A050997 -id:A050997 - cp's OEIS frontend

A325264 Numbers whose omega-sequence sums to 7.

30, 36, 42, 64, 66, 70, 78, 100, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 196, 222, 225, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).

			The sequence of terms together with their prime indices and omega-sequences begins:
   30: {1,2,3} (3,3,1)
   36: {1,1,2,2} (4,2,1)
   42: {1,2,4} (3,3,1)
   64: {1,1,1,1,1,1} (6,1)
   66: {1,2,5} (3,3,1)
   70: {1,3,4} (3,3,1)
   78: {1,2,6} (3,3,1)
  100: {1,1,3,3} (4,2,1)
  102: {1,2,7} (3,3,1)
  105: {2,3,4} (3,3,1)
  110: {1,3,5} (3,3,1)
  114: {1,2,8} (3,3,1)
  130: {1,3,6} (3,3,1)
  138: {1,2,9} (3,3,1)
  154: {1,4,5} (3,3,1)
  165: {2,3,5} (3,3,1)
  170: {1,3,7} (3,3,1)
  174: {1,2,10} (3,3,1)
  182: {1,4,6} (3,3,1)
  186: {1,2,11} (3,3,1)
  190: {1,3,8} (3,3,1)
  195: {2,3,6} (3,3,1)
  196: {1,1,4,4} (4,2,1)

Positions of 7's in A325249.
Numbers with omega-sequence summing to m: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7).
Cf. A056239, A060687, A062770, A118914, A130091, A303555, A323023, A325238, A325265, A325277.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],Total[omseq[#]]==7&]

A132214 Numbers that are sums of seventh powers of two distinct primes.

Original entry on oeis.org

2315, 78253, 80312, 823671, 825730, 901668, 19487299, 19489358, 19565296, 20310714, 62748645, 62750704, 62826642, 63572060, 82235688, 410338801, 410340860, 410416798, 411162216, 429825844, 473087190, 893871867, 893873926

Offset: 1

Jonathan Vos Post, Aug 13 2007

This is to 7th powers as A130555 is to 6th powers, A130292 is to fifth powers, A130873 is to 4th powers and A120398 is to cubes. These can never be prime, as the polynomial x^7 + y^7 factors over Z. Note however that A132215, which is the analog for eighth powers, can be prime, as seen also in A132216.

			a(1) = 2^7 + 3^7 = 2315 = 5 * 463.

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

Cf. A000040, A001015, A050997, A120398, A122616, A130873, A130555, A132215, A132216.

P:= select(isprime, [2,seq(i,i=3..100,2)]): nP:= nops(P):
N:= 2^7 + P[-1]^7:
sort(convert(select(`<=`, {seq(seq(P[i]^7+P[j]^7,j=i+1..nP),i=1..nP-1)},N),list)); # Robert Israel, Jul 01 2024

Select[Sort[ Flatten[Table[Prime[n]^7 + Prime[k]^7, {n, 15}, {k, n - 1}]]], # <= Prime[15^7] &]
Union[Total/@(Subsets[Prime[Range[10]],{2}]^7)] (* Harvey P. Dale, Jan 03 2012 *)

{A001015(A000040(i)) + A001015(A000040(j)) for i > j}.

A138405 a(n) = prime(n)^5 - prime(n)^2.

Original entry on oeis.org

28, 234, 3100, 16758, 160930, 371124, 1419568, 2475738, 6435814, 20510308, 28628190, 69342588, 115854520, 147006594, 229342798, 418192684, 714920818, 844592580, 1350120618, 1804224310, 2073066264, 3077050158, 3939033754, 5584051528, 8587330848, 10510090300

Offset: 1

Artur Jasinski, Mar 19 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index to sequences related to prime powers

Cf. A001248, A030078, A050997, A138431.

[NthPrime((n))^5 - NthPrime((n))^2: n in [1..30] ]; // Vincenzo Librandi, Jun 17 2011

a = {}; Do[p = Prime[n]; AppendTo[a, p^5 - p^2], {n, 1, 50}]; a

forprime(p=2,1e3,print1(p^5-p^2", ")) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = A001248(n)*(A030078(n) - 1). - Elmo R. Oliveira, Jan 14 2023
From Alois P. Heinz, Jan 17 2023: (Start)
a(n) = 2 * A138431(n).
a(n) = A050997(n) - A001248(n). (End)

A157291 Decimal expansion of Zeta(5)/Zeta(10).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 26 2009

The product_{p = primes = A000040} (1+1/p^5), the fifth-power analog to A082020.

