A095841 -id:A095841 - 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

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

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

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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A095842 Prime powers having no partition into two prime powers.

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A208247 Numbers 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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