A095842 -id:A095842 - cp's OEIS frontend

A095840 Number of ways to write the n-th prime power as sum of two prime powers.

0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 3, 3, 2, 3, 4, 3, 3, 7, 2, 3, 3, 2, 3, 2, 2, 3, 7, 2, 2, 3, 2, 4, 4, 3, 2, 2, 2, 2, 2, 4, 2, 3, 1, 11, 3, 3, 4, 0, 2, 3, 1, 3, 3, 3, 1, 4, 1, 1, 2, 4, 2, 1, 3, 3, 2, 1, 3, 5, 1, 12, 3, 2, 1, 3, 2, 2, 3, 2, 4, 2, 1, 2, 2, 0, 1, 1, 3, 2, 4, 2, 3, 2, 0, 2, 3, 1, 2, 2, 2, 1, 3

Offset: 1

Reinhard Zumkeller, Jun 10 2004

nonn

a(n) = A071330(A000961(n)).

See A095842 and A095841 for prime powers having no more than one partition into two prime powers.

			A000961(8) = 3^2 = 9 = 1+8 = 2+7 = 4+5, therefore a(8)=3.

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

a095840 = a071330 . a000961  -- Reinhard Zumkeller, Jan 11 2013

A115231 Primes p which cannot be written in the form 2^i + q^j where i >= 0, j >= 1, q = odd prime.

Original entry on oeis.org

2, 3, 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, 3697, 3739, 3779

Offset: 1

Reinhard Zumkeller, Jan 17 2006

nonn

Union with A115232 gives all primes (A000040).

All terms > 3 are in A095842. - M. F. Hasler, Nov 20 2014

			A000040(35) = 149 = 2^7+3*7 = 2^6+5*17 = 2^5+3*3*13 =
2^4+7*19 = 2^3+3*47 = 2^2+5*29 = 2^1+3*7*7 = 2^0+2*2*37, therefore 149 is a term (A115230(35)=0).

David A. Corneth, Table of n, a(n) for n = 1..11532 (terms <= 10^6)

Cf. A095842, A115230, A115232, A115233, A000079, A061345.

maxp = 3779; Complement[pp = Prime[Range[PrimePi[maxp]]], Union[Sort[Reap[Do[p = 2^i + q^j; If[p <= maxp && PrimeQ[p], Sow[p]], {i, 0, Log[2, maxp]//Ceiling}, {j, 1, Log[3, maxp]//Ceiling}, {q, Rest[pp]} ]][[2, 1]]]]] (* Jean-François Alcover, Aug 03 2018 *)

upto(n) = {my(pr = primes(primepi(n)), found = List(), s); for(i = 0, logint(n, 2), s = 2^i; forprime(q = 3, n - 2^i, for(j = 1, logint(n - 2^i, q),
listput(found, s + q^j)))); listsort(found, 1); setminus(Set(pr), Set(found))} \\ David A. Corneth, Aug 03 2018

Recomputed (based on recomputation of A115230) by R. J. Mathar and Reinhard Zumkeller, Apr 29 2010.
Edited by N. J. A. Sloane, Apr 30 2010
2, 3 inserted by David A. Corneth, Aug 03 2018

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

A248412 Smallest prime p such that p - 2^e is also prime power (A053810) in exactly n cases for nonnegative integers e.

Original entry on oeis.org

149, 2, 5, 11, 83, 829, 3331, 32941, 176417, 854929, 2233531, 12699571, 47924959, 763597201

Offset: 0

Robert G. Wilson v, Oct 06 2014

nonn

first case when A115230 equals n.
149, 331, 373, 509, 701, 757, 809, 877, 907, 997, 1019, ...;
2, 3, 127, 163, 179, 191, 193, 223, 239, 251, 269, 311, ...;
5, 7, 23, 37, 47, 53, 59, 67, 71, 79, 97, 101, 103, ...;
11, 13, 17, 19, 29, 31, 41, 43, 61, 73, 89, 131, 137, ...;
83, 113, 139, 181, 199, 293, 353, 571, 593, 601, 619, ...;
829, 1217, 1487, 2131, 2341, 2551, 2971, 4051, 4261, ...;
3331, 12109, 14551, 17393, 18233, 22279, 22307, 22741, ...;
32941, 34369, 44029, 49433, 53633, 67189, 95717, 99833, ...;
176417, 304771, 314723, 314779, 349667, 414707, 451937, ...;
854929, 1297651, 1328927, 1784723, 2164433, 2488909, ...;
2233531, 6026089, 7475389, 7623229, 9644911, 10019551, ...;
12699571, 18464123, 52849879, 78127339, 79303579, 84397463, ...;
47924959, 153309649, 204797059, 248685923, 273865219, ...;
763597201, ...;
...

Cf. A115230, A244917, zeroth row A095842, first row A095841.

f[p_] := Length@ Table[q = p - 2^exp; If[ PrimeNu@ q == 1, {q}, Sequence @@ {}], {exp, 0, Floor@ Log2@ p}]; t = Table[0, {20}]; p = 2; While[p < 1000000000, a = f[p] +1; If[a < 101 && t[[a]] == 0, t[[a]] == p; Print[{a -1, p}]]; p = NextPrime@ p]; t

a(n) <= A244917(n) for n>0.

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A095840 Number of ways to write the n-th prime power as sum of two prime powers.

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A115231 Primes p which cannot be written in the form 2^i + q^j where i >= 0, j >= 1, q = odd prime.

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

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A248412 Smallest prime p such that p - 2^e is also prime power (A053810) in exactly n cases for nonnegative integers e.

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