			1.035897477277500224... = (1+1/2^5)*(1+1/3^5)*(1+1/5^5)*(1+1/7^5)*...

Cf. A013663, A013668, A050997, A082020.

Cf. A005117, A113850.

Maple
```
evalf(Zeta(5)/Zeta(10)) ;
```

RealDigits[Zeta[5]/Zeta[10],10,120][[1]] (* Harvey P. Dale, Apr 06 2013 *)

Equals A013663/A013668 = Product_{i>=1} (1+1/A050997(i)).
Equals Sum_{k>=1} 1/A005117(k)^5 = 1 + Sum_{k>=1} 1/A113850(k). - Amiram Eldar, May 22 2020
Equals 93555 * zeta(5) / Pi^10. - Vaclav Kotesovec, May 22 2020

A232105 Number of groups of order prime(n)^5.

Original entry on oeis.org

51, 67, 77, 83, 87, 97, 101, 107, 111, 125, 131, 145, 149, 155, 159, 173, 183, 193, 203, 207, 217, 227, 231, 245, 265, 269, 275, 279, 289, 293, 323, 327, 341, 347, 365, 371, 385, 395, 399, 413, 423, 433, 447, 457, 461, 467, 491, 515, 519, 529, 533, 543, 553

Offset: 1

Eric M. Schmidt, Nov 21 2013

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
G. Bagnera, La composizione dei Gruppi finiti il cui grado e la quinta potenza di un numero primo, Ann. Mat. Pura Appl. (3), 1 (1898), 137-228.
M. F. Newman, E. A. O'Brien and M. R. Vaughan-Lee, Groups and nilpotent Lie rings whose order is the sixth power of a prime, J. Algebra, 278 (2004), 383-401.
Index entries for sequences related to groups

Cf. A000001, A000679, A090091, A090130, A090140, A232106, A232107.

Cf. A050997.

A232105 := Concatenation([51, 67], List(Filtered([5..10^5], IsPrime), p -> 61 + 2 * p + 2 * Gcd(p-1, 3) + Gcd(p-1, 4))); # Muniru A Asiru, Nov 16 2017

def A232105(n) : p = nth_prime(n); return 51 if p==2 else 67 if p==3 else 61 + 2*p + 2*gcd(p - 1, 3) + gcd(p - 1, 4)

For a prime p > 3, the number of groups of order p^5 is 61 + 2p + 2 gcd(p - 1, 3) + gcd(p - 1, 4).

A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536, 544

Offset: 1

Amiram Eldar, Feb 19 2025

Subsequence of A301517 and A374459 and first differs from them at n = 21. A301517(21) = A374459(21) = 216 is not a term of this sequence.
Numbers having exactly one non-unitary prime factor and its multiplicity is odd.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m+1} with m >= 1, i.e., any number (including zero) of 1's and then a single odd number > 1.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/((p-1)*(p+1)^2) = 0.093382464285953613312...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A381311 within A190641.
Cf. A057521, A118914.
Subsequence of A268335, A301517 and A374459.
Subsequences: A030078, A050997, A056824, A065036, A079395, A092759, A138031, A178740, A179664, A179665, A179667, A179670, A179692, A179696, A179704, A189975, A189984, A190378, A190383, A190473.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 1 && OddQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] % 2 && e[1] > 1 && (#e == 1 || e[2] == 1));

A130292 Numbers that are sums of fifth powers of two distinct primes.

Original entry on oeis.org

275, 3157, 3368, 16839, 17050, 19932, 161083, 161294, 164176, 177858, 371325, 371536, 374418, 388100, 532344, 1419889, 1420100, 1422982, 1436664, 1580908, 1791150, 2476131, 2476342, 2479224, 2492906, 2637150, 2847392, 3895956

Offset: 1

Jonathan Vos Post, Aug 06 2007

This is to 5th powers as A120398 is to cubes and A130873 is to 4th powers.

			a(1) = prime(1)^5 + prime(2)^5 = 2^5 + 3^5 = 32 + 243 = 275.

Cf. A000040, A050997, A120398, A122616, A130873.

