A071330 -id:A071330 - 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

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

A347643 Number of partitions of n into at most 2 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 3, 5, 4, 5, 4, 5, 4, 4, 3, 5, 4, 5, 5, 5, 4, 6, 4, 8, 5, 7, 4, 7, 3, 6, 4, 6, 4, 6, 4, 6, 5, 6, 3, 8, 4, 8, 4, 6, 3, 9, 3, 7, 4, 6, 3, 8, 4, 7, 4, 8, 3, 9, 3, 8, 5, 7, 3, 10, 4, 8, 6

Offset: 0

Ilya Gutkovskiy, Sep 09 2021

nonn

Cf. A000961, A023893, A071330, A347644, A347645, A347646, A347647.

A137791 Number of ways to write n as sum of two positive numbers having no prime gaps in their factorization.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Feb 11 2008

nonn

			a(20) = #{1+19,2+18,3+17,4+16,5+15,7+13,8+12,9+11} = 8;
a(21) = #{2+19,3+18,4+17,5+16,6+15,8+13,9+12} = 7;
a(22) = #{3+19,4+18,5+17,6+16,7+15,9+13,11+11} = 7;
a(23) = #{4+19,5+18,6+17,7+16,8+15,11+12} = 6;
a(24) = #{1+23,5+19,6+18,7+17,8+16,9+15,11+13,12+12} = 8.

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

Cf. A073491, A002375, A061358, A071330, A137792, A137793.

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.

A307726 Number of partitions of n into 2 prime powers (not including 1).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 24 2019

			a(10) = 3 because we have [8, 2], [7, 3] and [5, 5].

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions

Cf. A000961, A023894, A061358, A071068, A071330, A071331, A246655, A280242, A307727.

# note that this requires A246655 to be pre-computed
f:= proc(n, k, pmax) option remember;
  local t, p, j;
  if n = 0 then return `if`(k=0, 1, 0) fi;
  if k = 0 then return 0 fi;
  if n > k*pmax then return 0 fi;
  t:= 0:
  for p in A246655 do
    if p > pmax then return t fi;
    t:= t + add(procname(n-j*p, k-j, min(p-1, n-j*p)), j=1..min(k, floor(n/p)))
  od;
  t
end proc:
map(f, [$0..100]); # Robert Israel, Apr 29 2019

Array[Count[IntegerPartitions[#, {2}], _?(AllTrue[#, PrimePowerQ] &)] &, 101, 0]

a(n) = [x^n y^2] Product_{k>=1} 1/(1 - y*x^A246655(k)).

A109829 Number of distinct prime factors of 149 - (n-th prime power less than 149).

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Jul 04 2005

a(n) > 1: A071331(2) = 149, A071330(149) = 0.

Join[{2},PrimeNu/@(149-Select[Range[0,148],PrimePowerQ])] (* Harvey P. Dale, Jan 27 2019 *)

a(n) = 149 - A000961(n) for n <= 48.

A208247 Numbers having exactly one partition into two prime powers.

Original entry on oeis.org

2, 3, 119, 127, 163, 179, 191, 193, 217, 219, 221, 223, 239, 251, 269, 311, 337, 343, 389, 403, 415, 419, 427, 431, 457, 491, 505, 547, 557, 569, 575, 581, 583, 597, 599, 613, 653, 659, 667, 671, 673, 683, 697, 719, 739, 749, 767, 779, 787, 799, 807, 817

Offset: 1

Reinhard Zumkeller, Jan 11 2013

nonn

A071330(a(n)) = 1.

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

A095841 = Intersection of A208247 and A000961.

a095841 n = a095841_list !! (n-1)
a095841_list = filter ((== 1) . a071330) a000961_list

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

is(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 || n==2 \\ faster; Charles R Greathouse IV, Nov 21 2014

A306433 Number of partitions of n into 2 distinct prime powers (not including 1).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 30 2019

nonn

			a(12) = 3 because we have [9, 3], [8, 4] and [7, 5].

Index entries for sequences related to partitions

Cf. A000961, A071330, A106244, A117929, A246655, A307726, A307825.

Table[Count[IntegerPartitions[n, {2}], _?(And[UnsameQ @@ #, AllTrue[#, PrimePowerQ[#] &]] &)], {n, 0, 95}]

a(n) = [x^n y^2] Product_{k>=1} (1 + y*x^A246655(k)).

cp's OEIS Frontend

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

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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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A347643 Number of partitions of n into at most 2 prime powers (including 1).

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A137791 Number of ways to write n as sum of two positive numbers having no prime gaps in their factorization.

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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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A307726 Number of partitions of n into 2 prime powers (not including 1).

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A109829 Number of distinct prime factors of 149 - (n-th prime power less than 149).

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A208247 Numbers having exactly one partition into two prime powers.

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