A095874 -id:A095874 - cp's OEIS frontend

A192135 Prime powers p^e with p < e.

8, 16, 32, 64, 81, 128, 243, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 59049, 65536, 78125, 131072, 177147, 262144, 390625, 524288, 531441, 1048576, 1594323, 1953125, 2097152, 4194304, 4782969, 5764801, 8388608, 9765625, 14348907

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

Donovan Johnson, Table of n, a(n) for n = 1..10000

Complement to A074583 with respect to A000961.

Cf. A095874, A192187, A257278.

A192135 := proc(nmax) local s ,i,p,e ; s := {} ; for i from 1 do p := ithprime(i) ; if p^(p+1) > nmax then break; end if; for e from p+1 do if p^e > nmax then break; end if; s := s union {p^e} ; end do: end do: sort(s) ; end proc:
A192135(20000000) ; # R. J. Mathar, Jul 09 2011

seq[lim_] := Module[{s = {}, p = 2}, While[p^p <= lim, AppendTo[s, p^Range[p+1, Log[p, lim]]]; p = NextPrime[p]]; Sort[Flatten[s]]]; seq[10^7] (* Amiram Eldar, Apr 14 2025 *)

a(n) = A000961(A192187(n)).
A095874(a(n)) =  A192187(n).
Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^p*(p-1)) = 0.26859872089648243789... . - Amiram Eldar, Apr 14 2025

A095841 Prime powers having exactly one partition into two prime powers.

Original entry on oeis.org

2, 3, 127, 163, 179, 191, 193, 223, 239, 251, 269, 311, 337, 343, 389, 419, 431, 457, 491, 547, 557, 569, 599, 613, 653, 659, 673, 683, 719, 739, 787, 821, 839, 853, 883, 911, 929, 953, 967, 977, 1117, 1123, 1201, 1229, 1249, 1283, 1289, 1297, 1303, 1327, 1381, 1409, 1423, 1439, 1451, 1471, 1481, 1499, 1607, 1663, 1681

Offset: 1

Reinhard Zumkeller, Jun 10 2004

nonn

A095840(A095874(a(n))) = 1.

A071330(a(n)) = 1.

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

Cf. A000961, A095842.
Intersection of A208247 and A000961.
Cf. A071330, A095840, A095874.

a095841 n = a095841_list !! (n-1)
a095841_list = filter ((== 1) . a071330) a000961_list
-- Reinhard Zumkeller, Jan 11 2013

N:= 10^4: # to get all terms <= N
primepows:= {1,seq(seq(p^n, n=1..floor(log[p](N))),
    p=select(isprime,[2,seq(2*k+1,k=1..(N-1)/2)]))}:
npp:= nops(primepows):
B:= Vector(N,datatype=integer[4]):
for n from 1 to npp do for m from n to npp do
   j:= primepows[n]+primepows[m];
   if j <= N then B[j]:= B[j]+1 fi;
od od:
select(t -> B[t] = 1, primepows); # Robert Israel, Nov 21 2014

max = 2000; ppQ[n_] := n == 1 || PrimePowerQ[n]; pp = Select[Range[max], ppQ]; lp = Length[pp]; Table[pp[[i]] + pp[[j]], {i, 1, lp}, {j, i, lp}] // Flatten // Select[#, ppQ[#] && # <= max&]& // Sort // Split // Select[#, Length[#] == 1&]& // Flatten (* Jean-François Alcover, Mar 04 2019 *)

is(n)=if(n<127,return(n==2||n==3)); isprimepower(n) && sum(i=2,n\2,isprimepower(i)&&isprimepower(n-i))==1 \\ naive; Charles R Greathouse IV, Nov 21 2014

is(n)=if(!isprimepower(n), return(0)); my(s); forprime(p=2, n\2, if(isprimepower(n-p) && s++>1, return(0))); for(e=2, log(n)\log(2), forprime(p=2, sqrtnint(n\2, e), if(isprimepower(n-p^e) && s++>1, return(0)))); s+(!!isprimepower(n-1))==1 || n==2 \\ faster; Charles R Greathouse IV, Nov 21 2014

has(n)=my(s); forprime(p=2, n\2, if(isprimepower(n-p) && s++>1, return(0))); for(e=2, log(n)\log(2), forprime(p=2, sqrtnint(n\2, e), if(isprimepower(n-p^e) && s++>1, return(0)))); s+(!!isprimepower(n-1))==1
list(lim)=my(v=List([2])); forprime(p=2,lim,if(has(p), listput(v,p))); for(e=2,log(lim)\log(2), forprime(p=2,lim^(1/e), if(has(p^e), listput(v,p^e)))); Set(v) \\ Charles R Greathouse IV, Nov 21 2014

A095842 Prime powers having no partition into two prime powers.

