cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A108605 Semiprimes with prime sum of factors: twice the lesser of the twin prime pairs.

Original entry on oeis.org

6, 10, 22, 34, 58, 82, 118, 142, 202, 214, 274, 298, 358, 382, 394, 454, 478, 538, 562, 622, 694, 838, 862, 922, 1042, 1138, 1198, 1234, 1282, 1318, 1618, 1642, 1654, 1714, 1762, 2038, 2062, 2098, 2122, 2182, 2302, 2458, 2554, 2578, 2602, 2638, 2854, 2902
Offset: 1

Views

Author

Zak Seidov, Jun 12 2005

Keywords

Comments

All terms are even. (Cf. formula.)
The definition implies that the sum of factors is the sum over the prime factors with multiplicity, as in A001414. - R. J. Mathar, Nov 28 2008
The sum of factors of a semiprime pq is p+q, which can only be prime if {p, q} = {2, odd prime}. Requiring the sum to be prime then implies that the semiprime is twice the lesser of a twin prime pair. - M. F. Hasler, Apr 07 2015
Subsequence of A288814, each term being of the form A288814(p), where p is the greatest of a pair of twin primes. - David James Sycamore, Aug 29 2017

Examples

			58=2*29 and 2+29 is prime.

Links

Crossrefs

Cf. A001358 semiprimes, A001359 lesser of twin primes, A101605 3-almost primes, A108606 semiprimes with prime sum of digits, A108607 intersection of A108605 and A108606.

Programs

  • Mathematica
    Select[Range[2, 3000, 2], !IntegerQ[Sqrt[ # ]]&&Plus@@(Transpose[FactorInteger[ # ]])[[2]]==2&&PrimeQ[Plus@@(Transpose[FactorInteger[ # ]])[[1]]]&]
Select[Range[2,3000,2],PrimeOmega[#]==PrimeNu[#]==2&&PrimeQ[Total[ FactorInteger[ #][[;;,1]]]]&] (* Harvey P. Dale, Apr 10 2023 *)
  • PARI
    list(lim)=my(v=List(),p=2); forprime(q=3,lim\2+1, if(q-p==2, listput(v,2*p)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 05 2017

Formula

a(n)=2*p, with p and 2+p twin primes: a(n)=2*A001359(n).

Extensions

Changed division by 2 to multiplication by 2 in formula related to A001359. - R. J. Mathar, Nov 28 2008

A108606 Semiprimes with prime sum of digits.

Original entry on oeis.org

14, 21, 25, 34, 38, 49, 58, 65, 74, 85, 94, 106, 111, 115, 119, 122, 133, 142, 146, 155, 166, 201, 203, 205, 209, 214, 218, 221, 247, 254, 265, 274, 278, 287, 289, 298, 302, 319, 326, 335, 346, 355, 362, 371, 377, 382, 386, 391, 395, 403, 407, 427, 445, 454
Offset: 1

Views

Author

Zak Seidov, Jun 12 2005

Keywords

Comments

34 is the smallest term in common with A108605.

Examples

			34 = 2*17 (semiprime) and 2 + 17 = 19 is prime.

Links

Crossrefs

Cf. A001358 (semiprimes), A101605 (3-almost primes), A108605 (semiprimes with prime sum of factors), A108607 (intersection of A108605 and A108606).

Programs

  • Mathematica
    A108606=Select[Range[1000], Plus@@(Transpose[FactorInteger[ # ]])[[2]]==2&& PrimeQ[Plus@@IntegerDigits[ # ]]&]
DeleteCases[ParallelTable[If[PrimeOmega[n]==2&&PrimeQ[Total[IntegerDigits[n]]],n,a],{n,0,126181}],a] (* J.W.L. (Jan) Eerland, Dec 21 2021 *)
  • PARI
    select(isA108606(n)={bigomega(n)==2&&isprime(sumdigits(n))},[1..1000]) \\ J.W.L. (Jan) Eerland, Dec 23 2021
  • Python
    from sympy import isprime, factorint
def ok(n): return isprime(sum(map(int, str(n)))) and sum(factorint(n).values()) == 2
print([k for k in range(455) if ok(k)]) # Michael S. Branicky, Aug 22 2022

A108608 5-almost primes whose sum of factors is a prime.

Original entry on oeis.org

48, 80, 108, 176, 252, 300, 368, 405, 420, 464, 468, 500, 567, 660, 675, 684, 848, 891, 944, 980, 1020, 1116, 1136, 1140, 1323, 1332, 1377, 1424, 1428, 1452, 1539, 1548, 1575, 1616, 1700, 1716, 1740, 1820, 1860, 1875, 1932, 2096, 2156, 2196, 2295, 2300
Offset: 1

Views

Author

Zak Seidov, Jun 12 2005

Keywords

Examples

			48=2*2*2*2*3 (5-almost prime) and 2+2+2+2+3=11 is a prime.

Links

Crossrefs

Cf. A107707, A108607, A108609 (resp.) 3, 4, 6 (resp.)-almost primes whose sum of factors is a prime.

Programs

  • Mathematica
    Select[Range[2500],PrimeOmega[#]==5&&PrimeQ[Total[Times@@@FactorInteger[#]]]&] (* Harvey P. Dale, Apr 07 2022 *)
  • PARI
    is(n)=my(f=factor(n));sum(i=1,#f~,f[i,2])==5 && isprime(sum(i=1,#f~,f[i,1]*f[i,2])) \\ Charles R Greathouse IV, Oct 11 2013

A108609 6-almost primes whose sum of factors is a prime.

Original entry on oeis.org

96, 224, 360, 416, 486, 504, 600, 608, 792, 810, 992, 1176, 1184, 1224, 1368, 1376, 1400, 1890, 1952, 2040, 2088, 2184, 2232, 2250, 2336, 2528, 2600, 2754, 2760, 2904, 2952, 3080, 3104, 3296, 3384, 3480, 3510, 3640, 3726, 4064, 4158, 4248, 4312, 4392
Offset: 1

Views

Author

Zak Seidov, Jun 12 2005

Keywords

Examples

			96=2*2*2*2*2*3 (6-almost prime) and 2+2+2+2+2+3=13 is a prime.

Links

Crossrefs

Cf. A107707, A108607, A108608 (resp.) 3, 4, 5 (resp.)-almost primes whose sum of factors is a prime.

Programs

  • Mathematica
    Select[Range[8000], Last[Plus@@FactorInteger[ # ]]==6&&PrimeQ[Plus@@Times@@ Transpose[FactorInteger[ # ]]]&]
sfp6Q[n_]:=Module[{pf=Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]}, Length[ pf]==6&&PrimeQ[Total[pf]]]; Select[Range[4400],sfp6Q] (* Harvey P. Dale, Jul 01 2018 *)
  • PARI
    is(n)=my(f=factor(n));sum(i=1,#f~,f[i,2])==6 && isprime(sum(i=1,#f~,f[i,1]*f[i,2])) \\ Charles R Greathouse IV, Oct 11 2013
Showing 1-4 of 4 results.

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