A051674 -id:A051674 - cp's OEIS frontend

A257278 Prime powers p^m with p <= m.

4, 8, 16, 27, 32, 64, 81, 128, 243, 256, 512, 729, 1024, 2048, 2187, 3125, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 59049, 65536, 78125, 131072, 177147, 262144, 390625, 524288, 531441, 823543, 1048576, 1594323, 1953125, 2097152, 4194304, 4782969, 5764801, 8388608, 9765625

Offset: 1

M. F. Hasler, Apr 28 2015

nonn

Might be called "high" powers of primes. Motivated by challenges for which low powers of large primes provide somewhat trivial solutions, cf. A257279. The definition also avoids the question of the whether prime itself is to be considered as a prime power or not, cf. A000961 vs. A025475. In view of the condition p <= n, up to 10^10, only powers of the primes 2, 3, 5 and 7 (namely, less than 10) can occur.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000961, A025475, A257279.
Cf. A000040, A051674 (subsequence).
Subsequence of A122494 and A192135 (p < m, subsequence).
Cf. A257572, A257573.

import Data.Set (singleton, deleteFindMin, insert)
a257278 n = a257278_list !! (n-1)
a257278_list = f (singleton (4, 2)) 27 (tail a000040_list) where
   f s pp ps@(p:ps'@(p':_))
     | qq > pp   = pp : f (insert (pp * p, p) s) (p' ^ p') ps'
     | otherwise = qq : f (insert (qq * q, q) s') pp ps
     where ((qq, q), s') = deleteFindMin s
-- Reinhard Zumkeller, May 01 2015

seq[lim_] := Module[{s = {}, p = 2}, While[p^p <= lim, AppendTo[s, p^Range[p, Log[p, lim]]]; p = NextPrime[p]]; Sort[Flatten[s]]]; seq[10^7] (* Amiram Eldar, Apr 14 2025 *)

L=List();lim=10;forprime(p=1,lim,for(n=p,lim*log(lim)\log(p),listput(L,p^n)));listsort(L);L

a(n) = A257572(n) ^ A257573(n). - Reinhard Zumkeller, May 01 2015

Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^(p-1)*(p-1)) = 0.55595697220270661763... - Amiram Eldar, Oct 24 2020

A381205 a(n) is the cardinality of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

Original entry on oeis.org

0, 2, 2, 1, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 1, 3, 2, 4, 2, 2, 3, 3, 3, 2, 2, 3, 3, 4, 2, 4, 2, 3, 4, 3, 2, 4, 2, 3, 3, 3, 2, 3, 3, 4, 3, 3, 2, 4, 2, 3, 4, 2, 3, 4, 2, 3, 3, 4, 2, 2, 2, 3, 4, 3, 3, 4, 2, 4, 2, 3, 2, 4, 3, 3, 3, 4, 2, 4

Offset: 1

Paolo Xausa, Feb 17 2025

The prime factorization of 1 is the empty set, so a(1) = 0 by convention.

			a(16) = 2 because 12 = 2^3, the set of these bases and exponents is {2, 3} and its size is 2.
a(31500) = 5 because 31500 = 2^2*3^2*5^3*7^1, the set of these bases and exponents is {1, 2, 3, 5, 7} and its size is 5.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A051674 (positions of ones), A381201, A381202, A381203, A381204, A381212.

a:= n-> nops({map(i-> i[], ifactors(n)[2])[]}):
seq(a(n), n=1..90);  # Alois P. Heinz, Feb 18 2025

A381205[n_] := If[n == 1, 0, Length[Union[Flatten[FactorInteger[n]]]]];
Array[A381205, 100]

a(n) = my(f=factor(n)); #setunion(Set(f[,1]), Set(f[,2])); \\ Michel Marcus, Feb 18 2025

from sympy import factorint
def a(n): return len(set().union(*(set(pe) for pe in factorint(n).items())))
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Feb 18 2025

A129286 a(n) = A129152(n) / 5^5, where A129152 is the trajectory of 5^6 under A003415, the arithmetic derivative.

