such that p(n)^5 - k is prime where p(n) is the n-th prime.

Original entry on oeis.org

1, 2, 4, 20, 4, 2, 18, 18, 16, 6, 2, 6, 24, 12, 36, 22, 10, 8, 8, 24, 20, 86, 22, 6, 18, 42, 26, 6, 50, 52, 20, 12, 48, 2, 196, 68, 18, 14, 16, 16, 18, 2, 10, 6, 16, 38, 2, 36, 6, 2, 16, 42, 18, 42, 40, 34, 22, 2, 38, 4, 36, 52, 26, 132, 36, 28, 24, 74, 46, 36, 4, 16, 8, 24, 80, 16

Offset: 1

Carl R. White, Sep 14 2007

			p(10)=29, 29^5 = 20511149; for odd k and n > 1, p(n)^r - k is even and thus not prime, so we only need consider even k.
for k = 2: 20511149 - 2 = 20511147, which is 3 * 23 * 297263 and not prime.
for k = 4: 20511149 - 4 = 20511145, which is 5 * 4102229, also not prime.
for k = 6: 20511149 - 6 = 20511141, which is prime, so 6 is the smallest number that can be subtracted from 20511149 to make another prime.
Hence a(10) = 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A050997, A054271, A091666, A133517, A133518, A133519, A133520, A133522, (A001223).

#-NextPrime[#,-1]&/@(Prime[Range[80]]^5) (* Harvey P. Dale, Sep 27 2020 *)

cp's OEIS Frontend

A182944 Square array A(i,j), i >= 1, j >= 1, of prime powers prime(i)^j, by descending antidiagonals.

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A189975 Numbers with prime factorization pqr^3 for distinct p, q, r.

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A131992 a(n) = 1 + prime(n) + prime(n)^2 + prime(n)^3 + prime(n)^4.

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A133536 Sum of fifth powers of two consecutive primes.

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A138404 a(n) = prime(n)^5 - prime(n).

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A115975 Numbers of the form p^k, where p is a prime and k is a Fibonacci number.

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A122103 Sum of the fifth powers of the first n primes.

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A131993 1 + prime(n) + prime(n)^2 + prime(n)^3 + prime(n)^4 + prime(n)^5.