Original entry on oeis.org

1, 149, 331, 373, 509, 701, 757, 809, 877, 907, 997, 1019, 1087, 1259, 1549, 1597, 1619, 1657, 1759, 1777, 1783, 1867, 1973, 2293, 2377, 2503, 2579, 2683, 2789, 2843, 2879, 2909, 2999, 3119, 3163, 3181, 3187, 3299, 3343, 3433, 3539, 3643

Offset: 1

Reinhard Zumkeller, Jun 10 2004

nonn

A095840(A095874(a(n))) = 0.
A071330(a(n)) = 0.
Here, "prime powers" is used in the relaxed sense, including 1. The numbers 96721, 121801, 192721, 205379, 226981,... seem to be the smallest composite terms of this sequence, which establishes the difference with the subsequence A115231. - M. F. Hasler, Nov 20 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Subsequence of A071331.

Cf. A000961, A095841.

a095842 n = a095842_list !! (n-1)
a095842_list = filter ((== 0) . a071330) a000961_list
-- Reinhard Zumkeller, Jan 11 2013

isprimepower(n)=ispower(n,,&n);isprime(n)||n==1;
isA095842(n)=if(!isprimepower(n),return(0));forprime(p=2,n\2,if(isprimepower(n-p),return(0)));forprime(p=2,sqrtint(n\2),for(e=1,log(n\2)\log(p),if(isprimepower(n-p^e),return(0))));!isprimepower(n-1)
\\ Charles R Greathouse IV, Jul 06 2011

A117331 Lexicographically earliest permutation of prime powers such that the exponents of succeeding terms increase at most by 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 8, 11, 13, 17, 19, 23, 25, 27, 16, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 81, 32, 64, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset: 1

Reinhard Zumkeller, Mar 08 2006

nonn

A025474(A095874(a(n+1))) - A025474(A095874(a(n))) <= 1;
A117332(n) = A095874(a(n));
a(A117333(n)) = A000961(n).

			a(13)..a(16): 23,5^2,3^3,2^4;
a(38)..a(43): 113,11^2,5^3,3^4,2^5,2^6;
a(239)..a(248): 1367,37^2,11^3,5^4,3^5,3^6,2^7,2^8,2^9,2^10.

Eric Weisstein's World of Mathematics, Prime Power

A192188 Positions of prime powers p^e with e <= p within A000961.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

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

Complement of A192187.
a(n) = A095874(A074583(n)).
A000961(a(n)) = A074583(n).

A322981 If n is the k-th prime power > 1, a(n) = k, otherwise a(n) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 0, 5, 6, 7, 0, 8, 0, 9, 0, 0, 10, 11, 0, 12, 0, 0, 0, 13, 0, 14, 0, 15, 0, 16, 0, 17, 18, 0, 0, 0, 0, 19, 0, 0, 0, 20, 0, 21, 0, 0, 0, 22, 0, 23, 0, 0, 0, 24, 0, 0, 0, 0, 0, 25, 0, 26, 0, 0, 27, 0, 0, 28, 0, 0, 0, 29, 0, 30, 0, 0, 0, 0, 0, 31, 0, 32, 0, 33, 0, 0, 0, 0, 0, 34, 0, 0, 0, 0, 0, 0, 0, 35

Offset: 1

Antti Karttunen, Jan 01 2019

nonn

This is a ("corrected") variant of A095874, which uses the list of "powers of primes" A000961 instead of prime powers A246655. - M. F. Hasler, Jun 16 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000961 (powers of primes, including 1), A010055, A025528, A069513, A095874 (analog based on A000961), A246655 (prime powers > 1).