Original entry on oeis.org

5, 6, 11, 12, 28, 60, 152, 388, 780, 2036, 4076, 8156, 16316, 32636, 66232, 169612, 339228, 1244808, 4856004, 14568120, 56648484, 190791072, 866874960, 4084880112, 16342519392, 73816544592, 305811402048, 1732931293824, 10145637246528, 52715820454848

Offset: 0

Reinhard Zumkeller, Apr 07 2007

nonn

Cf. A129284, A129285, A051674.

a129286 n = a129286_list !! n
a129286_list = iterate a129283 5  -- Reinhard Zumkeller, Nov 01 2013

a(n+1) = A129283(a(n)), a(0) = 5.

a(n) = A090635(n+3) / 4, i.e., the trajectory of 20 under A003415, divided by the common factor 4. - M. F. Hasler, Nov 27 2019

A007965 a(n) = n^prime(n) - prime(n)^n.

Original entry on oeis.org

-1, -1, 118, 13983, 48667074, 13055867207, 232630103648534, 144115171092292831, 8862938117851348434466, 99999999999999579292766699799, 191943424957750455095669944886980, 8505622499821102144569548732108989899311

Offset: 1

Dennis S. Kluk (mathemagician(AT)ameritech.net)

sign

			a(3) = 3^5 - 5^3 = 243 - 125 = 118.
a(4) = 16384 - 2401 = 13983.

Matthew House, Table of n, a(n) for n = 1..96

Cf. A000312, A051674.

lst={};Do[p=Prime[n];AppendTo[lst, n^p-p^n], {n, 4!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 19 2008 *)

vector(13, n, n^prime(n) - prime(n)^n) \\ Michel Marcus, Aug 05 2015

More terms from Labos Elemer, Jul 09 2001

A125136 Triangle read by rows in which row n gives list of prime factors of p^p + 1 where p = prime(n).

Original entry on oeis.org

5, 2, 2, 7, 2, 3, 521, 2, 2, 2, 113, 911, 2, 2, 3, 23, 89, 199, 58367, 2, 7, 13417, 20333, 79301, 2, 3, 3, 45957792327018709121, 2, 2, 5, 108301, 1049219, 870542161121, 2, 2, 2, 3, 47, 139, 1013, 1641281, 52626071, 1522029233, 2, 3, 5, 233, 6864997

Offset: 1

N. J. A. Sloane, Jan 21 2007

Product over the n-th row of the table is A051674(n) + 1. The number of elements in the n-th row is A115973(n). - R. J. Mathar, Jan 22 2007

(p + 1) divides p^p + 1 for odd prime p. - Alexander Adamchuk, Jan 22 2007

			Rows read
  5;
  2, 2, 7;
  2, 3, 521;
  2, 2, 2, 113, 911;
  2, 2, 3, 23, 89, 199, 58367;
  2, 7, 13417, 20333, 79301;
  2, 3, 3, 45957792327018709121;
  2, 2, 5, 108301, 1049219, 870542161121;
  2, 2, 2, 3, 47, 139, 1013, 1641281, 52626071, 1522029233;
  2, 3, 5, 233, 6864997, 9487923853, 5639663878716545087233;
  2, 2, 2, 2, 2, 373, 1613, 62869, 145577, 35789156484227, 2706690202468649;
  etc.

Sam Wagstaff, Factorizations of p^p + 1 for most p < 180

Cf. A088730, A125135.

Cf. A007571 = largest factor of n^n + 1.

pfs := proc(n) local ifs,a,e,b ; ifs := ifactors(n)[2] ; a := [] ; for b from 1 to nops(ifs) do for e from 1 to op(2,op(b,ifs)) do a := [op(a),op(1,op(b,ifs))] ; od ; od ; RETURN(a) ; end; A125136 := proc(nmax) local a,p,n,pp ; a := [] ; p := 2 ; while nops(a) < nmax do a := [op(a),op(pfs(p^p+1))] ; p := nextprime(p) ; od ; RETURN(a) ; end; A125136(40) ; # R. J. Mathar, Jan 22 2007

lpf[n_]:=Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]; lpf/@(#^#+1&/@ Prime[Range[10]])//Flatten (* Harvey P. Dale, Oct 18 2020 *)

More terms from Alexander Adamchuk and R. J. Mathar, Jan 22 2007

A129252 Smallest prime factor p of n such that p^p is a divisor of n, a(n)=1 if no such factor exists.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Apr 07 2007

nonn

			For n = 108 = 2^2 * 3^3, it is 2 that is the smallest prime factor p satisfying p^p | 108, thus a(108) = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A020639, A048103 (positions of 1's), A008578, A051674, A129251, A359550, A368333, A380528.