Select[Sort[ Flatten[Table[Prime[n]^5 + Prime[k]^5, {n, 15}, {k, n - 1}]]], # <= Prime[15^5] &]

A132215 Numbers that are sums of eighth powers of two distinct primes.

Original entry on oeis.org

6817, 390881, 397186, 5765057, 5771362, 6155426, 214359137, 214365442, 214749506, 220123682, 815730977, 815737282, 816121346, 821495522, 1030089602, 6975757697, 6975764002, 6976148066, 6981522242, 7190116322, 7791488162

Offset: 1

Jonathan Vos Post, Aug 13 2007

This is to 8th powers as A132214 is to 7th powers, A130555 is to 6th powers, A130292 is to fifth powers, A130873 is to 4th powers and A120398 is to cubes. These CAN be prime, as the polynomial x^8 + y^8 is irreducible over Z, as seen in A132216. The first such example is a(11) = A132216(1) = 2^8 + 13^8 = 256 + 815730721 = 815730977, which is prime.

A subset of A003380. - R. J. Mathar, May 11 2008

			a(1) = 2^8 + 3^8 = 256 + 6561 = 6817 = 17 * 401.

Cf. A000040, A001016, A050997, A120398, A122616, A130873, A130555, A132214, A132216.

Select[Sort[ Flatten[Table[Prime[n]^8 + Prime[k]^8, {n, 15}, {k, n - 1}]]], # <= Prime[15^8] &]
Total/@Subsets[Prime[Range[10]]^8,{2}]//Sort (* Harvey P. Dale, Jun 27 2017 *)

{A001016(A000040(i)) + A001016(A000040(j)) for i > j}.

A132216 Primes that are sums of eighth powers of two distinct primes.

Original entry on oeis.org

815730977, 124097929967680577, 6115597639891380737, 144086718355753024097, 524320466699664691937, 3377940044732998170977, 10094089678769799935777, 30706777728209453204417, 58310148000746221725857

Offset: 1

Jonathan Vos Post, Aug 13 2007

These primes exist because the polynomial x^8 + y^8 is irreducible over Z. Note that 2^8 + n^8 can be prime for composite n beginning 21, 55, 69, 77, 87, 117.

			a(1) = 2^8 + 13^8 = 256 + 815730721 = 815730977, which is prime.
a(2) = 2^8 + 137^8 = 124097929967680577, which is prime.
a(3) = 2^8 + 223^8 = 6115597639891380737, which is prime.
a(4) = 2^8 + 331^8 = 144086718355753024097, which is prime.
a(5) = 2^8 + 389^8 = 524320466699664691937, which is prime.
a(6) = 2^8 + 491^8 = 3377940044732998170977, which is prime.
a(7) = 2^8 + 563^8 = 10094089678769799935777, which is prime.

Cf. A000040, A001016, A050997, A120398, A122616, A130873, A130555, A132214, A132215.

Primes in A132215. {A001016(A000040(i)) + A001016(A000040(j)) for i > j and are elements of A000040}.

More terms from Jon E. Schoenfield, Jul 16 2010

A137489 Numbers with 26 divisors.

Original entry on oeis.org

12288, 20480, 28672, 45056, 53248, 69632, 77824, 94208, 118784, 126976, 151552, 167936, 176128, 192512, 217088, 241664, 249856, 274432, 290816, 299008, 323584, 339968, 364544, 397312, 413696, 421888, 438272, 446464, 462848, 520192, 536576

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^25 (5th powers of A050997, subset of A010813) or p*q^12, where p and q are distinct primes. - R. J. Mathar, Mar 01 2010

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283.

Select[Range[900000],DivisorSigma[0,#]==26&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

is(n)=numdiv(n)==26 \\ Charles R Greathouse IV, Jun 19 2016

A000005(a(n))=26.

cp's OEIS Frontend

A325264 Numbers whose omega-sequence sums to 7.

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A132214 Numbers that are sums of seventh powers of two distinct primes.

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A138405 a(n) = prime(n)^5 - prime(n)^2.

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A157291 Decimal expansion of Zeta(5)/Zeta(10).

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A232105 Number of groups of order prime(n)^5.

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A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).

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A130292 Numbers that are sums of fifth powers of two distinct primes.

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A132215 Numbers that are sums of eighth powers of two distinct primes.

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