A075308 -id:A075308 - cp's OEIS frontend

A076704 Numbers k of the form p^q where both p and q are prime and all digits of k are odd.

9, 1331, 357911, 5177717, 5735339, 9393931, 17171515157399, 335571975137771, 7979737131773191, 13337513771953951, 13137917533317175739371379, 33159599371999557199755557, 1593395573971551557179777111133, 131755773357537951113179771515713, 315113377779977515359339551539771

Offset: 1

Zak Seidov, Oct 26 2002

Up to 10^17, there are only 10 odd-digit prime powers of prime numbers. a(1) = 3^2, a(2) = 11^3, a(3) = 71^3, a(4) = 173^3, a(5) = 179^3, a(6) = 211^3, a(7) = 25799^3, a(8) = 69491^3, a(9) = 199831^3, and a(10) = 237151^3.
The only candidates for even-digit prime powers of prime numbers are of the form 2^n, and below 2^10000 there are only 2, 4, 8, 64, and 2048, two of which are not raised to prime powers.
a(11) <= 13137917533317175739371379 and a(12) <= 33159599371999557199755557. - Jinyuan Wang, Mar 02 2020

Giovanni Resta, Table of n, a(n) for n = 1..36 (terms < 10^57)

Cf. A014261, A053810, A075308.

pp = Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[10^17]]}, {i, 1, PrimePi[ Floor[ Log[ Prime[n], 10^17]]]}]]]; Do[ If[ Union[ OddQ[ IntegerDigits[ pp[[n]]]]] == {True}, Print[ pp[[n]]]], {n, 1, Length[pp]}]

lista(nn) = {my(k, v=List([])); forprime(p=2, nn, forprime(q=2, logint(nn, p), if(Set(digits(k=p^q)%2)==[1], listput(v, k)))); Set(v); } \\ Jinyuan Wang, Mar 02 2020

Edited and extended by Robert G. Wilson v, Oct 31 2002
Corrected and edited by Elliott Line, Jul 11 2013
Better definition from Jon E. Schoenfield, Nov 19 2018
Terms a(11) and beyond from Giovanni Resta, Mar 03 2020

A076707 Ordered differences without repetitions between two successive prime powers of prime numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 12, 16, 17, 18, 30, 38, 41, 46, 54, 72, 74, 94, 120, 128, 138, 139, 168, 186, 199, 240, 248, 250, 260, 271, 286, 288, 312, 316, 354, 356, 370, 408, 424, 432, 496, 546, 552, 582, 600, 602, 618, 678, 720, 768, 792, 836, 840, 876, 890, 894, 912

Offset: 1

Zak Seidov, Oct 26 2002

nonn

Several entries are represented by at least two differences: 4 (which equals 8-4 & 125-121), 168, 312, 600, 768, 792, 912, 1848, 2472, etc.

			250 = 161051 - 160801 = 11^5 - 401^2.

Cf. A053810, A075308, A077257, A077258.

pp = Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[10^16]]}, {i, 1, PrimePi[ Floor[ Log[ Prime[n], 10^16]]]}]]]; l = Length[pp]; b = Take[pp, -l + 1] - Take[pp, l - 1]; Take[ Union[a], 57]

Edited and corrected by Robert G. Wilson v, Oct 31 2002

A077257 Differences between two successive prime powers of prime numbers (A076707) in more than one way.

Original entry on oeis.org

4, 168, 312, 600, 768, 792, 912, 1848, 2472, 3048, 3192, 3288, 3528, 3720, 4008, 4920, 5160, 5208, 5928, 6072, 6792, 6840, 6888, 7080, 7512, 7728, 7800, 8520, 8760, 10632, 11400, 11880, 11928, 12792, 13200, 13440, 13560, 14280, 14640, 15960

Offset: 1

Zak Seidov and Robert G. Wilson v, Oct 31 2002

nonn

			4 = 8-4 = 125-121, 168 = 529-361 = 1849-1681, 312 = 841-529 = 1681-1369.
It is interesting that 529 is a member of the last two examples.
6888 is the first one to be represented in just three ways.
4920 is the first one to be represented in four ways.

Cf. A053810, A075308, A076707, A077258.

pp = Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[10^16]]}, {i, 1, PrimePi[ Floor[ Log[ Prime[n], 10^16]]]}]]]; l = Length[pp]; b = Sort[ Take[pp, -l + 1] - Take[pp, l - 1]]; Union[ b[[ Select[ Range[355], b[[ # ]] == b[[ # + 1]] &]]]]

A077258 Least number which can be represented by the difference between two successive prime powers of prime numbers (A076707) in exactly n ways.