Differs from A327936 for the first time at n=108.

Array[If[IntegerQ@ #, #, 1] &@ First@ SelectFirst[FactorInteger[#], #1 <= #2 & @@ # &] &, 120] (* Michael De Vlieger, Oct 01 2019 *)

A129252(n) = { my(f = factor(n)); for(k=1, #f~, if(f[k, 2]>=f[k, 1], return(f[k, 1]))); (1); }; \\ Antti Karttunen, Oct 01 2019

A129252(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(p)); if(pp > n, return(1))); }; \\ Antti Karttunen, Feb 09 2025

a(n) = 1 iff A129251(n) = 0.
a(A048103(n)) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 - 1/p^p) + Sum_{p prime} ((1/p^(p-1)) * Product_{primes q < p} (1-1/q^q)) = 1.30648526015949409005... . - Amiram Eldar, Nov 07 2022

Data section extended to a(120) by Antti Karttunen, Oct 01 2019

A129284 a(n) = A129150(n) / 4, where A129150(n) = n-th arithmetic derivative of 2^3.

Original entry on oeis.org

2, 3, 4, 8, 20, 44, 92, 188, 380, 856, 2148, 5024, 17616, 58768, 176320, 755904, 3305920, 13885184, 69634816, 348174336, 2385273856, 14652403712, 102566830080, 849285738496, 6035962949632, 44017806979072, 308166534991872, 2380768960708608, 23410894780694528

Offset: 0

Reinhard Zumkeller, Apr 07 2007

nonn

In general, the trajectory of p^(p+1) under A003415 has a common factor p^p, and divided by p^p it gives the trajectory of p under A129283: n -> n + n'. Here we have the case p = 2, see A129151 and A129152 for p = 3 and 5. - M. F. Hasler, Nov 28 2019

Paolo P. Lava, Table of n, a(n) for n = 0..75

Cf. A129285, A129286, A051674.

a129284 n = a129150 n `div` 4  -- Reinhard Zumkeller, Nov 01 2013, corrected by M. F. Hasler, Nov 29 2019

A129284_upto(n)=A129150_upto(n)\4 \\ M. F. Hasler, Nov 29 2019

a(n+1) = A129283(a(n)), a(0) = 2.

a(18)-a(28) from Paolo P. Lava, Apr 16 2012

Edited by M. F. Hasler, Nov 27 2019

A129285 a(n) = A129151(n) / 27.

Original entry on oeis.org

3, 4, 8, 20, 44, 92, 188, 380, 856, 2148, 5024, 17616, 58768, 176320, 755904, 3305920, 13885184, 69634816, 348174336, 2385273856, 14652403712, 102566830080, 849285738496, 6035962949632, 44017806979072, 308166534991872, 2380768960708608, 23410894780694528

Offset: 0

Reinhard Zumkeller, Apr 07 2007

nonn

Cf. A129284 (essentially the same), A129283 (n + n'), A129286, A051674, A129150, A090636 (trajectory of 15 under arithmetic derivative A003415).

a129285 n = a129151 n `div` 27  -- Reinhard Zumkeller, Nov 01 2013 [Moved here from A129284, first erroneous line deleted. - M. F. Hasler, Nov 27 2019]

a(n+1) = A129283(a(n)), a(0) = 3.
a(n) = A129284(n+1). - Eric M. Schmidt, Oct 22 2013
Thus a(n) = A129150(n+1) / 4 = A090636(n+3) / 4. - M. F. Hasler, Nov 27 2019

A383300 Numbers k such that primorial base expansion of k has the primorial base expansion of k' as its suffix, where k' stands for the arithmetic derivative of k (A003415).

