A095840 -id:A095840 - cp's OEIS frontend

A071330 Number of decompositions of n into sum of two prime powers.

0, 1, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 3, 4, 3, 4, 3, 5, 3, 5, 4, 4, 2, 5, 3, 5, 4, 5, 3, 6, 3, 7, 5, 7, 4, 7, 2, 6, 4, 6, 3, 6, 3, 6, 5, 6, 2, 8, 3, 8, 4, 6, 2, 9, 3, 7, 4, 6, 2, 8, 3, 7, 4, 7, 3, 9, 2, 8, 5, 7, 2, 10, 3, 8, 6, 7, 3, 9, 2, 9, 4, 7, 4, 11, 3, 9, 4, 7, 3, 12, 4, 8, 3, 7, 2

Offset: 1

Reinhard Zumkeller, May 19 2002

a(2*n) > 0 (Goldbach's conjecture).

a(A071331(n)) = 0; A095840(n) = a(A000961(n)).

			10 = 1 + 3^2 = 2 + 2^3 = 3 + 7 = 5 + 5, therefore a(10) = 4;
11 = 2 + 3^2 = 3 + 2^3 = 4 + 7, therefore a(11) = 3;
12 = 1 + 11 = 3 + 3^2 = 2^2 + 2^3 = 5 + 7, therefore a(12) = 4;
a(149)=0, as for all x<149: if x is a prime power then 149-x is not.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000961, A002375, A010055, A061358, A071331, A109829.

a071330 n = sum $
   map (a010055 . (n -)) $ takeWhile (<= n `div` 2) a000961_list
-- Reinhard Zumkeller, Jan 11 2013

primePowerQ[n_] := Length[ FactorInteger[n]] == 1; a[n_] := (r = 0; Do[ If[ primePowerQ[k] && primePowerQ[n-k], r++], {k, 1, Floor[n/2]}]; r); Table[a[n], {n, 1, 95}](* Jean-François Alcover, Nov 17 2011, after Michael B. Porter *)

ispp(n) = (omega(n)==1 || n==1)
A071330(n) = {local(r);r=0;for(i=1,floor(n/2),if(ispp(i) && ispp(n-i),r++));r} \\ Michael B. Porter, Dec 04 2009

a(n)=my(s); forprime(p=2,n\2,if(isprimepower(n-p), s++)); for(e=2,log(n)\log(2), forprime(p=2, sqrtnint(n\2,e), if(isprimepower(n-p^e), s++))); s+(!!isprimepower(n-1))+(n==2) \\ Charles R Greathouse IV, Nov 21 2014

A282062 Expansion of (x + Sum_{p prime, k>=1} x^(p^k))^2.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Feb 05 2017

Number of ways to write n as an ordered sum of two prime powers (1 included).

			a(8) = 5 because we have  [7, 1], [5, 3], [4, 4], [3, 5] and [1, 7].

Robert Israel, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Prime Power

Cf. A000961, A071330, A071331, A073610, A095840, A280242, A282064.

N:= 100: # to get a(0)..a(N)
P:= select(isprime, [$2..N]):
g:= x + add(add(x^(p^k),k=1..floor(log[p](N))),p=P):
S:= series(g^2,x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Feb 10 2017

nmax = 95; CoefficientList[Series[(x + Sum[Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}])^2, {x, 0, nmax}], x]

G.f.: (x + Sum_{p prime, k>=1} x^(p^k))^2.

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

A280242 Expansion of (Sum_{k>=2} floor(1/omega(k))*x^k)^2, where omega(k) is the number of distinct prime factors (A001221).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 29 2016

nonn

Number of ordered ways of writing n as the sum of two prime powers (1 excluded).

			a(6) = 3 because we have [4, 2], [3, 3] and [2, 4].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime Power

Cf. A001221, A071330, A071331, A073610, A095840, A246655.

nmax = 97; CoefficientList[Series[(Sum[Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}])^2, {x, 0, nmax}], x]

G.f.: (Sum_{k>=2} floor(1/omega(k))*x^k)^2.

cp's OEIS Frontend

A071330 Number of decompositions of n into sum of two prime powers.

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A282062 Expansion of (x + Sum_{p prime, k>=1} x^(p^k))^2.

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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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A280242 Expansion of (Sum_{k>=2} floor(1/omega(k))*x^k)^2, where omega(k) is the number of distinct prime factors (A001221).

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