Original entry on oeis.org

1, 4, 6888, 4920, 187117320

Offset: 1

Zak Seidov and Robert G. Wilson v, Oct 31 2002

			1 = 9-8, 4 = 8-4 & 125-121, 6888 = 332929 - 326041 = 744769 - 737881 = 2968729 - 2961841, 4920 = 44521 - 39601 = 63001 - 58081 = 380689 - 375769 = 1515361 - 1510441 and
187117320 = 9725896737769 - 9725709620449 = 21883150711249 - 21882963593929 = 60786363426721 - 60786176309401 = 243145173030769 - 243144985913449 = 2188305808807561 - 2188305621690241.

Cf. A053810, A075308, A076707, A077257.

pp = Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[10^16]]}, {i, 1, PrimePi[ Floor[ Log[ Prime[n], 10^16]]]}]]]; l = Length[pp]; b = Sort[ Take[pp, -l + 1] - Take[pp, l - 1]];

A076047 Primes which are the difference between two successive nontrivial prime powers (A025475).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 41, 139, 151, 199, 271, 307, 751, 1217, 3343, 3617, 4001, 4241, 40841, 97169, 117017, 203897, 746153, 123090449, 137542193, 256534591, 21249911167, 88109383889, 112332648583, 85726065193313, 226411321073393

Offset: 1

Zak Seidov and Robert G. Wilson v, Oct 29 2002

nonn

I have searched through prime powers up to 2^8532. It is very unlikely that there are any other values between the ones listed here, but no prime has been proved to be absent from this sequence. - David Wasserman, Mar 31 2005

			3 = 128 - 125 = 2^7 - 5^3; 7 = 16 - 9 = 32768 - 32761; 17 = 49 - 32 = 81 - 64 = 529 - 512; 4241 = 528529 - 524288 = 727^2 - 2^19.

Cf. A025475, A053810, A075308.

pp = Sort[ Flatten[ Table[ Prime[n]^i, {n, 1, PrimePi[ Sqrt[10^16]]}, {i, 1, Log[ Prime[n], 10^16]}]]]; l = Length[pp]; d = Sort[Take[pp, -l + 1] - Take[pp, l - 1]]; a = {}; Do[ If[ PrimeQ[ d[[n]]], a = Append[a, d[[n]]]], {n, 1, l - 1}]; Union[a] a = {}; Do[ If[ PrimeQ[ pp[[n + 1]] - pp[[n]]], a = Append[a, pp[[n + 1]] - pp[[n]]]], {n, 1, Length[pp] - 1}]; Union[a]

More terms from David Wasserman, Mar 31 2005

A076700 Number of prime powers of prime numbers less than 10^n.

Original entry on oeis.org

3, 7, 18, 40, 88, 208, 513, 1345, 3605, 9965, 27984, 79815, 230184, 669541, 1961782, 5781027, 17121898, 50926700, 152037454, 455379086, 1367866472, 4119420084, 12434683582, 37613680284, 113995431428, 346090115288, 1052421037283, 3205047207003, 9774084239631

Offset: 1

Zak Seidov, Oct 26 2002

nonn

			a(2) = 7 because 4, 8, 9, 25, 27, 32 and 49 are the only prime powers of prime numbers less than 100.

Chai Wah Wu, Table of n, a(n) for n = 1..34

Cf. A053810, A075308.

pp = Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[10^17]]}, {i, 1, PrimePi[ Floor[ Log[ Prime[n], 10^17]]]}]]]; a = {}; k = 1; Do[ While[ pp[[k]] <= 10^n, k++ ]; a = Append[a, k-1], {n, 1, 17}]

from sympy import primepi, integer_nthroot, primerange
def A076700(n): return int(sum(primepi(integer_nthroot(10**n, p)[0]) for p in primerange((10**n).bit_length()))) # Chai Wah Wu, Sep 12 2024

Edited and extended by Robert G. Wilson v, Oct 31 2002
a(18), a(19) from Ray Chandler, Oct 30 2008
a(20)-a(29) from Donovan Johnson, Mar 12 2013

A076702 Distinct-digit prime powers of prime numbers.

Original entry on oeis.org

4, 8, 9, 25, 27, 32, 49, 125, 128, 169, 243, 289, 361, 529, 841, 961, 1369, 1849, 2048, 2187, 2197, 2809, 3125, 3481, 3721, 4913, 5041, 5329, 6241, 6859, 7921, 8192, 12769, 16807, 18769, 24389, 32041, 32761, 36481, 37249, 39601, 54289, 68921

Offset: 1

Zak Seidov, Oct 26 2002

Up to 10^10, there are only 108 distinct-digit prime powers of prime numbers; the last one being 2^29 = 536870912.

