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 1-6 of 6 results.

A158339 Semiprimes that are the sum of four successive semiprimes.

Original entry on oeis.org

39, 94, 106, 118, 146, 158, 185, 201, 221, 254, 302, 365, 427, 473, 485, 519, 537, 589, 633, 655, 707, 723, 749, 767, 842, 851, 869, 901, 1003, 1145, 1205, 1211, 1219, 1247, 1263, 1337, 1349, 1603, 1646, 1681, 1703, 1731, 1797, 1891, 1903, 1937, 2005, 2019
Offset: 1

Views

Author

Zak Seidov, Mar 16 2009

Keywords

Examples

			a(1)=39=6+9+10+14, or A001358(15)=A001358(2)+A001358(3)+A001358(4)+A001358(5).

Links

Crossrefs

Semiprimes that are the sum of k successive semiprimes: A131610 (k=3), A092192 (k=2), A001358 (k=1).

Programs

  • Mathematica
    Select[Total/@Partition[Select[Range[600],PrimeOmega[#]==2&],4,1], PrimeOmega[ #]==2&] (* Harvey P. Dale, Aug 14 2014 *)
  • PARI
    issemi(n)=bigomega(n)==2
list(lim)=if(lim<39, return([])); my(v=List(), u=v, x=lim\4+log(lim)*4\1+9); forprime(p=2,x\2, forprime(q=2,min(x\p,p), listput(u,p*q))); u=Set(u); while(u[#u-2]+u[#u-1]+u[#u]+x+1<=lim, while(!issemi(x++),); u=concat(u,x)); for(i=1,#u-3, u[i]+=u[i+1]+u[i+2]+u[i+3]); u[#u-2]=u[#u-1]=u[#u]=1; forprime(p=2,lim\2, forprime(q=2,min(lim\p,p), listput(v,p*q))); setintersect(Set(v), u) \\ Charles R Greathouse IV, Mar 05 2017

A254712 Semiprimes that are the sum of five successive semiprimes.

Original entry on oeis.org

69, 82, 166, 417, 451, 545, 614, 679, 717, 731, 763, 779, 799, 813, 851, 1047, 1057, 1077, 1101, 1119, 1138, 1262, 1293, 1346, 1371, 1603, 1639, 1651, 1678, 1739, 1761, 1817, 1897, 1959, 1983, 2049, 2078, 2177, 2263, 2446, 2498, 2533, 2661, 2705, 2734, 2746, 2762
Offset: 1

Views

Author

Zak Seidov, Feb 06 2015

Keywords

Examples

			69=A001358(24)=A001358(3)+...+A001358(7)= 9 + 10 + 14 + 15 + 21,
82=A001358(27)=A001358(4)+...+A001358(8)= 10 + 14 + 15 + 21 + 22.

Links

Crossrefs

Cf. A001358, A092192, A131610, A158339.

Programs

  • Mathematica
    Module[{sps=Select[Range[750],PrimeOmega[#]==2&]},Select[Total/@ Partition[ sps,5,1],PrimeOmega[ #] ==2&]] (* Harvey P. Dale, Nov 11 2021 *)

A370162 Semiprimes that are the sum of two successive semiprimes and also the sum of three successive semiprimes.

Original entry on oeis.org

134, 597, 614, 898, 982, 998, 1649, 2045, 2078, 2126, 2386, 2705, 2855, 2935, 3394, 3418, 3899, 5533, 5686, 5959, 6982, 7721, 8567, 8986, 9182, 9722, 9998, 10342, 10587, 10862, 10942, 11015, 11363, 11602, 11667, 11962, 13238, 13606, 14054, 14138, 14506, 14614, 15658, 15802, 15898, 16138, 16382
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Feb 26 2024

Keywords

Examples

			a(3) = 614 is a term because 614 = 2 * 307 is a semiprime, A001358(98) = 305 = 5 * 61 and A001358(99) = 309 = 3 * 103 are two successive semiprimes whose sum is 614, and A001358(65) = 203 = 7 * 29, A001358(66) = 205 = 5 * 41 and A001358(67) = 206 = 2 * 103 are three successive semiprimes whose sum is 614.

Links

Crossrefs

Cf A001358, A118717. Intersection of A092192 and A131610.

Programs

  • Maple
    N:= 10^5: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..N/2,2)]):
nP:= nops(P):
SP:= 0:
for i from 1 while P[i]^2 <= N do
  m:= ListTools:-BinaryPlace(P, N/P[i]);
  SP:= SP, op(P[i]*P[i..m]);
od:
SP:= sort([SP]):
SS:= ListTools:-PartialSums(SP):
SS2:= {seq(SS[i]-SS[i-2],i=3..nops(SS))}:
SS3:= {seq(SS[i]-SS[i-3],i=4..nops(SS))}:
A:=SS2 intersect SS3 intersect convert(SP,set):
sort(convert(A,list));

A266451 Semiprimes that are the sum of six consecutive semiprimes.