Cf. A049084.

up_to = 16384;
A322981list(up_to) = { my(v=vector(up_to), k=0); for(n=1,up_to,if(isprimepower(n),k++; v[n] = k, v[n] = 0)); (v); };
v322981 = A322981list(up_to);
A322981(n) = v322981[n];

A322981(n)=if(isprimepower(n),sum(i=1,exponent(n),primepi(sqrtnint(n,i)))) \\ M. F. Hasler, Jun 16 2021

a(n) = A010055(n) * A025528(n) = A069513(n) * A025528(n).
a(n) = A025528(A069513(n)*n), when assuming that A025528(0) = 0.
a(A000961(1+n)) = n for all n >= 1.

A344826 Integers k such that k/A097621(k) is an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, 891, 1584, 1782, 3564, 4032, 4455, 4752, 7920, 8910, 17820, 20160, 22275, 23760, 44550, 49896, 86400, 89100, 100800, 118800, 249480, 349272, 399168, 694008, 1097712, 1746360, 1778400, 1995840, 2181168, 2774304, 2794176, 3470040

Offset: 1

Michel Marcus, May 29 2021, after a suggestion from Allan C. Wechsler

nonn

Allan C. Wechsler remarks that one can derive larger terms from existing terms. For instance, k = 5552064 has q = k/A097621(k) = 18. So multiplying 5552064 by 31 = A000961(18) will give a new term with q = 31.

More precisely, if k = a(n) has q = A343886(k) and m = A000961(q) such that gcd(k, m) = 1, then k*m is also a term. We could call "primitive" those terms not derived from a smaller term in this way. All the listed terms are primitive, but a({35, 36, 38, 42, 43}) allow the sequence to be extended to five larger non-primitive terms. The second and fourth one, having q = 17 resp. q = 23, both lead to a whole chain of many new terms. - M. F. Hasler, Jun 15 2021

Ray Chandler, Table of n, a(n) for n = 1..187 (terms 1..90 from Michel Marcus, terms 91..133 from Lars Blomberg)

Cf. A000961, A095874, A097621, A127724, A343886 (the ratios k/A097621(k)).

f(n) = if(isprimepower(n), sum(i=1, logint(n, 2), primepi(sqrtnint(n, i)))+1, n==1); \\ A095874
ff(n) = my(fr=factor(n)); for (k=1, #fr~, fr[k,1] = f(fr[k,1]^fr[k,2]); fr[k,2] = 1); factorback(fr); \\ A097621
isok(k) = denominator(k/ff(k)) == 1;

mappp(nn) = {my(map = Map()); mapput(map, 1, 1); my(nb=1); for (n=2, nn, if (isprimepower(n), nb++; mapput(map, n, nb));); map;}
ff(n, map) = my(fr=factor(n)); for (k=1, #fr~, fr[k, 1] = mapget(map, fr[k, 1]^fr[k, 2]); fr[k, 2] = 1); factorback(fr); \\ A097621
wa(na, nb) = {my(map = mappp(nb)); for (k=na, nb, if (denominator(k/ff(k, map)) == 1, print1(k, ", ")););}
wa(1, 10^8)

is_A344826(n)=!(n%A097621(n))
extend(n)=n*if(gcd(n, n=A000961(n/A097621(n)))==1,n) \\ Return the larger non-primitive term "derived" from a term n = a(k) with gcd(n,q) = 1, cf. COMMENTS, or zero if gcd(n,q) > 1, i.e., it cannot be "extended" that way. This allows the production of (infinitely?) many new terms from the existing ones. - M. F. Hasler, Jun 15 2021

A192187 Positions of prime powers p^e with p < e within A000961.

Original entry on oeis.org

7, 11, 19, 28, 33, 45, 69, 71, 118, 153, 199, 341, 360, 605, 894, 1079, 1881, 1962, 2294, 3591, 6060, 6636, 7772, 12371, 16229, 23151, 33292, 43580, 44126, 82268, 121021, 146014, 155922, 296348, 334771, 398221, 564689, 650688, 931905, 1078556, 2064590, 2454882

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

Complement of A192188.
A000961(a(n)) = A192135(n).
a(n) = A095874(A192135(n)).

a(22)-a(42) from Donovan Johnson, Jul 09 2011

A248906 Binary representation of prime power divisors of n: Sum_{p^k | n} 2^(A065515(p^k)-1).