Ray Chandler, Table of n, a(n) for n = 1..108 (full sequence)

Cf. A053810, A075308.

pp = Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[10^10]]}, {i, 1, PrimePi[ Floor[ Log[ Prime[n], 10^10]]]}]]]; a = {}; Do[ d = IntegerDigits[ pp[[n]]]; If[ Length[d] == Length[ Union[d]], a = Append[a, pp[[n]]]], {n, 1, 9965}]; a

Edited by Robert G. Wilson v, Oct 31 2002

A076705 Prime powers of prime numbers such that the sum of its digits is also prime power of prime number.

Original entry on oeis.org

4, 8, 9, 27, 121, 125, 243, 1331, 4489, 10201, 12769, 24649, 37249, 66049, 80089, 96721, 113569, 139129, 167281, 175561, 177147, 214369, 259081, 358801, 371293, 413449, 426409, 436921, 552049, 579121, 591361, 635209, 823543, 1026169

Offset: 1

Zak Seidov, Oct 26 2002

Up to 10^7, there are 513 prime powers of prime numbers. Of these, 79 are such that the sum of their digits is also prime power of prime number. Up to 10^14 there are 43915.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A053810, A075308.

N:= 2000000: # for terms <= N
R:= NULL:
p:= 1:
do
  p:= nextprime(p);
  if p^2 > N then break fi;
  q:= 1;
  do
    q:= nextprime(q);
    x:= p^q;
    if x > N then break fi;
    R:= R, x;
  od;
od:
S:= {R}:
sort(convert(select(s -> member(convert(convert(s,base,10),`+`),S),S), list));  # Robert Israel, Apr 06 2020

pp = Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[10^14]]}, {i, 1, PrimePi[ Floor[ Log[ Prime[n], 10^14]]]}]]]; a = {}; Do[ If[ Position[pp, Plus @@ IntegerDigits[ pp[[n]] ]] != {}, a = Append[a, pp[[n]] ]], {n, 1, 669541}]

Edited and corrected by Robert G. Wilson v, Oct 31 2002

A076703 Palindromic prime powers of prime numbers.

Original entry on oeis.org

4, 8, 9, 121, 343, 1331, 10201, 94249, 1030301, 900075181570009, 10022212521222001, 12124434743442121, 12323244744232321, 12341234943214321

Offset: 1

Zak Seidov, Oct 26 2002

Up to 10^14, there are only 9 palindromic prime powers of prime numbers.

Cf. A053810, A075308, subset of A072037.

a = {}; Do[pp = Prime[n]^Prime[i]; d = IntegerDigits[pp]; If[d == Reverse[d], a = Append[a, pp]], {n, 1, PrimePi[ Sqrt[10^21]]}, {i, 1, PrimePi[ Floor[ Log[ Prime[n], 10^21]]]}]; Sort[a]

Edited and corrected by Robert G. Wilson v, Oct 31 2002

A076706 Primes which are the difference between two successive prime powers of prime numbers.

Original entry on oeis.org

2, 3, 5, 17, 41, 139, 199, 271, 1217, 3617, 4001, 4241, 117017, 746153, 137542193

Offset: 1

Zak Seidov, Oct 26 2002

nonn

			3 = 128 - 125 = 2^7 - 5^3; 4241 = 528529 - 524288 = 727^2 - 2^19.

Cf. A053810, A075308.

pp = Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[10^16]]}, {i, 1, PrimePi[ Floor[ Log[ Prime[n], 10^16]]]}]]]; a = {}; Do[ If[ PrimeQ[ pp[[n + 1]] - pp[[n]]], a = Append[a, pp[[n + 1]] - pp[[n]]]], {n, 1, Length[pp] - 1}]; Union[a]

Edited and extended by Robert G. Wilson v, Oct 31 2002

cp's OEIS Frontend

A076704 Numbers k of the form p^q where both p and q are prime and all digits of k are odd.

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A076707 Ordered differences without repetitions between two successive prime powers of prime numbers.

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A077257 Differences between two successive prime powers of prime numbers (A076707) in more than one way.

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A077258 Least number which can be represented by the difference between two successive prime powers of prime numbers (A076707) in exactly n ways.

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A076047 Primes which are the difference between two successive nontrivial prime powers (A025475).

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A076700 Number of prime powers of prime numbers less than 10^n.

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A076702 Distinct-digit prime powers of prime numbers.

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A076705 Prime powers of prime numbers such that the sum of its digits is also prime power of prime number.

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