Original entry on oeis.org

58, 91, 123, 142, 161, 205, 278, 473, 566, 614, 706, 718, 802, 838, 851, 889, 1079, 1211, 1226, 1238, 1262, 1286, 1385, 1415, 1633, 1714, 1819, 1891, 1945, 2005, 2123, 2147, 2194, 2217, 2327, 2374, 2427, 2563, 2594, 2653, 2771, 2815, 2854, 2947, 2987, 3118, 3133, 3151, 3199, 3214, 3305, 3379
Offset: 1

Views

Author

Zak Seidov, Dec 29 2015

Keywords

Examples

			58 = A001358(21) = A001358(1) + ... + A001358(6) = 4+6+9+10+14+15,
91 = A001358(31) = A001358(3) + ... + A001358(8) = 9+10+14+15+21+22.

Links

Crossrefs

Cf. A001358, A092192, A131610, A158339, A254712.

Programs

  • Maple
    N:= 10^4: # to get all terms where the 6 consecutive semiprimes <= N
P:= select(isprime, [2,seq(i,i=3..N/2,2)]): nP:= nops(P):
SP:= NULL:
for i from 1 to nP do
  for j from 1 to nP while P[i]*P[j] <= N do od:
  SP:= SP, op(map(`*`,P[i],P[1..j-1]));
od:
SP:= sort(convert({SP},list)): nSP:= nops(SP):
select(numtheory:-bigomega=2, [seq(convert(SP[i..i+5],`+`),i=1..nSP-5)]): # Robert Israel, Nov 19 2017
  • Mathematica
    Select[(Total[#] & /@ Partition[Select[Range[4, 9999], 2 == PrimeOmega[#] &], 6, 1]), 2 == PrimeOmega[#] &]

A370685 Semiprimes that are also the sums of two, three and four successive semiprimes.

Original entry on oeis.org

2045, 2705, 2855, 14614, 18838, 28437, 31299, 43603, 68807, 76841, 77386, 88041, 108415, 116822, 194605, 213679, 218729, 252094, 255202, 269653, 290449, 294683, 302761, 305362, 310799, 339382, 348242, 361055, 398111, 445066, 445174, 459761, 464567, 489809, 496081, 501386, 515981, 534777, 544405
Offset: 1

Views

Author

Robert Israel, Feb 26 2024

Keywords

Examples

			a(3) = 2855 is a term because 2855 = 5 * 571 is a semiprime, A001358(423) = 1418 = 2*709 and A001358(424) = 1437 = 3 * 479 are two successive semiprimes whose sum is 2855, A001358(285) = 949 = 13 * 73, A001358(286) = 951 = 3 * 317 and A001358(287) = 955 = 5 * 191 are three successive semiprimes whose sum is 2855, and A001358(216) = 707 = 7 * 101, A001358(217) = 713 = 23 * 31, A001358(218) = 717 = 3 * 239, A001358(219) = 718 = 2 * 359 are four successive semiprimes whose sum is 2855.

Links

Crossrefs

Cf. A001358, A370162. Intersection of A092192, A131610 and A158339.

Programs

  • Maple
    N:= 10^6: # for terms <= N
P:= select(isprime, [2, seq(i, i=3..N/2, 2)]):
nP:= nops(P):
SP:= 0:
for i from 1 while P[i]^2 <= N do
  m:= ListTools:-BinaryPlace(P, N/P[i]);
  SP:= SP, op(P[i]*P[i..m]);
od:
SP:= sort([SP]):
SS:= ListTools:-PartialSums(SP):
SS2:= {seq(SS[i]-SS[i-2], i=3..nops(SS))}:
SS3:= {seq(SS[i]-SS[i-3], i=4..nops(SS))}:
SS4:= {seq(SS[i]-SS[i-4], i=5..nops(SS))}:
A:=SS2 intersect SS3 intersect SS4 intersect convert(SP, set):
A:= sort(convert(A, list)):

A265438 Smallest semiprime that is the sum of n consecutive semiprimes.

Original entry on oeis.org

4, 10, 25, 39, 69, 58, 133, 122, 249, 209, 185, 219, 254, 327, 458, 377, 473, 579, 745, 589, 951, 898, 1047, 843, 917, 1382, 1157, 1243, 1247, 1678, 1514, 1895, 1703, 1707, 2138, 2147, 2599, 2157, 2509, 2515, 2519, 2642, 2771, 3566, 4126, 3317, 3599, 3891, 4198, 3755, 4369, 4223, 4227
Offset: 1

Views

Author

Zak Seidov, Dec 09 2015

Keywords

Comments

The sequence is non-monotonic. But are all the terms distinct?
A092190 is a subsequence. More precisely, a(A092189(k)) = A092190(k). - Altug Alkan, Dec 13 2015

Examples

			a(1) = 4 = A001358(1),
a(2) = 10 = A001358(3) = A092192(1) = A001358(1)+A001358(2) = 4+6,
a(3) = 25 = A001358(9) = A131610(1),
a(4) = 39 = A001358(15) = A158339(1),
a(5) = 69 = A001358(24) = A254712(1),
a(6) = 58 = A001358(21) = A266451(1).

Links

Crossrefs

Cf. A001358, A092190, A092192, A131610, A158339, A254712.
Showing 1-6 of 6 results.

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