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331

Offset: 1

Antti Karttunen, May 15 2025

nonn

a(n) = A348283(n) for n=1 and n=3..334432. a(334433) = 4784261, which is not present in A348283 (see examples). - R. J. Mathar and Antti Karttunen, May 16 2025

			0 is a term as A003415(0) = 0.
1 is a term as A003415(1) = 0, whose primorial base expansion is here understood as an empty sequence of digits, thus it is a suffix of A049345(1) = 1.
3, like all odd primes, is a term as A003415(3) = 1, with A049345(3) = 11 and A049345(1) = 1.
4 and 27 are terms as they are in A051674 (the nonzero fixed points of A003415).
4784261 is a term as A003415(4784261) = 189671, with A049345(4784261) = 96411121 and A049345(189671) = 6411121. 4784261 is the first term > 1 of this sequence that is not in A348283. See more examples in A383301.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to primorial base.

Cf. A003415, A049345, A235224.
Disjoint union of {1}, A348283\{2} and A383301.
Cf. A006005, A051674 (other subsequences).
Subsequence of A383299.
Cf. also A383933.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA383300(n) = if(n<2, 1, my(p=2, k=A003415(n)); while(k, if((k%p)!=(n%p), return(0)); n = n\p; k = k\p; p = nextprime(1+p)); (1));

A002110(n) = prod(i=1,n,prime(i));
A235224(n) = { my(s=0, p=2); while(n, s++; n = n\p; p = nextprime(1+p)); (s); };
isA383300(n) = { my(ad=A003415(n)); (n%A002110(A235224(ad))==ad); };

{k such that A003415(k) is equal to k modulo A002110(A235224(A003415(k)))}.

A082872 a^n + b^n + c^n + ..., where abc* ... is the prime factorization of n.

Original entry on oeis.org

1, 4, 27, 32, 3125, 793, 823543, 768, 39366, 9766649, 285311670611, 539633, 302875106592253, 678223089233, 30531927032, 262144, 827240261886336764177, 775103122, 1978419655660313589123979, 95367433737777, 558545874543637210

Offset: 1

Jason Earls, May 25 2003

n*log_10(2) + log_10(log_2(n)) <= length(a(n)) <= n*log_10(n). - Martin Renner, Jan 18 2012

If m = p^k is a power of a prime then a(n) = sum(p^m,i=1..k) = k*p^m is composite. - Martin Renner, Jan 31 2013

			a(6) = a(2*3) = 2^6 + 3^6 = 793.
a(8) = a(2*2*2) = 2^8 + 2^8 + 2^8 = 768.

T. D. Noe, Table of n, a(n) for n = 1..100

Cf. A082813, A082814, A051674.

A082872 := proc(n)
    local ps;
    if n= 1 then
        1;
    else
        ps := ifactors(n)[2] ;
        add( op(2,p)*op(1,p)^n,p=ps) ;
    end if;
end proc: # R. J. Mathar, Mar 12 2014

Table[f = FactorInteger[n]; Total[Flatten[Table[Table[f[[i, 1]], {f[[i, 2]]}], {i, Length[f]}]]^n], {n, 25}] (* T. D. Noe, Feb 01 2013 *)
Table[Total[Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]^n],{n,30}] (* Harvey P. Dale, Jun 10 2016 *)

cp's OEIS Frontend

A257278 Prime powers p^m with p <= m.

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A381205 a(n) is the cardinality of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

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A129286 a(n) = A129152(n) / 5^5, where A129152 is the trajectory of 5^6 under A003415, the arithmetic derivative.

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A007965 a(n) = n^prime(n) - prime(n)^n.

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A125136 Triangle read by rows in which row n gives list of prime factors of p^p + 1 where p = prime(n).

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A129252 Smallest prime factor p of n such that p^p is a divisor of n, a(n)=1 if no such factor exists.

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A129284 a(n) = A129150(n) / 4, where A129150(n) = n-th arithmetic derivative of 2^3.

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A082872 a^n + b^n + c^n + ..., where abc* ... is the prime factorization of n.