Original entry on oeis.org

0, 1, 2, 5, 8, 3, 16, 37, 66, 9, 128, 7, 256, 17, 10, 549, 1024, 67, 2048, 13, 18, 129, 4096, 39, 8200, 257, 16450, 21, 32768, 11, 65536, 131621, 130, 1025, 24, 71, 262144, 2049, 258, 45, 524288, 19, 1048576, 133, 74, 4097, 2097152, 551, 4194320, 8201

Offset: 1

Franklin T. Adams-Watters, Mar 06 2015

nonn

			The prime power divisors of 12 are 2, 3, and 4. These are indices 1, 2, and 3 in the list of prime powers, so a(12) = 2^(1-1) + 2^(2-1) + 2^(3-1) = 7.

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

Cf. A246655, A065515, A034729, A000961.

Cf. A095874, A210208, A073093, A000079.

a248906 = sum . map ((2 ^) . subtract 2 . a095874) . tail . a210208_row
-- Reinhard Zumkeller, Mar 07 2015

al(n) = my(r=vector(n),pps=[p| p <- [1..n], isprimepower(p)],p2); for(k=1,#pps,p2=2^(k-1);forstep(j=pps[k],n,pps[k],r[j]+=p2));r

Additive with a(p^k) = Sum_{j=1..k} 2^(A065515(p^j)-1).
a(A051451(k)) = 2^k - 1.
a(n) = Sum_{k=1..A073093(n)} 2^(A095874(A210208(n,k))-2). - Reinhard Zumkeller, Mar 07 2015

A305974 a(1) = 1; for n > 1, if n = p^k for some prime p and exponent k >= 1, then a(n) = -k, otherwise a(n) = 1+A085970(n).

Original entry on oeis.org

1, -1, -1, -2, -1, 2, -1, -3, -2, 3, -1, 4, -1, 5, 6, -4, -1, 7, -1, 8, 9, 10, -1, 11, -2, 12, -3, 13, -1, 14, -1, -5, 15, 16, 17, 18, -1, 19, 20, 21, -1, 22, -1, 23, 24, 25, -1, 26, -2, 27, 28, 29, -1, 30, 31, 32, 33, 34, -1, 35, -1, 36, 37, -6, 38, 39, -1, 40, 41, 42, -1, 43, -1, 44, 45, 46, 47, 48, -1, 49, -4, 50, -1, 51, 52

Offset: 1

Antti Karttunen, Jul 02 2018

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000961, A065515, A085970, A095874, A305975 (rgs-transform).

up_to = 65537;
partialsums(f,up_to) = { my(v = vector(up_to), s=0); for(i=1,up_to,s += f(i); v[i] = s); (v); }
v065515 = partialsums(n -> (omega(n)<=1), up_to);
A065515(n) = v065515[n];
A085970(n) = (n - A065515(n));
A305974(n) = if(1==n,n,my(e = isprimepower(n)); if(e,-e,1+A085970(n)));

a(1) = 1; for n > 1, if n = p^k for some prime p and exponent k >= 1, then a(n) = -k, otherwise [when n is not a prime power], a(n) = 1+A085970(n) = running count from 2 onward.

cp's OEIS Frontend

A192135 Prime powers p^e with p < e.

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A095841 Prime powers having exactly one partition into two prime powers.

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A095842 Prime powers having no partition into two prime powers.

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Haskell

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A117331 Lexicographically earliest permutation of prime powers such that the exponents of succeeding terms increase at most by 1.

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A192188 Positions of prime powers p^e with e <= p within A000961.

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A322981 If n is the k-th prime power > 1, a(n) = k, otherwise a(n) = 0.

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PARI

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A344826 Integers k such that k/A097621(k) is an integer.

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PARI

PARI

PARI

A192187 Positions of prime powers p^e with p < e within A000961.

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Extensions

A248906 Binary representation of prime power divisors of n: Sum_{p^k | n} 2^(A065515(p^k)-1).

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